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Today, sculptured parts are widely used in aeronautical, automotive, and many other industries. To make high-quality sculptured surfaces through efficient toolpaths in three-axis CNC machining is a major technical challenge. In this work, a new toolpath generation method for three-axis CNC milling, the Steepest-Directed and Iso-Cusped (SDIC) toolpath generation scheme, is introduced. The method integrates the steepest-directed and iso-cusped toolpaths in three-axis CNC programming. The steepest-directed toolpaths ensure the best geometry matching between cutter and surface for high cutting efficiency, and form a frame to control the trend in direction of the iso-cusped toolpaths. The iso-cusped toolpaths eliminate overmachining on the surface. Their combination ensures high sculptured part productivity, less redundant milling, and good surface quality. Machining examples of a half-cylinder part and a sculptured part are used to illustrate and to compare a range of SDIC toolpath patterns. The proposed SDIC toolpath scheme produces shorter toolpaths with higher machining efficiency. With the help of global optimization algorithms, this method of optimal toolpath generation can be applied to three-axis CNC machining of sculptured surfaces. [PUBLICATION ABSTRACT]
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Abstract
Today, sculptured parts are widely used in aeronautical, automotive, and many other industries. To make high-quality sculptured surfaces through efficient toolpaths in three-axis CNC machining is a major technical challenge. In this work, a new toolpath generation method for three-axis CNC milling, the Steepest-Directed and Iso-Cusped (SDIC) toolpath generation scheme, is introduced. The method integrates the steepest-directed and iso-cusped toolpaths in three-axis CNC programming. The sfeepesf-d/recfedtoolpaths ensure the best geometry matching between cutter and surface for high cutting efficiency, and form a frame to control the trend in direction of the iso-cusped toolpaths. The iso-cusped toolpaths eliminate overmachining on the surface. Their combination ensures high sculptured part productivity, less redundant milling, and good surface quality. Machining examples of a half-cylinder part and a sculptured part are used to illustrate and to compare a range of SDIC toolpath patterns. The proposed SDIC toolpath scheme produces shorter toolpaths with higher machining efficiency. With the help of global optimization algorithms, this method of optimal toolpath generation can be applied to three-axis CNC machining of sculptured surfaces.
Keywords: Toolpath Generation, CNC Programming, Sculptured Part Machining, Three-Axis Machining, Finish Machining
Introduction
Mechanical parts with sculptured surfaces are widely used in aeronautical, automotive, and domestic electronic industries due to their unique functions and appearance. Machining these parts, or their dies and molds, is mostly carried out with three-axis CNC machines. Over the years, significant progress has been made in sculptured part modeling and toolpath generation of three-axis CNC machining.
However, due to the diversity and complexity of sculptured surfaces, a general method of generating three-axis CNC machining toolpaths that guarantees high-quality finished surfaces and also ensures high machining efficiency remains a major technical challenge. This is particularly true for the case of finish machining. The objective of finish machining is to remove the unevenly distributed excess material produced in rough machining to form a finished surface satisfying the specified accuracy and tolerance. Tolerance determines how much finish machining is needed, which in turn determines the amount of finishing and polishing operations. The ideal toolpath is one that produces satisfactory surface quality with a minimum of machining time. At present, research on finish machining of sculptured surfaces primarily focuses on the surface quality through various toolpath generation methods that guarantee acceptable cusps on the finished surface. However, many of these approaches generate inefficient toolpaths causing longer machining time.
In this work, a new method integrates the steepest-directed and iso-cusped toolpath generation methods to machine a sculptured surface to a specified surface tolerance with a minimum of machining time. The iso-cusped toolpath generation scheme ensures that extremities of cusps on the machined surface fall either on the design surface or the tolerance surface (a surface offset from the design surface by a given tolerance). This ensures that cusps all across the surface have the same magnitude in height and avoids overmachining some areas of the surface (see Figure 1). In addition, the steepest-directed toolpath generation scheme ensures the best geometry fitting between tool and surface and produces the maximum cutting efficiency. Examples are given to demonstrate the proposed method. Discussions on some implementation issues and on the merits of the approach are also presented.
