- Email: [email protected]
A Practical Approach to Generating Accurate Iso-Cusped Tool Paths for Three-Axis CNC Milling of Sculptured Surface Parts
Journal of Journal Manufacturing of Manufacturing ProcessesProcesses Vol. 8/No. Vol. 1 8/No. 1 2006 2006
A Practical Approach to Generating Accurate Iso-Cusped Tool Paths for Three-Axis CNC Milling of Sculptured Surface Parts Zezhong C. Chen ([email protected]) and Dejun Song, Dept. of Mechanical and Industrial Engineering, Concordia University, Montreal, Quebec, Canada
Abstract
surface finish are difficult to achieve in sculptured surface machining and machining time usually takes a significant portion of the total manufacturing time. Because tool path patterns determine how the cutting tool machines the surfaces, well-planned paths can both increase machining efficiency and ensure surface quality. Therefore, CNC tool path generation has become a focus in the research on sculptured surface machining for the past two decades. In sculptured surface machining, the cutting tool removes the stock material of the workpiece along the generated paths and leaves some excess material in ridges—cusps—between the adjacent paths on the surface. Usually the cusps need to be manually ground down for the required surface accuracy. The maximum height of the cusps determines the surface accuracy. A recognized tool path pattern— iso-cusped tool paths—is aimed at achieving evenly distributed cusps with a constant height equal to the part tolerance (see Figure 1). Therefore, the part quality is improved by this consistent surface accuracy, and the manufacturing time is reduced with less redundant machining and manual grinding. However, due to the geometric complexity of sculptured surfaces, cusp heights (the shortest distance between a cusp point and the design surface) are difficult to calculate, and the geometries of the cusps cannot be represented easily. As a result, the major technical challenges of planning iso-cusped tool paths for sculptured surface machining include how to determine the heights of the cusps and how to determine the tool path intervals. In past decades, researchers have proposed different approaches to addressing these challenges (Feng and Li 2002; Huang and Oliver 1994; Lin and
In three-axis computer numerically controlled (CNC) machining of sculptured surface parts, the tool path pattern is crucial to surface quality and machining time. A well-known tool path pattern—iso-cusped tool paths—over sculptured surfaces can save manufacturing time by eliminating redundant machining and reducing manual grinding, while other tool path patterns cannot; thus, this pattern has been accepted in industry. However, accurate iso-cusped tool paths are difficult to plan for three-axis CNC machining of sculptured surfaces when using bull-nose or flat end mills, and the solutions currently available are impractical and/or inaccurate; consequently, these tool paths are rarely used in industry. To solve this problem, this work proposes a new, practical approach to generating accurate iso-cusped tool paths in the process planning for sculptured surface parts. The main features of this work include: (1) defining a planar virtual cutting edge with a closed-form formula in order to accurately represent the geometries of cusps, and (2) calculating cutter contact points of iso-cusped tool paths based on the relationship between these paths and the cutting edges. This approach can be easily implemented into computer-aided manufacturing (CAM) software systems to promote the usage of isocusped tool paths in the manufacturing industry.
Keywords: Iso-Cusped Tool Path, Tool Path Generation, Sculptured Surface Machining, Three-Axis CNC Milling, Computer-Aided Process Planning
Introduction Sculptured surface parts have become an important type of mechanical parts in the automotive, aeronautical, and die-and-mold industries. Machining these parts with the required surface quality and minimum machining time is necessary and profitable, thus it is in high demand from industry. In the manufacturing process of sculptured surface parts, the most important operation is the three-axis CNC milling of the sculptured surfaces because high part accuracy and
29
Journal of Manufacturing Processes Vol. 8/No. 1 2006
Figure 1 Iso-Cusped Tool Paths for Planar Surface Machining Using a Ball End Mill
Figure 2 Model of Cutter-Swept Surfaces in Three-Axis Milling
Koren 1996; Sarma and Dutta 1997; Suresh and Yang 1994). A method proposed by Suresh and Yang (1994) is based on extrapolations from an initial tool path on the design surface. They first found a curve of cusp points—a cusp curve—on the tolerance surface, which is an offset of the design (nominal) surface by the part tolerance. They then approximately located the next iso-cusped tool path on the design surface from the calculated cusp curve (see Figure 1). This method assumes that only flat end mills are used because of their simple geometry. However, the method is impractical for ball and bull-nose end mills because a huge amount of computation is needed. Similarly, Sarma and Dutta (1997) proposed a method to calculate approximate iso-cusped tool paths by offsetting the previous tool paths on the design surface one by one, according to a specified cusp height. But the implementation of these methods in CAM software systems is limited by either the huge computing load or the tool path inaccuracy. Feng and Li (2002) took advantage of the simplicity of the shape swept by ball end mills in three-axis milling to represent the geometry of the cusps (see Figure 1), and employed a local optimization method to locate cusp points. This method can only plan iso-cusped tool paths for ball end mills in sculptured surface machining and cannot solve the problems when using bull-nose or flat end mills, which are more efficient in cutting. In CNC milling of a sculptured surface, a cutting tool removes stock material along planned paths, forming irregular shapes of the machined surface. From the geometric perspective, the concept of cutter-swept surfaces has been proposed to represent the geometries of the cusps for the newly formed
shapes, and the intersection of two neighboring surfaces can represent a cusp curve of these shapes (see Figure 1). Research by Altintas and Spence (1991); Blackmore, Leu, and Wang (1997); Chiou and Lee (2002); Chung et al. (1996); Roth et al. (2001); and Wang and Wang (1986) has been carried out on representing cutter-swept surfaces to simulate three-axis CNC machining processes. Chung et al. (1996) built a model of a cutter-swept surface in a three-axis tool motion between two cutter contact (CC) points by first finding the 3-D silhouette curve on the cutting surface of the tool and then deriving the mathematical formula of these surfaces (see Figure 2). But in this method, high-order, nonlinear systems have to be solved, which is unstable and time-consuming. Roth et al. (2001) presented a method to represent the shapes of cutter-swept surfaces formed by endmills in five-axis CNC machining. In this method, the tool is assumed to have many inserts, and an imprint point is located on each insert by using a modified principle of silhouette curves. Some cutter-swept surfaces have been used as examples; however, these surfaces are represented with a large number of points and the computing time is long. The present work proposes a simple representation of cutter-swept surfaces for the geometries of the cusps and a practical approach to generating accurate iso-cusped tool paths by using this representation. First, based on the cutter-swept surface model for bull-nose end mills in three-axis CNC milling, a planar virtual cutting edge is introduced to simplify the geometric representation of the cusp. Second, a generic, closed-form formula of the planar virtual cutting edge is derived for all types of end mills to calculate iso-cusped tool paths accurately. To dem-
30
Journal of Manufacturing Processes Vol. 8/No. 1 2006
onstrate its versatility and feasibility, this approach is then applied to two sculptured surface parts to generate iso-cusped tool paths. Finally, these parts are machined with bull-nose end mills on a DECKEL MAHO 60T milling machine, and the machined surfaces are measured to verify the validity of this approach.
Planar Virtual Cutting Edge When a sculptured surface part is machined, part accuracy and surface finish are determined by the cusps on the surfaces, these being closely related to the cutting tool, tool paths, and surface geometry. To represent the geometries of the cusps, a cutter-swept surface model for bull-nose end mills is introduced.
