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Numerical control machining of triangulated sculptured surfaces in a stereo lithography format with a generalized cutter A unified approach to the generation of numerical control tool paths for triangulated sculptured surfaces in a stereo lithography format using a generalized cutter is presented. This is important because the use of a stereo lithography format for representing a computer-aided design model has been widely accepted in industry for quite some time. It is not only just because of an application such as rapid prototyping (RP), which specifically requires the use of it, but also it is due to the fact that complex stereo lithography models can now be created directly by the digitization and reverse engineering process. Although many computer-aided design/computer-aided manufacturing software systems support the translator of stereo lithography files, only a few papers have addressed the issue of numerical control machining directly from a stereo lithography file. A general computing algorithm to generate tool paths by using a generalized automatically programmed tools cutter is presented. It is general in the sense that it can be applied to various cutters including ball, flat and fillet end-mills. To reduce the computation time, an efficient method for the region query of a tessellated mesh is also presented. Simulations as well as real machining examples are given to illustrate the effectiveness of the proposed method. 1. Introduction Currently, most computer-aided design (CAD) systems use parametric surfaces to represent the geometry of a CAD model. To transfer models between different CAD/computer-aided manufacturing (CAM) systems for various designs or manufacturing processes, neutral data files such as Initial Graphics Exchange Specification (IGES) are used extensively (US Product Data Association 1996). IGES describes the possible information to be used in building a CAD model, the parameters for the definition of model entities and the relationships between different entities. However, the translation of CAD models using IGES is not always easy because most CAD systems use different internal representations and the conversion is not always straightforward and error free. In contrast to IGES, the stereo lithography (STL) format is simple and its implementation is easy (Jacobs and Reid 1992, Kochan 1993). Basically, an STL file contains only triangles and their normal vectors. The STL file is not intended to replace IGES, which contains more design-related information, nevertheless the information contained in STL is sufficient for many downstream manufacturing activities such as rapid prototyping, numerical control (NC) machining and even finite-element analysis. Because of its simplicity and use in various engineering fields, STL translation today is supported by most CAD/CAM systems. In the past, large STL files had been a burden to memory allocation and computation speed. However, as the central processing unit cranks up more power and memory chips become less expensive, this is no longer a barrier for the transfer and processing of STL files. Furthermore, the latest three-dimensional scanning technology also helps the rapid growth of the reverse engineering application in which very large and complex models are created and stored in STL files (Chuang et al. 2002). It is generally agreed that the use of triangulated surfaces and STL files for design and manufacturing applications will become increasingly popular. In the past, many path-planning approaches for three-axis machining have been studied (Dragomatz and Mann 1997). The tool-path generation methods can be categorized into two types: Cartesian and parametric (Zeid 1991). The former is generated from cross-section planes that intersect the machined surfaces. The latter generates NC tool paths along constant u or v surface curves and the cutter location (CL) point is usually computed from the offset of the machined surface (Kishinami et al. 1987, Tang et al. 1995, Choi et al. 1997, Lee 2003). The parametric method has the advantage of utilizing accurate surface information, but it might not be suitable for machining a compound surface consisting of surface patches and is susceptible to concave gouging (Choi and Jerard 1998). On the other hand, the Cartesian method is good at generating gouging-free tool paths but it lacks the ability to generate pencil cuts or cornering cuts (Dragomatz and Mann 1997).Therefore, in the machining of a real-world part, both Cartesian and parametric path generation strategies are used interchangeably. STL machining primarily employs the Cartesian machining strategy since it does not contain the full surface information. In Cartesian path planning, CLs are calculated when the cutter touches the machined surface. One of the most robust methods in Cartesian path planning is the Z-map method (Choi et al. 1988, Choi 1991, Saito and Takahashi 1991, Lin and Liu 1998). The Z-map method computes the interference-free CLs from a grid data set. The precision of machining is dependent on the density of the grid data. There is usually a need for a large memory space to be allocated for the grid data. Hwang and colleagues presented a method to generate interference-free tool paths from tessellated surfaces by using flat, ball and fillet end-mills (Hwang 1992, Hwang and Chang 1998). However, the method treats each cutter separately, and algorithms were developed for limited cutter types. In practice, however, there are more different types of cutter that are being used. For example, a tapered and sharp end mill is often used for marking thin grooves. It would be tedious and cumbersome to develop separate algorithms and codes for all the needed cutters. The present paper presents a unified approach to the tool path generation directly from the triangulated surface of an STL model by using a generalized automatically programmed tools (APT) cutter (Kral 1986). The topology definition of an APT cutter is usually used for NC verification, but most tool path generation approaches are developed for specific types of cutters and are not general (Chung et al. 1998, Chiou and Lee 1999). The method presented here is general in that it can be applied to all types of cutter represented by the APT cutter, which includes the frequently-used ball, flat and fillet end-mills, and more (figure 3 and 4). From this research result, only one systematic and unified algorithm is needed, which is very compatible to the principle of the APT generalized cutter. To reduce the computation time when dealing with a large STL file, an efficient method for the region query has presented。 