Related Work
The term "toolpath" refers to the path of cutter contacting (CC) points or path of cutter locations (CLs) (see Figure 1). Toolpaths are vital to surface quality and machining efficiency in sculptured part machining. In finish machining, current research on CNC programming has focused on methods of automatic toolpath generation that can guarantee accurate sculptured surfaces. These methods can be classified into several major categories: constant parameter, nonconstant parameter, iso-cusped, and steepest-directed tree toolpath generation methods.
In constant-parameter approaches (Loney and Ozsoy 1987; Broomhead and Edkins 1986; Suh and Lee 1990; Huang and Oliver 1994), constant-parameter curves are retrieved from the parametric design of a sculptured part as the paths of CC points. A toolpath interval is the constant parameter increment of two adjacent toolpaths. Its value is determined by keeping the maximum cusp height between adjacent paths consistent within the specified surface tolerance. Similarly, Sata et al. (1981) took a conservative parameter deviation between two iso-parameter curves as the toolpath interval. However, in all these methods, cusp height varies significantly across the part surface because of nonuniformity between the surface parameters and the Cartesian coordinates of the CC points (Bobrow 1985).
In nonconstant-parameter toolpath generation, Huang and Oliver used the intersection of parallel and/or nonparallel guide planes with the part surface to create planar toolpaths. This reduced the computing load and provided flexibility on plane adjustment.
In iso-cusped toolpath generation, Suresh and Yang (1994) were able to generate toolpaths that yielded cusps of constant height across the surface. In a similar way, S arma and Dutta (1997) offset the part surface by a specified tolerance so that the cusp extremities lay on the offset surface. From the known path of CC points, the cusp curve could be computed on the offset surface, and the new toolpath could then be generated on the part surface. This method was effective in eliminating redundant machining but did not necessarily generate the most efficient toolpaths.
A "steepest directed tree" toolpath generation scheme was introduced in the authors' previous work (Maeng, Ly, Vickers 1996). The approach represents a sculptured surface using a surface mesh. Toolpaths are generated along several steepest-slope trajectories on the surface, passing through the nodes of the surface mesh. These key toolpaths start from the bottom of the surface and gradually merge as they move upward. The approach ensures that the total toolpath length is minimal but does not eliminate overmachining the surface.
Due to the geometric complexity of sculptured surfaces and the difficulty in calculating cusp heights, there has not been any comprehensive solution to this problem. In this work, an approach is presented that encompasses the steepest-directed and iso-cusped toolpath generation methods.
In addition to surface quality, machining efficiency is another important issue for the finish machining of sculptured parts. Toolpath length has been found to be a major influencing factor to the total machining time and machining efficiency (Dong, Li, Vickers 1993; Huang and Oliver 1994; Maeng, Ly, Vickers 1996; Chen, Vickers, Dong 2003). A new toolpath generation method that can address both issues is needed.
Steepest-Directed and Iso-Cusped Toolpath Generation Method (SDIC)
Among the various toolpath generation methods discussed previously, constant parameter, nonconstant parameter, iso-cusped, and steepest-directed tree, two toolpath generation schemes, the iso-cusped and the steepest-directed tree (or its generalization, the steepest-direction scheme), are of particular interest and are used as the foundation of this work.
In this work, a flat or torus end mill is used for machining the convex curved surface. The cutter machines the surface from the bottom up using the side of the cutter due to higher cutting efficiency, as identified by Vickers and Quan (1989) and many other researchers.
Steepest-Directed Toolpath
The general scheme of the steepest-directed toolpath is illustrated in Figure 2. A steepest-directed toolpath refers to a path of CC points, at which the tangent vector of the path is in the steepest direction of the surface. It has been proven that in three-axis CNC milling, if a cutter is moved along the steepest direction at any CC point, the cutter profile fits the surface better than movement in any other direction (Chen, Dong, Vickers 2001). In oilier words, the approach removes the largest amount of raw material in the step and thus gives the most efficient machining. Calculation of the steepest direction on a surface is given below.