Figure 3 Virtual Cutting Edge on a Cutter-Swept Surface
Cutter-Swept Surface Model To produce a sculptured surface on a three-axis milling machine, CNC tool paths are planned to cover the whole surface. Each tool path is approximately represented with a number of CC points. During machining, the tool cuts through these CC points in a point-to-point way and moves from one tool path to another. A model of the cutter-swept surface illustrated in Figure 2 is introduced in the following paragraphs. In this model, a general cutting tool—a bull-nose end mill with a torus-shaped cutting surface—is chosen for machining. When this end mill moves along the line segment connecting two adjacent CC points, the cutter-swept surface is conceivably formed (see Figure 2). This surface is the envelope of the cutting surfaces when the tool has crossed all of the positions of this motion. The main geometric features of the cutter-swept surface include: (a) this surface is tangent to the cutting surface when the tool is at the two CC points, with a 3-D curve called a silhouette or imprint curve (Chung et al. 1994); and (b) this surface is a ruled surface whose directrices are the 3-D silhouette curves, and whose ruling is the tool feed direction. In Figure 2, the tool feed direction refers to a vector starting from the tip of the cutter (cutter location) at the first CC point to that at the second CC point. Hence, the cutter-swept surface can be represented by the silhouette curves and the tool feed direction. However, it is neither easy to calculate the 3-D silhouette curve nor convenient to implement it in planning accurate iso-cusped tool paths. To get these paths, the geometries of the cusps and the points of the cusp curves should be found. While a cusp curve is just
the intersection of two cutter-swept surfaces (see Figure 1), the computation for this curve is expensive if these surfaces are represented by 3-D silhouette curves. As a result, cusp curves are usually found by approximation, and the 3-D silhouette curves are impractical for accurate iso-cusped tool paths planning. Definition of Virtual Cutting Edge To solve the above problem, a new, planar directrice is used to represent the cutter-swept surface. A cusp point then can be simply found as the intersection between two planar directrices of the adjacent cutter-swept surfaces. To find the planar directrice, a horizontal plane, ⍀, is made through the CC point to intersect the cutter-swept surface at a curve, which is the planar directrice (see Figure 3). Thus the cutter-swept surface can be formed by sliding the directrice, instead of the 3-D silhouette curve, along the tool feed direction. Because the stock material that is above and inside the cutter-swept surface is removed, this planar directrice is called a virtual cutting edge in this work. However, it is difficult to directly formulate the virtual cutting edge according to its geometric definition. This work presents an insight into the metal-cutting process of three-axis CNC milling in order to derive the mathematic formula of the virtual cutting edge. Suppose a bull-nose end mill cuts a surface along a tool feed direction from a CC point A on a vertical three-axis milling machine (see Figure 4). To construct the virtual cutting edge on the horizontal plane, imagine that the torus-shaped cutting surface of the tool is decomposed into a family of cutting circles perpendicular to the tool axis, for example, C1, C2, 31
Journal of Manufacturing Processes Vol. 8/No. 1 2006
Figure 5 Illustration of Part and Cutter Coordinate Systems in Surface Machining Figure 4 (a) Family of Cutting Circles, and (b) Cutting Circles Relocated on Horizontal Plane
etry of the cusp. In the next section, the virtual cutting edge is formulated for an accurate representation of the geometry of the cusp.
C3, and C4 in Figure 4. In this tool motion, the stock material of the workpiece actually is removed simultaneously by the cutting circles. When the tool starts from the CC point A, the cutting circle C1 on the horizontal plane removes the stock material inside the circle physically. As the tool moves along the tool feed direction, the cutting circles C2, C3, and C4 are raised onto the horizontal plane ⍀ sequentially and relocated at different centers, B2, B3, and B4, respectively. Because each cutting circle can remove the stock material inside itself, all of the stock material on the plane and inside the cutting circles is removed gradually when the cutting circles reach the plane one by one. Figure 4b shows the material removed by the cutting circles. Thus the geometry of the cusp on the horizontal plane is the shape formed by the relocated cutting circles and represented as the geometric envelope of the circles. Therefore, the envelope is the mathematical model of the virtual cutting edge, representing the geom-
Formulation of Virtual Cutting Edge Suppose on a three-axis vertical CNC milling machine a bull-nose end mill with a radius of R1 and a corner radius of R2 cuts the surface, S, of a part at a point A(Ax,p, Ay,p, Az,p) in the part coordinate system (xp, yp, zp) as illustrated in Figure 5. In the part coordinate system, the surface normal np(nx,p, ny,p, nz,p) and the tool feed direction Fp(Fx,p, Fy,p, Fz,p) at point A are calculated, and the cutter location CL(CLx,p, CLy,p, CLz,p), which is the origin, O, of the cutter coordinate system, can be calculated as follows: ⎧CLx , p = Ax , p + R2 ⋅ nx , p + 2nx , p 2 ⋅ ( R1 − R2 ) nx , p + n y , p ⎪ ⎪ ny , p ⎨CLy , p = Ay , p + R2 ⋅ n y , p + n2 + n2 ⋅ ( R1 − R2 ) x,p y,p ⎪ ⎪CLz , p = Az , p + R2 ⋅ nz , p − R2 ⎩
32
(1)
Journal of Manufacturing Processes Vol. 8/No. 1 2006
To assist in the formulation of the virtual cutting edge, the cutter coordinate system, (x, y, z), is built by translating the part coordinate system from the point Op to the tip of the cutter O (see Figure 5); hence, the transformation matrix from the part coordinate system to the cutter coordinate system (Mortenson 1987) is ⎡1 ⎢0 T = [ ] ⎢⎢0 ⎢ ⎣0
0 0 −CLx , p ⎤ 1 0 −CLy , p ⎥ ⎥ 0 1 −CLz , p ⎥ ⎥ 0 0 1 ⎦
(2) Figure 6 Notations in Formulation of Virtual Cutting Edges
Then by using this transformation matrix, the coordinates of point A, its surface normal, np, and the tool feed direction, Fp, are transformed into the cutter coordinate system as A(Ax, Ay, Az), n(nx, ny, nz), and F(Fx, Fy, Fz), respectively, because it is simpler to derive the mathematical formula of the virtual cutting edge in the cutter coordinate system. In the cutter coordinate system, first make a horizontal plane, ⍀, passing through the CC point A and then calculate the z coordinate of this plane as z0 = R2 ⋅ (1 − cos α 0 )
Therefore, the cutting circle C can be expressed in the following equation: Fx ⎧ ⎪ x = F ⋅ ( z0 − z ) + ⎡⎣ R1 + R2 ⋅ ( sin α − 1) ⎤⎦ ⋅ cos β z ⎪ Fy ⎪ ⋅ ( z0 − z ) + ⎡⎣ R1 + R2 ⋅ ( sin α − 1) ⎤⎦ ⋅ sin β ⎨y = Fz ⎪ ⎪ z = z0 ⎪ ⎩
(3)
where ␣0 = arccos(nz) (see Figure 6). The cutting circle on the horizontal plane ⍀ is represented as ⎧ x = ⎡⎣ R1 + R2 ⋅ ( sin α 0 − 1) ⎤⎦ ⋅ cos β ⎪⎪ ⎨ y = ⎣⎡ R1 + R2 ⋅ ( sin α 0 − 1) ⎦⎤ ⋅ sin β ⎪ ⎪⎩ z = R2 ⋅ (1 − cos α 0 )
When the variable z is changed, the formula of all cutting circles on the horizontal plane ⍀ is found. Based on the formula, the envelope of the cutting circles can be formulated according to the definition of envelope (Abraham 1970) as
(4)
∂x ∂y ∂x ∂y ⋅ − ⋅ =0 ∂z ∂β ∂β ∂z
Between the horizontal plane ⍀ and the bottom of the tool, a cutting circle C is centered at (0, 0, z) in the cutter coordinate system, and its radius is R1 2 + R2 ⭈ (sin␣–1), where sin α = R 2 − ( R2 − z ) R2 . When the tool moves along the tool feed direction F, a family of cutting circles will pass through the horizontal plane ⍀. The center of a cutting circle C relocated on the horizontal plane ⍀ can be calculated as Fx ⋅ ( z0 − z ) Fz Fy Fz
(7)
By solving Eq. (7), the relationship between the variable z and  can be found. So the envelope of the cutting circles (the virtual cutting edge) in the cutter coordinate system can be found with the following equation:
2
⎧ ⎪x = ⎪ ⎨ ⎪y = ⎪⎩
(6)
⎧ ⎪ Fx 2 ⎪ x = ⋅ ( z0 − z ) + R1 − R2 + 2 ⋅ R2 ⋅ z − z F z ⎪ ⎪⎪ Fy ⋅ ( z0 − z ) + R1 − R2 + 2 ⋅ R2 ⋅ z − z 2 ⎨y = F z ⎪ ⎪ ⎛ Fz R2 − z ⎪ ⎜ arcsin β = −θ + ⋅ ⎪ 2 2 ⎜ Fx + Fy 2 ⋅ R2 ⋅ z − z 2 ⎪⎩ ⎝
( (
(5)
⋅ ( z0 − z )
33
) ⋅ cos β ) ⋅ sin β ⎞ ⎟ ⎟ ⎠
(8)
Journal of Manufacturing Processes Vol. 8/No. 1 2006
where sin θ = Fx
Fx2 + Fy2 and cos θ = Fy
Fx2 + Fy2 . After
the virtual cutting edge in the cutter coordinate is found, it is transformed back into the part coordinate system to plan iso-cusped tool paths. In this transformation, the matrix is ⎡1 ⎢0 T = [ ] ⎢⎢0 ⎢ ⎣0
0 0 CLx ⎤ 1 0 CLy ⎥ ⎥ 0 1 CLz ⎥ ⎥ 0 0 1 ⎦
(9)
Based on Eq. (8), the virtual cutting edge at any CC point can be calculated, so the geometry of the cusp and cusp points can be found and accurate isocusped tool paths can be generated. In a special case, if the tool feed direction is horizontal, the virtual cutting edge becomes the profile of the tool; the cusp point is thus the intersection of the profile of the tool at two neighboring CC points in two adjacent tool paths, respectively.
Figure 7 Iso-Cusped Tool Paths for a Sculptured Surface
to the definition of iso-cusped tool paths, if the height of the cusp is equal to the part tolerance, the cusp point should be on the tolerance surface. Therefore, the relationship is that the intersection of the two virtual cutting edges on the plane ⍀ is the cusp point, and the cusp point should be on the contour of the tolerance surface. Applying this relationship, a procedure is provided to plan accurate iso-cusped tool paths for sculptured surface machining.
Relationship Between Iso-Cusped Tool Paths and Virtual Cutting Edges The iso-cusped tool paths are defined as: when a tool cuts along two adjacent tool paths, the heights of the cusps between the paths remain constant, usually equal to the part tolerance (see Figures 2 and 7). To plan accurate iso-cusped tool paths, the cusp point of two adjacent virtual cutting edges and its height to the nominal surface should be found precisely. This work renders an illustration for the relationship between iso-cusped tool paths and virtual cutting edges. Based on the sculptured surface, a tolerance surface is defined as an offset of the surface by the part tolerance. Assuming two iso-cusped tool paths on the sculptured surface, a horizontal plane ⍀ is made to intersect the two iso-cusped tool paths at two CC points, A and B (see Figure 8a). Meanwhile, this plane ⍀ intersects with the sculptured surface and the tolerance surface at two contours, respectively. When the tool passes through the CC point A on the first tool path, a cutter-swept surface is formed; thus, the horizontal plane ⍀ cuts through the cutter-swept surface at a virtual cutting edge. Similarly, the cutterswept surface at the CC point B on the second tool path is intersected by the horizontal plane for the adjacent virtual cutting edge (see Figure 8b). According
Procedure of Iso-Cusped Tool Path Planning Theoretically, the above relationship holds at every CC point on an accurate iso-cusped tool path. To apply this relationship to iso-cusped tool paths planning, the procedure is provided step by step. (1) Based on a sculptured surface, the first tool path should be designated and then is referred as the reference tool path for the next iso-cusped tool path. (2) According to the part tolerance, the number of the horizontal planes at the CC points is determined. On each horizontal plane, find the contours of the design surface and tolerance surfaces and the intersection point between the plane and the reference tool path (called the reference point). (3) Starting from the lowest contour of the design surface, find the virtual cutting edge at the reference point. Next, assume a point on the same contour and close to the reference point as the CC point for the next iso-cusped
34
Journal of Manufacturing Processes Vol. 8/No. 1 2006
Figure 9 Convex NURBS Surface on a Sculptured Surface Part
(6) Once the next iso-cusped tool path is found, this tool path is regarded as the reference tool path and the CC points on this tool path become the reference points. (7) Repeat Steps 3 to 6 to find the iso-cusped tool paths and then check if all the iso-cusped tool paths have covered the whole surface. If positive, end the program. By implementing the procedure, a software system has been made using MATLAB to generate accurate iso-cusped tool paths of sculptured surfaces.
Applications To demonstrate its versatility and feasibility, this new approach is applied to two sculptured surface parts using bull-nose end mills. The first sculptured surface part is a typical convex NURBS surface designed with the CATIA V5 CAD/CAM system (see Figure 9). The software system is employed to generate accurate iso-cusped tool paths for the NURBS surface machining. In the process planning for the surface milling, 6061T aluminum is selected for this part. A highspeed-steel (HSS) flat end mill with a diameter of 12.7 mm and a HSS bull-nose end mill with a diameter of 12.7 mm and a corner radius of 1.58 mm are chosen for rough and finishing machining, respectively. For demonstration purposes, the specified part tolerance is 0.2 mm. The cutting feed rate is set as 500 mm/ min., and the spindle speed is 1000 rpm to prevent chatter in machining. After inputting the design of the convex NURBS surface and the information determined in the process planning, the system takes a couple of minutes to generate the iso-cusped tool paths. Then the CNC machining of this part is simulated with the CATIA system, and the result is shown in Figure 10a. Finally, the surface is machined on a DECKEL MAHO 60T CNC milling machine, and the machined part is shown in Figure 10b.