Figure 2Steepest-Directed Toolpaths for a Sculptured SurfaceFigure 3Steepest Direction at a Cutter Contacting Point on a Part Surface
Theoretically, the tangent vector of the steepest-directed toolpath at any point is in the form of Eq. (10). However, in practice, the steepest-directed toolpath is fit by a set of CC points, and each of the CC points is the intersection between a surface contour and the vertical plane, ?, created by its previous CC point. When the surface contours are closely layered, the vector from a lower CC point to the following upper CC point approximates the steepest direction of the surface at the lower CC point. Moreover, no matter where the starting point is, the steepest-directed toolpath will always reach the apex of the local region.
Iso-Cusped Toolpath
Iso-cusped toolpaths are such that if a tool machines along two adjacent toolpaths, the heights of the cusps left on the surface are equal (see Figure 4). Therefore, iso-cusped toolpaths can prevent redundant machining and increase productivity without sacrificing surface quality.
Suppose a design surface, a tolerance surface, a set of surface contours, and the iso-cusped toolpaths of the surface are given. Any adjacent CC points on a surface contour occur at the intersection of the surface contour and adjacent iso-cusped toolpaths. These CC points are determined in such a way that when the cutter is located at these CC points, the adjacent cutter profiles intersect at the tolerance surface. Extremities of the cusps therefore occur at the design and the tolerance surfaces.
The process for calculating CC points to form an iso-cusped toolpath is developed below. The toolpath can be offset from either an initial toolpath (usually the steepest-directed toolpath) or previous iso-cusped paths. Given an initial CC point, an adjacent CC point is arbitrarily selected on the same surface contour. If the resulting cusp height is greater than the specified surface tolerance, the adjacent point is moved along the contour in the direction of the initial point. This occurs iteratively until the intersection occurs on the tolerance surface. Thereafter, CC points on higher contour lines are calculated until the apex is reached. These CC points form the isocusped toolpath and produce a reference to offset the next path.
To illustrate how to calculate the CC points in the new iso-cusped toolpath, a flat end mill and a convex half-cylinder part surface are taken as an example. Because of the specialty of the tool and surface, the CC points of the iso-cusped toolpath can be computed with closed-form expressions. In general, an iterative optimization algorithm is adopted.
According to these formulae and the CC points in the toolpath of reference, the corresponding CC points of the new iso-cusped toolpath can be quickly calculated; therefore, the new iso-cusped toolpath can be found by computing a set of CC points.
Integration of Steepest-Directed and Iso-Cusped Toolpaths
The steepest-directed toolpaths are productive because the machining capacity of the tool reaches maximum when it cuts along the steepest pathway. However, in this approach, a series of adjacent steepest-directed toolpaths may cause redundant machining. The iso-cusped toolpaths are efficient because redundant machining is eliminated but geometry matching between tool and surface is not optimized. An optimum situation may be obtained using a series of steepest-directed toolpaths as a frame or skeleton across the surface and offsetting iso-cusped toolpaths from this frame.
In Figure 6, two steepest-directed toolpaths starting from the corners of the surface are shown as the frame with the iso-cusped toolpaths covering the enclosed region. It is clear from this figure that the iso-cusped toolpaths are determined sequentially from each side and form an interlaced pattern on the surface. The steepest-directed toolpaths establish a frame that controls the trends of directions of the iso-cusped toolpaths, so the possibility of their straying too far away from the steepest directions is reduced. Within each enclosed region of the frame, the iso-cusped toolpaths are evenly distributed, and the extent of repetitive machining is significantly decreased.
Procedure of SDIC Toolpath Generation
To implement the SDIC method on sculptured parts in finish machining, the general procedure includes the following steps:
1. Make a contour map of the sculptured surface in its parametric design space, and generate a frame with several steepest-directed toolpaths to divide the surface. Each steepest-directed toolpath will be represented by a set of CC points occurring sequentially on surface contoursstarting from the lowest and going to the highest contour for the path. Consequently, the steepest-directed toolpaths divide the part surface into several patches.
2. Consider one of these patches with one or two steepest-directed toolpaths as its boundaries, and select one of the steepest-directed toolpath(s) as a toolpath of reference, and the other steepest-directed toolpath or boundary as a checking toolpath.
3. Start from the lowest CC point on the toolpath of reference and calculate the corresponding CC point of the new iso-cusped toolpath. If this new CC point is beyond the checking toolpath, the new CC point will be replaced with the intersection between the surface contour and the checking toolpath.