Figure 8 Illustration of Relationship Between Iso-Cusped Tool Paths and Virtual Cutting Edges
tool path and then determine the virtual cutting edge at the assumed CC point. (4) Find the intersection point between the two virtual cutting edges and then check if the intersection point is inside, on, or outside the contour of the tolerance surface. If the intersection point is outside, adjust the assumed CC point closer to the reference point along the contour of the design surface. If the intersection point is inside, move the assumed CC point farther away from the reference point along the contour. Otherwise, the CC point is found for the next iso-cusped tool path. (5) Repeat Steps 3 and 4 to find the CC points for the next iso-cusped tool path from the lower contour to the upper contour of the design surface. 35
Journal of Manufacturing Processes Vol. 8/No. 1 2006
Figure 10 (a) Simulation of Surface Milling Using Iso-Cusped Tool Paths, and (b) Machined NURBS Surface Part Figure 11 (a) NURBS Surface Part Measured on a Mitutoyo CMM, and (b) Measured Surface Points
To verify the fidelity of the iso-cusped tool paths generated with this new approach, the machined surface is measured at five different altitudes on a Mitutoyo LEGEX 9106 coordinate measurement machine (CMM), whose accuracy is 0.1 micro (see Figure 11a). The five altitudes are 15, 20, 25, 30, and 35 mm along the z-axis, and on each altitude about 3,000 points are measured and shown in Figure 11b. In this figure, the virtual cutting edges and the cusp points are clearly shown, and five cusp points are selected on each of the example cusp curves (A, B, C, and D) to verify the cusp height. The results of the cusp heights are listed in Table 1, and the deviations of the cusp heights from the part tolerance are computed. Among the 20 locations, the maximum deviation of the cusp height occurs at the point 1 in the cusp curve A, at which the cusp height is 9.5% less than the specified surface tolerance. The main reason for this is the machining error in the finish machining of this part. As a whole, the cusp heights at the different locations can be regarded as con-
stant; therefore, the iso-cusped tool paths on the first part are verified to be valid. The second sculptured part is designed as a compound NURBS surface containing concave and convex shapes, and the software system is used to generate the iso-cusped tool paths shown in Figure 12a. Under the same machining conditions as those in producing the first part, the surface is machined with the generated iso-cusped tool paths, and the finished surface is shown in Figure 12b. In the two examples, the heights of the cusps between the iso-cusped tool paths generated using this new approach are close to the part tolerance; therefore, the surface accuracy is consistent across the entire surfaces without redundant machining. This is the advantage of iso-cusped tool paths. The examples also demonstrate the versatility and feasibility of this new iso-cusped tool path generation approach. 36
Journal of Manufacturing Processes Vol. 8/No. 1 2006
Table 1 Cusp Heights at Different Locations and Deviation Percentages of Cusp Heights
Unit, mm Point 1 2 3 4 5
Cusp Curve A Cusp Deviation Height (%) 0.181 –9.5 0.193 –3.5 0.187 –6.5 0.191 –4.5 0.188 –6.0
Cusp Curve B Cusp Deviation Height (%) 0.193 –3.5 0.191 –4.5 0.188 –6.0 0.192 –4.0 0.187 –6.5
Cusp Curve C Cusp Deviation Height (%) 0.205 2.5 0.197 –1.5 0.189 –5.5 0.186 –7.0 0.184 –8.0
Cusp Curve D Cusp Deviation Height (%) 0.188 –6.0 0.192 –4.0 0.195 –2.5 0.187 –6.5 0.187 –6.5
Figure 13 Inefficient Iso-Cusped Tool Paths for Surface Machining
upper-left corner of the part are inefficient. To address this problem, integration of the steepest-ascending tool path pattern and the iso-cusped tool path pattern provides an optimal solution for sculptured surface machining (Chen, Vickers, and Dong 2003).
Conclusions A new, practical approach is proposed to generate accurate iso-cusped tool paths for sculptured surface machining on three-axis CNC machines using all types of end mills. The major contributions of this work include simplifying the geometric representation of the cusp with virtual cutting edges and locating the cusp points accurately. In this work, computing the intersection of two planar virtual cutter edges is easier and faster than computing the intersection between two cutter-swept surfaces in the existing methods. Two typical examples have shown the versatility and feasibility of this newly proposed approach. Therefore, this approach to generating isocusped tool paths for sculptured surface machining can be implemented in CAD/CAM software systems to benefit the manufacturing industry.
Figure 12 (a) CATIA Simulation Result of Second Part, and (b) Machined Part
Although iso-cusped tool paths can eliminate redundant machining on sculptured surfaces, some tool paths may be quite inefficient in machining. In Figure 13, the iso-cusped tool paths are generated on a semi-cylindrical part; however, the tool paths on the 37
Journal of Manufacturing Processes Vol. 8/No. 1 2006
Acknowledgments The financial support of this work from the Natural Science and Engineering Research Council of Canada (NSERC) and a Concordia University Startup Grant to Dr. Z.C. Chen is gratefully acknowledged.
Lin, R. and Koren, Y. (1996). “Efficient tool-path planning for machining free-form surfaces.” ASME Journal of Engg. for Industry (v118), pp20-28. Mortenson, M.E. (1987). Geometric Modeling. New York: John Wiley & Sons. Roth, D.; Bedi, S.; Ismail, F.; and Mann, S. (2001). “Surface swept by a toroidal cutter during 5-axis machining.” Computer-Aided Design (v33), pp57-63. Sarma, R. and Dutta, D. (1997). “The geometry and generation of NC tool paths.” Journal of Mechanical Design (v119), pp253-258. Suresh, K. and Yang, D.C.H. (1994). “Constant scallop height machining of free-form surfaces.” Journal of Engg. for Industry (v116), pp253-259. Wang, W.P. and Wang, K.K. (1986). “Geometric modeling for swept volume of moving solids.” IEEE Computer Graphics and Applications (v6, n12), pp8-17.
References Abraham, G. (1970). Introduction to Differential Geometry. Addison Wesley Publishing Co. Altintas, Y. and Spence, A. (1991). “End milling force algorithm for CAD systems.” Annals of the CIRP (v40), pp31-34. Blackmore, D.; Leu, M.C.; and Wang, L.P. (1997). “The sweep-envelope differential equation algorithm and its application to NC machining verification.” Computer-Aided Design (v29, n9), pp629-637. Chen, Z.C.; Vickers, G.W.; and Dong, Z. (2003). “Integrated steepestdirected and iso-cusped tool path generation for three-axis CNC machining of sculptured parts.” Journal of Manufacturing Systems (v22, n3), pp190-201. Chiou, C.J. and Lee, Y.S. (2002). “Swept surface determination for five-axis numerical control machining.” Int’l Journal of Machine Tools and Manufacture (v42), pp1497-1507. Chung, Y.C.; Park, J.W.; Shin, H.; and Choi, B.K. (1996). “Modeling the surface swept by a generated cutter for NC verification.” Computer-Aided Design (v30, n8), pp587-594. Feng, H.Y. and Li, H. (2002). “Constant scallop-height tool path generation for three-axis sculptured surface machining.” Computer-Aided Design (v34, n9), pp647-654. Huang, Y. and Oliver, J.H. (1994). “Non-constant parameter NC tool path generation on sculptured surfaces.” Int’l Journal of Advanced Mfg. Technology (v9), pp281-290.
Authors’ Biographies Dr. Zezhong C. Chen is an assistant professor in the Dept. of Mechanical and Industrial Engineering at Concordia University, Montreal, Canada. His research interests include multi-axis CNC machining, high-performance/speed machining, intelligence and optimization for manufacturing, and computer-aided design and automation. His research is sponsored by the Natural Science and Engineering Research Council of Canada (NSERC), Pratt & Whitney Canada, and Consortium for Research and Innovation in Aerospace in Quebec (CRIAQ). He received his PhD in mechanical engineering at the University of Victoria in 2002. He is a senior member of the Society of Manufacturing Engineers. Dr. Dejun Song received his PhD in mechanical engineering at the Beijing University of Aeronautics & Astronautics, China. His research interests include multi-axis CNC machining, computer simulation and visualization, optimized computer geometric modeling, and product life cycle management.
38
A Practical Approach to Generating Accurate Iso-Cusped Tool Paths for Three-Axis CNC Milling of Sculptured Surface Parts Zezhong C. Chen ([email protected]) and Dejun Song, Dept. of Mechanical and Industrial Engineering, Concordia University, Montreal, Quebec, Canada
Abstract
surface finish are difficult to achieve in sculptured surface machining and machining time usually takes a significant portion of the total manufacturing time. Because tool path patterns determine how the cutting tool machines the surfaces, well-planned paths can both increase machining efficiency and ensure surface quality. Therefore, CNC tool path generation has become a focus in the research on sculptured surface machining for the past two decades. In sculptured surface machining, the cutting tool removes the stock material of the workpiece along the generated paths and leaves some excess material in ridges—cusps—between the adjacent paths on the surface. Usually the cusps need to be manually ground down for the required surface accuracy. The maximum height of the cusps determines the surface accuracy. A recognized tool path pattern— iso-cusped tool paths—is aimed at achieving evenly distributed cusps with a constant height equal to the part tolerance (see Figure 1). Therefore, the part quality is improved by this consistent surface accuracy, and the manufacturing time is reduced with less redundant machining and manual grinding. However, due to the geometric complexity of sculptured surfaces, cusp heights (the shortest distance between a cusp point and the design surface) are difficult to calculate, and the geometries of the cusps cannot be represented easily. As a result, the major technical challenges of planning iso-cusped tool paths for sculptured surface machining include how to determine the heights of the cusps and how to determine the tool path intervals. In past decades, researchers have proposed different approaches to addressing these challenges (Feng and Li 2002; Huang and Oliver 1994; Lin and
In three-axis computer numerically controlled (CNC) machining of sculptured surface parts, the tool path pattern is crucial to surface quality and machining time. A well-known tool path pattern—iso-cusped tool paths—over sculptured surfaces can save manufacturing time by eliminating redundant machining and reducing manual grinding, while other tool path patterns cannot; thus, this pattern has been accepted in industry. However, accurate iso-cusped tool paths are difficult to plan for three-axis CNC machining of sculptured surfaces when using bull-nose or flat end mills, and the solutions currently available are impractical and/or inaccurate; consequently, these tool paths are rarely used in industry. To solve this problem, this work proposes a new, practical approach to generating accurate iso-cusped tool paths in the process planning for sculptured surface parts. The main features of this work include: (1) defining a planar virtual cutting edge with a closed-form formula in order to accurately represent the geometries of cusps, and (2) calculating cutter contact points of iso-cusped tool paths based on the relationship between these paths and the cutting edges. This approach can be easily implemented into computer-aided manufacturing (CAM) software systems to promote the usage of isocusped tool paths in the manufacturing industry.
Keywords: Iso-Cusped Tool Path, Tool Path Generation, Sculptured Surface Machining, Three-Axis CNC Milling, Computer-Aided Process Planning
Introduction Sculptured surface parts have become an important type of mechanical parts in the automotive, aeronautical, and die-and-mold industries. Machining these parts with the required surface quality and minimum machining time is necessary and profitable, thus it is in high demand from industry. In the manufacturing process of sculptured surface parts, the most important operation is the three-axis CNC milling of the sculptured surfaces because high part accuracy and
29
Journal of Manufacturing Processes Vol. 8/No. 1 2006
Figure 1 Iso-Cusped Tool Paths for Planar Surface Machining Using a Ball End Mill
Figure 2 Model of Cutter-Swept Surfaces in Three-Axis Milling
Koren 1996; Sarma and Dutta 1997; Suresh and Yang 1994). A method proposed by Suresh and Yang (1994) is based on extrapolations from an initial tool path on the design surface. They first found a curve of cusp points—a cusp curve—on the tolerance surface, which is an offset of the design (nominal) surface by the part tolerance. They then approximately located the next iso-cusped tool path on the design surface from the calculated cusp curve (see Figure 1). This method assumes that only flat end mills are used because of their simple geometry. However, the method is impractical for ball and bull-nose end mills because a huge amount of computation is needed. Similarly, Sarma and Dutta (1997) proposed a method to calculate approximate iso-cusped tool paths by offsetting the previous tool paths on the design surface one by one, according to a specified cusp height. But the implementation of these methods in CAM software systems is limited by either the huge computing load or the tool path inaccuracy. Feng and Li (2002) took advantage of the simplicity of the shape swept by ball end mills in three-axis milling to represent the geometry of the cusps (see Figure 1), and employed a local optimization method to locate cusp points. This method can only plan iso-cusped tool paths for ball end mills in sculptured surface machining and cannot solve the problems when using bull-nose or flat end mills, which are more efficient in cutting. In CNC milling of a sculptured surface, a cutting tool removes stock material along planned paths, forming irregular shapes of the machined surface. From the geometric perspective, the concept of cutter-swept surfaces has been proposed to represent the geometries of the cusps for the newly formed
shapes, and the intersection of two neighboring surfaces can represent a cusp curve of these shapes (see Figure 1). Research by Altintas and Spence (1991); Blackmore, Leu, and Wang (1997); Chiou and Lee (2002); Chung et al. (1996); Roth et al. (2001); and Wang and Wang (1986) has been carried out on representing cutter-swept surfaces to simulate three-axis CNC machining processes. Chung et al. (1996) built a model of a cutter-swept surface in a three-axis tool motion between two cutter contact (CC) points by first finding the 3-D silhouette curve on the cutting surface of the tool and then deriving the mathematical formula of these surfaces (see Figure 2). But in this method, high-order, nonlinear systems have to be solved, which is unstable and time-consuming. Roth et al. (2001) presented a method to represent the shapes of cutter-swept surfaces formed by endmills in five-axis CNC machining. In this method, the tool is assumed to have many inserts, and an imprint point is located on each insert by using a modified principle of silhouette curves. Some cutter-swept surfaces have been used as examples; however, these surfaces are represented with a large number of points and the computing time is long. The present work proposes a simple representation of cutter-swept surfaces for the geometries of the cusps and a practical approach to generating accurate iso-cusped tool paths by using this representation. First, based on the cutter-swept surface model for bull-nose end mills in three-axis CNC milling, a planar virtual cutting edge is introduced to simplify the geometric representation of the cusp. Second, a generic, closed-form formula of the planar virtual cutting edge is derived for all types of end mills to calculate iso-cusped tool paths accurately. To dem-
30
Journal of Manufacturing Processes Vol. 8/No. 1 2006
onstrate its versatility and feasibility, this approach is then applied to two sculptured surface parts to generate iso-cusped tool paths. Finally, these parts are machined with bull-nose end mills on a DECKEL MAHO 60T milling machine, and the machined surfaces are measured to verify the validity of this approach.
Planar Virtual Cutting Edge When a sculptured surface part is machined, part accuracy and surface finish are determined by the cusps on the surfaces, these being closely related to the cutting tool, tool paths, and surface geometry. To represent the geometries of the cusps, a cutter-swept surface model for bull-nose end mills is introduced.