4. Take the next higher CC point on the toolpath of reference and execute Step 3. This loop continues until all CC points of the new iso-cusped toolpath are found.
5. Check whether or not the surface patch is fully covered with iso-cusped toolpaths. If yes, terminate toolpath planning for this patch and move to next step; otherwise, reassign the checking toolpath as the toolpath of reference and the newly calculated iso-cusped toolpath as the checking toolpath. Then go to Step 3.
6. If all the patches have been covered with toolpaths, toolpath planning is finished; if not, continue toolpath planning for the next patch and go to Step 3.
7. Rearrange the toolpaths in a certain order according to the requirement of machining and plot all toolpaths for evaluation and checking.
SDIC Application
The SDIC approach can be implemented for any type of part surface and cutting tool with three-axis CNC machining.
SDIC Toolpath Patterns
To illustrate the SDIC approach, a simple machining example was selected as shown in Figure 7. The workpiece is a horizontal half-cylinder (30 mm radius and 100 mm long). A surface machining tolerance of 0.05 mm and a flat end-milling cutter of 10 mm radius are used. The cylindrical curved surface is used to better illustrate the toolpath variation for different surface curvatures. A sculptured surface example is presented in the following section to demonstrate the generic applicability of the approach.
In the numerical solution to the problem, 20 contours were generated on the half-cylinder, and five of them are shown in Figure 7. To illustrate a typical toolpath frame, four steepest-directed toolpaths were evenly distributed along the cylinder. The steepestdirected toolpath in this case is the intersectional are normal to the cylinder axis. After the SDIC approach was applied, the resulting SDIC toolpaths were generated and shown in Figure 8.
In Figure 8a, the SDIC toolpaths are in the cylinder parametric space with two parameters (sectional circular angle [theta], cylinder length v). Among the SDIC toolpaths, the dashed lines are the steepest-directed toolpaths, and the solid curves are the iso-cusped toolpaths. Also, the cylinder contours are the horizontal lines. These SDIC toolpaths are then transformed into Cartesian coordinates. In Figure 8b, the SDIC toolpaths are shown in 3-D, and the steepestdirected toolpaths (the dashed lines) form the toolpath frame, and the iso-cusped toolpaths take major part of the SDIC toolpaths.
Figure 7Horizontal Half-Cylinder and Flat End Milling CutterFigure 8SDIC Toolpaths of Half-Cylinder Part
Because the number of steepest-directed toolpaths may be varied over a wide range, a variety of different SDIC toolpath patterns may be obtained. The range of steepest-directed toolpath of this part is between 1 and 52 for the given tool and surface tolerance. The two extreme cases of the SDIC toolpath scheme include the following:
* Only one steepest-directed toolpath: In this case the first toolpath is steepest-directed and thereafter the SDIC toolpath scheme becomes the unconstrained iso-cusped toolpath scheme. All but one toolpath is generated based on the isocusp rule.
* Fifty-two steepest-directed toolpaths: This is the other extreme, where the number of the steepest-directed toolpath reaches the maximum, because the surface tolerance is just met at the base of the half-cylinder and all other areas are overmachined. Under this condition, the SDIC toolpath scheme actually becomes the steepest-directed toolpath generation method.
Following the procedure of SDIC toolpath generation and setting the number of steepest-directed toolpaths as 1, the SDIC toolpaths are calculated and shown in Figure 9. Similarly, Figures 10 and 77 illustrate the SDIC toolpaths when the steepest-directed toolpaths number is 6 and 52, respectively.
Optimum SDIC Toolpath
Four SDIC toolpath patterns given in Figures 8 to 77 vary according to the number of steepestdirected toolpaths assumed. In Figure 9, the leftend toolpath is the only steepest-directed toolpath, and iso-cusped toolpaths cover the remainder of the cylinder. All toolpaths start from the base of the cylinder and proceed to the highest point on the cylinder surface. Due to the symmetric shape of the half-cylinder, the other side of the half-cylinder has mirror-imaged toolpaths. With only one steepest-directed toolpath, the frame is open at the right end. Directions of iso-cusped toolpaths are therefore not fully constrained and they tend to become horizontal at the right end, that is, they stray away from the vertical steepest direction. In contrast, the SDIC toolpath frame in Figure 10 shows six steepest-directed toolpaths with five isocusped patches. The iso-cusped toolpath within each patch is built up sequentially from both sides, forming an interlaced pattern. Increasing the number of steepest-directed toolpaths results in more and smaller-sized patches in which fewer iso-cusped toolpaths can be filled. At the extreme, all pathways become steepest-directed toolpaths, producing 52 steepest-directed toolpaths, as shown in Figure 11.