Figure 3 Virtual Cutting Edge on a Cutter-Swept Surface
Cutter-Swept Surface Model To produce a sculptured surface on a three-axis milling machine, CNC tool paths are planned to cover the whole surface. Each tool path is approximately represented with a number of CC points. During machining, the tool cuts through these CC points in a point-to-point way and moves from one tool path to another. A model of the cutter-swept surface illustrated in Figure 2 is introduced in the following paragraphs. In this model, a general cutting tool—a bull-nose end mill with a torus-shaped cutting surface—is chosen for machining. When this end mill moves along the line segment connecting two adjacent CC points, the cutter-swept surface is conceivably formed (see Figure 2). This surface is the envelope of the cutting surfaces when the tool has crossed all of the positions of this motion. The main geometric features of the cutter-swept surface include: (a) this surface is tangent to the cutting surface when the tool is at the two CC points, with a 3-D curve called a silhouette or imprint curve (Chung et al. 1994); and (b) this surface is a ruled surface whose directrices are the 3-D silhouette curves, and whose ruling is the tool feed direction. In Figure 2, the tool feed direction refers to a vector starting from the tip of the cutter (cutter location) at the first CC point to that at the second CC point. Hence, the cutter-swept surface can be represented by the silhouette curves and the tool feed direction. However, it is neither easy to calculate the 3-D silhouette curve nor convenient to implement it in planning accurate iso-cusped tool paths. To get these paths, the geometries of the cusps and the points of the cusp curves should be found. While a cusp curve is just
the intersection of two cutter-swept surfaces (see Figure 1), the computation for this curve is expensive if these surfaces are represented by 3-D silhouette curves. As a result, cusp curves are usually found by approximation, and the 3-D silhouette curves are impractical for accurate iso-cusped tool paths planning. Definition of Virtual Cutting Edge To solve the above problem, a new, planar directrice is used to represent the cutter-swept surface. A cusp point then can be simply found as the intersection between two planar directrices of the adjacent cutter-swept surfaces. To find the planar directrice, a horizontal plane, ⍀, is made through the CC point to intersect the cutter-swept surface at a curve, which is the planar directrice (see Figure 3). Thus the cutter-swept surface can be formed by sliding the directrice, instead of the 3-D silhouette curve, along the tool feed direction. Because the stock material that is above and inside the cutter-swept surface is removed, this planar directrice is called a virtual cutting edge in this work. However, it is difficult to directly formulate the virtual cutting edge according to its geometric definition. This work presents an insight into the metal-cutting process of three-axis CNC milling in order to derive the mathematic formula of the virtual cutting edge. Suppose a bull-nose end mill cuts a surface along a tool feed direction from a CC point A on a vertical three-axis milling machine (see Figure 4). To construct the virtual cutting edge on the horizontal plane, imagine that the torus-shaped cutting surface of the tool is decomposed into a family of cutting circles perpendicular to the tool axis, for example, C1, C2, 31
Journal of Manufacturing Processes Vol. 8/No. 1 2006
Figure 5 Illustration of Part and Cutter Coordinate Systems in Surface Machining Figure 4 (a) Family of Cutting Circles, and (b) Cutting Circles Relocated on Horizontal Plane
etry of the cusp. In the next section, the virtual cutting edge is formulated for an accurate representation of the geometry of the cusp.
C3, and C4 in Figure 4. In this tool motion, the stock material of the workpiece actually is removed simultaneously by the cutting circles. When the tool starts from the CC point A, the cutting circle C1 on the horizontal plane removes the stock material inside the circle physically. As the tool moves along the tool feed direction, the cutting circles C2, C3, and C4 are raised onto the horizontal plane ⍀ sequentially and relocated at different centers, B2, B3, and B4, respectively. Because each cutting circle can remove the stock material inside itself, all of the stock material on the plane and inside the cutting circles is removed gradually when the cutting circles reach the plane one by one. Figure 4b shows the material removed by the cutting circles. Thus the geometry of the cusp on the horizontal plane is the shape formed by the relocated cutting circles and represented as the geometric envelope of the circles. Therefore, the envelope is the mathematical model of the virtual cutting edge, representing the geom-
Formulation of Virtual Cutting Edge Suppose on a three-axis vertical CNC milling machine a bull-nose end mill with a radius of R1 and a corner radius of R2 cuts the surface, S, of a part at a point A(Ax,p, Ay,p, Az,p) in the part coordinate system (xp, yp, zp) as illustrated in Figure 5. In the part coordinate system, the surface normal np(nx,p, ny,p, nz,p) and the tool feed direction Fp(Fx,p, Fy,p, Fz,p) at point A are calculated, and the cutter location CL(CLx,p, CLy,p, CLz,p), which is the origin, O, of the cutter coordinate system, can be calculated as follows: ⎧CLx , p = Ax , p + R2 ⋅ nx , p + 2nx , p 2 ⋅ ( R1 − R2 ) nx , p + n y , p ⎪ ⎪ ny , p ⎨CLy , p = Ay , p + R2 ⋅ n y , p + n2 + n2 ⋅ ( R1 − R2 ) x,p y,p ⎪ ⎪CLz , p = Az , p + R2 ⋅ nz , p − R2 ⎩
32
(1)
Journal of Manufacturing Processes Vol. 8/No. 1 2006
To assist in the formulation of the virtual cutting edge, the cutter coordinate system, (x, y, z), is built by translating the part coordinate system from the point Op to the tip of the cutter O (see Figure 5); hence, the transformation matrix from the part coordinate system to the cutter coordinate system (Mortenson 1987) is ⎡1 ⎢0 T = [ ] ⎢⎢0 ⎢ ⎣0
0 0 −CLx , p ⎤ 1 0 −CLy , p ⎥ ⎥ 0 1 −CLz , p ⎥ ⎥ 0 0 1 ⎦
(2) Figure 6 Notations in Formulation of Virtual Cutting Edges
Then by using this transformation matrix, the coordinates of point A, its surface normal, np, and the tool feed direction, Fp, are transformed into the cutter coordinate system as A(Ax, Ay, Az), n(nx, ny, nz), and F(Fx, Fy, Fz), respectively, because it is simpler to derive the mathematical formula of the virtual cutting edge in the cutter coordinate system. In the cutter coordinate system, first make a horizontal plane, ⍀, passing through the CC point A and then calculate the z coordinate of this plane as z0 = R2 ⋅ (1 − cos α 0 )
Therefore, the cutting circle C can be expressed in the following equation: Fx ⎧ ⎪ x = F ⋅ ( z0 − z ) + ⎡⎣ R1 + R2 ⋅ ( sin α − 1) ⎤⎦ ⋅ cos β z ⎪ Fy ⎪ ⋅ ( z0 − z ) + ⎡⎣ R1 + R2 ⋅ ( sin α − 1) ⎤⎦ ⋅ sin β ⎨y = Fz ⎪ ⎪ z = z0 ⎪ ⎩
(3)
where ␣0 = arccos(nz) (see Figure 6). The cutting circle on the horizontal plane ⍀ is represented as ⎧ x = ⎡⎣ R1 + R2 ⋅ ( sin α 0 − 1) ⎤⎦ ⋅ cos β ⎪⎪ ⎨ y = ⎣⎡ R1 + R2 ⋅ ( sin α 0 − 1) ⎦⎤ ⋅ sin β ⎪ ⎪⎩ z = R2 ⋅ (1 − cos α 0 )
When the variable z is changed, the formula of all cutting circles on the horizontal plane ⍀ is found. Based on the formula, the envelope of the cutting circles can be formulated according to the definition of envelope (Abraham 1970) as
(4)
∂x ∂y ∂x ∂y ⋅ − ⋅ =0 ∂z ∂β ∂β ∂z
Between the horizontal plane ⍀ and the bottom of the tool, a cutting circle C is centered at (0, 0, z) in the cutter coordinate system, and its radius is R1 2 + R2 ⭈ (sin␣–1), where sin α = R 2 − ( R2 − z ) R2 . When the tool moves along the tool feed direction F, a family of cutting circles will pass through the horizontal plane ⍀. The center of a cutting circle C relocated on the horizontal plane ⍀ can be calculated as Fx ⋅ ( z0 − z ) Fz Fy Fz
(7)
By solving Eq. (7), the relationship between the variable z and  can be found. So the envelope of the cutting circles (the virtual cutting edge) in the cutter coordinate system can be found with the following equation:
2
⎧ ⎪x = ⎪ ⎨ ⎪y = ⎪⎩
(6)
⎧ ⎪ Fx 2 ⎪ x = ⋅ ( z0 − z ) + R1 − R2 + 2 ⋅ R2 ⋅ z − z F z ⎪ ⎪⎪ Fy ⋅ ( z0 − z ) + R1 − R2 + 2 ⋅ R2 ⋅ z − z 2 ⎨y = F z ⎪ ⎪ ⎛ Fz R2 − z ⎪ ⎜ arcsin β = −θ + ⋅ ⎪ 2 2 ⎜ Fx + Fy 2 ⋅ R2 ⋅ z − z 2 ⎪⎩ ⎝
( (
(5)
⋅ ( z0 − z )
33
) ⋅ cos β ) ⋅ sin β ⎞ ⎟ ⎟ ⎠
(8)
Journal of Manufacturing Processes Vol. 8/No. 1 2006
where sin θ = Fx
Fx2 + Fy2 and cos θ = Fy
Fx2 + Fy2 . After
the virtual cutting edge in the cutter coordinate is found, it is transformed back into the part coordinate system to plan iso-cusped tool paths. In this transformation, the matrix is ⎡1 ⎢0 T = [ ] ⎢⎢0 ⎢ ⎣0
0 0 CLx ⎤ 1 0 CLy ⎥ ⎥ 0 1 CLz ⎥ ⎥ 0 0 1 ⎦
(9)
Based on Eq. (8), the virtual cutting edge at any CC point can be calculated, so the geometry of the cusp and cusp points can be found and accurate isocusped tool paths can be generated. In a special case, if the tool feed direction is horizontal, the virtual cutting edge becomes the profile of the tool; the cusp point is thus the intersection of the profile of the tool at two neighboring CC points in two adjacent tool paths, respectively.
Figure 7 Iso-Cusped Tool Paths for a Sculptured Surface
to the definition of iso-cusped tool paths, if the height of the cusp is equal to the part tolerance, the cusp point should be on the tolerance surface. Therefore, the relationship is that the intersection of the two virtual cutting edges on the plane ⍀ is the cusp point, and the cusp point should be on the contour of the tolerance surface. Applying this relationship, a procedure is provided to plan accurate iso-cusped tool paths for sculptured surface machining.
Relationship Between Iso-Cusped Tool Paths and Virtual Cutting Edges The iso-cusped tool paths are defined as: when a tool cuts along two adjacent tool paths, the heights of the cusps between the paths remain constant, usually equal to the part tolerance (see Figures 2 and 7). To plan accurate iso-cusped tool paths, the cusp point of two adjacent virtual cutting edges and its height to the nominal surface should be found precisely. This work renders an illustration for the relationship between iso-cusped tool paths and virtual cutting edges. Based on the sculptured surface, a tolerance surface is defined as an offset of the surface by the part tolerance. Assuming two iso-cusped tool paths on the sculptured surface, a horizontal plane ⍀ is made to intersect the two iso-cusped tool paths at two CC points, A and B (see Figure 8a). Meanwhile, this plane ⍀ intersects with the sculptured surface and the tolerance surface at two contours, respectively. When the tool passes through the CC point A on the first tool path, a cutter-swept surface is formed; thus, the horizontal plane ⍀ cuts through the cutter-swept surface at a virtual cutting edge. Similarly, the cutterswept surface at the CC point B on the second tool path is intersected by the horizontal plane for the adjacent virtual cutting edge (see Figure 8b). According
Procedure of Iso-Cusped Tool Path Planning Theoretically, the above relationship holds at every CC point on an accurate iso-cusped tool path. To apply this relationship to iso-cusped tool paths planning, the procedure is provided step by step. (1) Based on a sculptured surface, the first tool path should be designated and then is referred as the reference tool path for the next iso-cusped tool path. (2) According to the part tolerance, the number of the horizontal planes at the CC points is determined. On each horizontal plane, find the contours of the design surface and tolerance surfaces and the intersection point between the plane and the reference tool path (called the reference point). (3) Starting from the lowest contour of the design surface, find the virtual cutting edge at the reference point. Next, assume a point on the same contour and close to the reference point as the CC point for the next iso-cusped
34
Journal of Manufacturing Processes Vol. 8/No. 1 2006
Figure 9 Convex NURBS Surface on a Sculptured Surface Part
(6) Once the next iso-cusped tool path is found, this tool path is regarded as the reference tool path and the CC points on this tool path become the reference points. (7) Repeat Steps 3 to 6 to find the iso-cusped tool paths and then check if all the iso-cusped tool paths have covered the whole surface. If positive, end the program. By implementing the procedure, a software system has been made using MATLAB to generate accurate iso-cusped tool paths of sculptured surfaces.
Applications To demonstrate its versatility and feasibility, this new approach is applied to two sculptured surface parts using bull-nose end mills. The first sculptured surface part is a typical convex NURBS surface designed with the CATIA V5 CAD/CAM system (see Figure 9). The software system is employed to generate accurate iso-cusped tool paths for the NURBS surface machining. In the process planning for the surface milling, 6061T aluminum is selected for this part. A highspeed-steel (HSS) flat end mill with a diameter of 12.7 mm and a HSS bull-nose end mill with a diameter of 12.7 mm and a corner radius of 1.58 mm are chosen for rough and finishing machining, respectively. For demonstration purposes, the specified part tolerance is 0.2 mm. The cutting feed rate is set as 500 mm/ min., and the spindle speed is 1000 rpm to prevent chatter in machining. After inputting the design of the convex NURBS surface and the information determined in the process planning, the system takes a couple of minutes to generate the iso-cusped tool paths. Then the CNC machining of this part is simulated with the CATIA system, and the result is shown in Figure 10a. Finally, the surface is machined on a DECKEL MAHO 60T CNC milling machine, and the machined part is shown in Figure 10b.