Figure 9SDIC Toolpaths with a Frame of One Steepest-Directed ToolpathFigure 10SDIC Toolpaths with a Frame of Six Steepest-Directed Toolpaths
The overall toolpath length, which is a function of machining time, was calculated for all valid toolpath patterns. A curve of toolpath length versus number of steepest-directed toolpath is shown in Figure 12. The curve shows that the minimum SDIC toolpath length of 4042 mm is generated with six steepest-directed toolpaths. The SDIC toolpath with 1, 4, and 52 steepest-directed toolpaths resulted in 6323, 4093, and 4899 mm toolpath lengths, respectively. This is a ratio of 1.56, 1.01, and 1.21 times the optimum toolpath length for these cases. In Figure 12, the SDIC toolpath with 1, 6, and 52 steepest-directed toolpaths corresponds with point 2, 3, and 4.
In comparison, if the least steepest-directed tree is taken, that is, the worst-case iso-cusped toolpath, which consists of horizontal toolpaths along the axis of the half-cylinder (see Figure 7), the toolpath length is 60,100 mm, which is a ratio of approximately 15 times the optimum toolpath length or 15 times longer machining time.
Sensitivity Study on SDIC Toolpaths
Because the relationship between the parameters, such as part surface, surface tolerance, tool shape and size, and the length of the SDIC toolpath, is very complex, a sensitivity study was carried out. The parameters that influence the toolpath length for the half-cylinder example, namely cylinder length, cylinder radius, cutter radius, and surface tolerance, are given in Tables 1 to 4.
Figure IISDIC Toolpaths with a Frame of 52 Steepest-Directed ToolpathsFigure 12Curve of Toolpath Length vs. Number of Steepest-Directed Toolpath
In Table 1, the cylinder length is varied from 100 to 200 mm, and in Table 2 the cylinder radius is changed from 20 to 40 mm. The percentage change in both length and radius of the half-cylinder causes a similar percentage change in the SDIC toolpath length. In Table 3, the tool radius is changed from 10 to 20 mm, and in Table 4 the tolerance is varied from 0.05 to 0.1 mm. The percentage change in both tool radius and surface tolerance causes less percentage change in the SDIC toolpath length. For example, a 50 percent change in tool radius or surface tolerance causes only a 16 percent change in SDIC toolpath length. However, for a given surface shape, the tool radius and surface tolerance are the only means of further reducing the optimized machining time.
An additional fact is that the optimized number of steepest-directed toolpath changes with cylinder length, but is almost independent of cylinder radius, tool radius, and surface tolerance.
SDICToolpath for an Example Sculptured Part
An example mechanical part with a convex sculptured surface that has considerable variations in surface curvature is used to illustrate the generated SDIC toolpaths. The convex free-form surface is shown in Figure 13.
Given a surface tolerance of 0.1 mm, the SDIC toolpaths are generated and shown in Figure 14, following the SDIC toolpath generation algorithm introduced previously.
* A contour map of the sculptured surface is first generated in its parametric design space. A frame with five steepest-directed toolpaths is then created to divide the surface. Each steepest-directed toolpath is represented by a set of CC points occurring sequentially on surface contours-starting from the bottom of the surface and going to the top. These steepest-directed toolpaths divide the part surface into five patches, as shown in Figure 14.
Table 2Cylinder Radius vs. Length of SDIC ToolpathsTable 3Tool Radius vs. Length of SDIC ToolpathsTable 4Surface Tolerance vs. Length of SDIC Toolpaths
* The iso-cusped toolpaths are then generated for each of the five patches with one or two steepest-directed toolpaths as its boundaries, and the steepest-directed toolpath on the right is used as a toolpath reference, and the other steepestdirected toolpath or boundary as a checking toolpath. The iso-cusped toolpaths start from the lowest CC point on the toolpath of reference. Calculations for the corresponding CC point of the new iso-cusped toolpath are conducted to form the climbing toolpaths following the algorithm given previously.