Figure 8 Illustration of Relationship Between Iso-Cusped Tool Paths and Virtual Cutting Edges
tool path and then determine the virtual cutting edge at the assumed CC point. (4) Find the intersection point between the two virtual cutting edges and then check if the intersection point is inside, on, or outside the contour of the tolerance surface. If the intersection point is outside, adjust the assumed CC point closer to the reference point along the contour of the design surface. If the intersection point is inside, move the assumed CC point farther away from the reference point along the contour. Otherwise, the CC point is found for the next iso-cusped tool path. (5) Repeat Steps 3 and 4 to find the CC points for the next iso-cusped tool path from the lower contour to the upper contour of the design surface. 35
Journal of Manufacturing Processes Vol. 8/No. 1 2006
Figure 10 (a) Simulation of Surface Milling Using Iso-Cusped Tool Paths, and (b) Machined NURBS Surface Part Figure 11 (a) NURBS Surface Part Measured on a Mitutoyo CMM, and (b) Measured Surface Points
To verify the fidelity of the iso-cusped tool paths generated with this new approach, the machined surface is measured at five different altitudes on a Mitutoyo LEGEX 9106 coordinate measurement machine (CMM), whose accuracy is 0.1 micro (see Figure 11a). The five altitudes are 15, 20, 25, 30, and 35 mm along the z-axis, and on each altitude about 3,000 points are measured and shown in Figure 11b. In this figure, the virtual cutting edges and the cusp points are clearly shown, and five cusp points are selected on each of the example cusp curves (A, B, C, and D) to verify the cusp height. The results of the cusp heights are listed in Table 1, and the deviations of the cusp heights from the part tolerance are computed. Among the 20 locations, the maximum deviation of the cusp height occurs at the point 1 in the cusp curve A, at which the cusp height is 9.5% less than the specified surface tolerance. The main reason for this is the machining error in the finish machining of this part. As a whole, the cusp heights at the different locations can be regarded as con-
stant; therefore, the iso-cusped tool paths on the first part are verified to be valid. The second sculptured part is designed as a compound NURBS surface containing concave and convex shapes, and the software system is used to generate the iso-cusped tool paths shown in Figure 12a. Under the same machining conditions as those in producing the first part, the surface is machined with the generated iso-cusped tool paths, and the finished surface is shown in Figure 12b. In the two examples, the heights of the cusps between the iso-cusped tool paths generated using this new approach are close to the part tolerance; therefore, the surface accuracy is consistent across the entire surfaces without redundant machining. This is the advantage of iso-cusped tool paths. The examples also demonstrate the versatility and feasibility of this new iso-cusped tool path generation approach. 36
Journal of Manufacturing Processes Vol. 8/No. 1 2006
Table 1 Cusp Heights at Different Locations and Deviation Percentages of Cusp Heights
Unit, mm Point 1 2 3 4 5
Cusp Curve A Cusp Deviation Height (%) 0.181 –9.5 0.193 –3.5 0.187 –6.5 0.191 –4.5 0.188 –6.0
Cusp Curve B Cusp Deviation Height (%) 0.193 –3.5 0.191 –4.5 0.188 –6.0 0.192 –4.0 0.187 –6.5
Cusp Curve C Cusp Deviation Height (%) 0.205 2.5 0.197 –1.5 0.189 –5.5 0.186 –7.0 0.184 –8.0
Cusp Curve D Cusp Deviation Height (%) 0.188 –6.0 0.192 –4.0 0.195 –2.5 0.187 –6.5 0.187 –6.5
Figure 13 Inefficient Iso-Cusped Tool Paths for Surface Machining
upper-left corner of the part are inefficient. To address this problem, integration of the steepest-ascending tool path pattern and the iso-cusped tool path pattern provides an optimal solution for sculptured surface machining (Chen, Vickers, and Dong 2003).
Conclusions A new, practical approach is proposed to generate accurate iso-cusped tool paths for sculptured surface machining on three-axis CNC machines using all types of end mills. The major contributions of this work include simplifying the geometric representation of the cusp with virtual cutting edges and locating the cusp points accurately. In this work, computing the intersection of two planar virtual cutter edges is easier and faster than computing the intersection between two cutter-swept surfaces in the existing methods. Two typical examples have shown the versatility and feasibility of this newly proposed approach. Therefore, this approach to generating isocusped tool paths for sculptured surface machining can be implemented in CAD/CAM software systems to benefit the manufacturing industry.
Figure 12 (a) CATIA Simulation Result of Second Part, and (b) Machined Part
Although iso-cusped tool paths can eliminate redundant machining on sculptured surfaces, some tool paths may be quite inefficient in machining. In Figure 13, the iso-cusped tool paths are generated on a semi-cylindrical part; however, the tool paths on the 37
Journal of Manufacturing Processes Vol. 8/No. 1 2006
Acknowledgments The financial support of this work from the Natural Science and Engineering Research Council of Canada (NSERC) and a Concordia University Startup Grant to Dr. Z.C. Chen is gratefully acknowledged.
Lin, R. and Koren, Y. (1996). “Efficient tool-path planning for machining free-form surfaces.” ASME Journal of Engg. for Industry (v118), pp20-28. Mortenson, M.E. (1987). Geometric Modeling. New York: John Wiley & Sons. Roth, D.; Bedi, S.; Ismail, F.; and Mann, S. (2001). “Surface swept by a toroidal cutter during 5-axis machining.” Computer-Aided Design (v33), pp57-63. Sarma, R. and Dutta, D. (1997). “The geometry and generation of NC tool paths.” Journal of Mechanical Design (v119), pp253-258. Suresh, K. and Yang, D.C.H. (1994). “Constant scallop height machining of free-form surfaces.” Journal of Engg. for Industry (v116), pp253-259. Wang, W.P. and Wang, K.K. (1986). “Geometric modeling for swept volume of moving solids.” IEEE Computer Graphics and Applications (v6, n12), pp8-17.
References Abraham, G. (1970). Introduction to Differential Geometry. Addison Wesley Publishing Co. Altintas, Y. and Spence, A. (1991). “End milling force algorithm for CAD systems.” Annals of the CIRP (v40), pp31-34. Blackmore, D.; Leu, M.C.; and Wang, L.P. (1997). “The sweep-envelope differential equation algorithm and its application to NC machining verification.” Computer-Aided Design (v29, n9), pp629-637. Chen, Z.C.; Vickers, G.W.; and Dong, Z. (2003). “Integrated steepestdirected and iso-cusped tool path generation for three-axis CNC machining of sculptured parts.” Journal of Manufacturing Systems (v22, n3), pp190-201. Chiou, C.J. and Lee, Y.S. (2002). “Swept surface determination for five-axis numerical control machining.” Int’l Journal of Machine Tools and Manufacture (v42), pp1497-1507. Chung, Y.C.; Park, J.W.; Shin, H.; and Choi, B.K. (1996). “Modeling the surface swept by a generated cutter for NC verification.” Computer-Aided Design (v30, n8), pp587-594. Feng, H.Y. and Li, H. (2002). “Constant scallop-height tool path generation for three-axis sculptured surface machining.” Computer-Aided Design (v34, n9), pp647-654. Huang, Y. and Oliver, J.H. (1994). “Non-constant parameter NC tool path generation on sculptured surfaces.” Int’l Journal of Advanced Mfg. Technology (v9), pp281-290.
Authors’ Biographies Dr. Zezhong C. Chen is an assistant professor in the Dept. of Mechanical and Industrial Engineering at Concordia University, Montreal, Canada. His research interests include multi-axis CNC machining, high-performance/speed machining, intelligence and optimization for manufacturing, and computer-aided design and automation. His research is sponsored by the Natural Science and Engineering Research Council of Canada (NSERC), Pratt & Whitney Canada, and Consortium for Research and Innovation in Aerospace in Quebec (CRIAQ). He received his PhD in mechanical engineering at the University of Victoria in 2002. He is a senior member of the Society of Manufacturing Engineers. Dr. Dejun Song received his PhD in mechanical engineering at the Beijing University of Aeronautics & Astronautics, China. His research interests include multi-axis CNC machining, computer simulation and visualization, optimized computer geometric modeling, and product life cycle management.
38