The sculptured surface machined using the generated SDIC toolpaths is given in Figure 15. The SDIC toolpath has a much shorter toolpath length and considerably reduced machining time over the iso-parametric and iso-cusped toolpaths.
Summary
The core of the SDIC method in three-axis CNC programming is the integration of steepest-directed and iso-cusped toolpaths. The steepest-directed toolpaths ensure the best geometry matching between cutter and surface for higher cutting efficiency, and form a frame to control the direction of the iso-cusped toolpaths. The iso-cusped toolpaths eliminate overmachining of the surface. An appropriate mixture of these two toolpath-generating schemes ensures high machining productivity, less redundant milling, and good quality surfaces. With the help of global optimization algorithms, this method of optimal toolpath generation can be applied to three-axis CNC machining of any sculptured surface.
A half-cylinder machining example is presented, and a range of SDIC toolpaths are generated and compared. The proposed SDIC toolpath scheme produces a shorter toolpath with higher machining efficiency. The optimized SDIC toolpath includes six patch-defining steepest-directed toolpaths and many filling iso-cusped toolpaths. The resulting SDIC toolpath is one fifth less than the steepest-directed toolpath scheme and 15 times less than the worst iso-cusped toolpath scheme.
Figure 14SDIC Toolpath for Machining the Example SurfaceFigure 15Machined Sculptured Surface Using SDIC Toolpath Generation Method
The proposed SDIC toolpath scheme was also demonstrated using a sculptured part. The shape of the sculptured surface in mesh form, the SDIC toolpaths generated for machining the sculptured surface, and the machined part are illustrated to further demonstrate the new method.
Acknowledgment
The graduate fellowship to Z. Chevy Chen from the University of Victoria and research funding to Geoffrey W. Vickers and Zuomin Dong from the Natural Science and Engineering Research Council (NSERC) of Canada are greatly appreciated. The authors would also like to thank Mr. Paul A. Wagener at the University of Victoria for the implementation and testing of the SDIC algorithm on sculptured surfaces.
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Z.C. Chen, Dept. of Mechanical and Industrial Engineering, Concordia University, Montreal, Quebec, Canada
G.W. Vickers and Z. Dong, Dept. of Mechanical Engineering, University of Victoria, Victoria, British Columbia, Canada
Authors' Biographies
Dr. Zezhong Chevy Chen received his PhD from the University of Victoria, Canada, in 2002. He is currently an assistant professor in the Dept. of Mechanical and Industrial Engineering at Concordia University, Montreal, Canada. His research interests include sculptured surface machining, multiple-axis CNC machining. CAD/CAM, optimization, and intelligent manufacturing. He has been doing research in these areas for many years and working with industiy on many projects.
Dr. Geoffrey W. Vickers received his PhD from the University of Manchester. UK. he is a professor of mechanical engineering and a senior ASI fellow at the University of Victoria, British Columbia, Canada. Dr. Vickers is specialized in the development and application of advanced manufacturing techniques to British Columbia and Canadian industries. His research has centered on the definition and numerically controlled machining of curved surfaces and laser-based reverse engineering. His research also has found applications in mold and die making, in ship building, and in the marine industiy. He won the Society of Manufacturing Engineers' "Educator of the Year" award for development of the educational-industrial interface.
Dr. Zuomin Dong received his PhD in mechanical engineering from the State University of New York at Buffalo in 1989. he is a professor of mechanical engineering at the University of Victoria, Canada. Dr. Dong's research interests include: integrated concurrent engineering design with virtual prototyping based global design optimization; automated CNC toolpath generation for machining sculptured parts using optimization and intelligent systems; automated tolerance analysis and synthesis; innovative fuel cell design and manufacturing; and fuel cell system design optimization. Dr. Dong is on the editorial boards of several technical journals, is a technical consultant, and is a director of several companies.
Copyright Society of Manufacturing Engineers 2003