2. Numerical control-path planning in three-axis machining In practical application, NC paths are generated for different machining procedures (Choi et al. 1994) (figure 1)： Figure 1. Different machining procedures in NC path planning. The main procedures include rough cut, semifinish cut, finish cut and undercut removal (often called pencil cut or corner cut). With large size cutter and high feed rate, the rough cut (usually with a flat end-mill) is designed to remove efficiently bulky redundant material. For a better cutting result, there are usually several semifinish cuts (usually with fillet end-mills or ball end-mills) preceding the finish cut. After the semifinish cut, a uniform thickness of material remains on the final surface before the finish cut (usually with a small ball end-mill) is used to remove this thin layer. At times there is a need to generate a pencil cut or corner cut after the finish cut because there is an undercut region along the corner edge (figure 2). An even smaller cutter is used in this ‘clean-up’ machining procedure to contour around corners or edges to remove uncut material. Based on the above discussion, a unified approach that can generate CLs for a generalized cutter is not only practical, but is also easier to implement and maintain. Since there are robust algorithms to triangulate parametric or implicit surfaces under a controlled tolerance, the algorithm developed here can serve as a ‘core engine’ for a general CAM package. Figure 2. Undercut region. 3. Generalized geometry of an APT cutter According to the definition of APT, the generalized cutter geometry shown in figure 3 can be described fully by the following parameters: Figure 3. Parameters for a generalized cutter geometry. d cutter diameter, which is twice the radial distance measured from the cutter axis to the intersection of the lower and upper line segments, r corner radius, e radial distance from the cutter axis to the centre of a corner circle; it is positive if its corner and centre are on the same side of the tool axis, otherwise it is negative, f distance from the endpoint to the centre of corner circle measured parallel to the tool axis. The cutter parameter values must be consistent among themselves and not violate certain restrictions so that permissible geometries are properly described (Kral 1986). Several selections of cutter shapes are shown in figure 4. Figure 4. Some selections of cutter shape based on APT definition. Some dependent parameters are derived as follows. They are used to help escribe the computation of CL points. R=+（Lc-tan1）tan2--------（1） where the radius, R, is the maximum boundary of the cutter projecting on the part surface. The boundary will be used to find the intersected triangles in the projected region. From the geometric profile of the cutter, the radius of ring circles R1 and R2 can be computed as follows: R1=（u+）/2-------------（2） Where R2=e+（vsin(22)+）/2--------（3） And V=（（R-e）/tan2）-（Le-f） where L=Lc-f+----------------（4） The distance, L, measured from the centre of ring circle with radius R2 is computed as follows: L=Lc-f+ The two distinct points on the cutter profile with distances of R1 and R2, respectively,from the tool axis divide the cutter profile into three different regions. On the top is a frustum of cone with radius R, R2 and height L, the median part is a torus of ring radius e and corner circle radius r, and the bottom is a circle cone of radius R1 and height R1 tan_1. Generally, a cutter profile needs not contain all the three regions. As shown in figure 4(a),the shape becomes a cylinder; in figure 4(c), the torus becomes a semisphere; in figure 4(d), it is a taper cutter. 4. Algorithm for generating cutter locations A traditional solid modelling approach to NC machining requires the generation of offset surfaces to approximate the CL surfaces. Although it is simple in concept, there are several shortcomings. First, the generation of offset surfaces in itself is not a trivial problem. In solid modelling, a boundary representation (B-Rep) model is the most popular representational form. The offset of trimmed non-uniform rational B-splines (NURBS) surfaces is a complex and computationally expensive operation. Second, the offset of multiple trimmed surfaces can easily create complex selfintersection and global-intersection (with adjacent surfaces) problems. Third, the uniform offset of surfaces is only useful to the generation of CL points for ballend mills. The offset of CL surfaces for fillet-end mills is a more difficult problem, not to mention the more general case of APT cutters. Overall, the traditional solid modelling approach to NC tool-path generation by surface offsetting is complex in calculation and inefficient in computation. Furthermore, in Cartesian machining, given a parametric part surface, the tool path is generated from the intersection of the offsetting part surface and a series of vertical planes parallel to the tool axis. Non-linear equation solving may be involved for finding the intersection curves. For an STL model, however,since the part surface is already triangulated, the tool path generation is to compute the CLs from the polyhedral surface. In most cases, only linear operations are needed. As shown in figure 5, the point of the cutter contacting the part surface is called a cutter-contact (CC) point, and the endpoint of a cutter is defined as a CL point. During machining,the CC point is not fixed, but the x–y locations of the CL point can be arbitrarily determined (most times falling on fixed grid points). The only unknown to be found is the z component of the CL point. Therefore, the tool path usually consists of a sequenced series of CL points. When the tool axis moves to a two-dimensional point (xc, yc), there exists a region on the part surface, enclosed by a two-dimensional circle with radius R, and centred at (xc, yc). It is called a CC region (figure 5). Figure 5. CC point, CL point and CC region. This paper presents an algorithm to compute the inference-free CL point from those triangular facets of the part surface that are overlapped with the CC region. When a cutter makes contact with a triangular polyhedron, the CC point may be located at a vertex, a facet or an edge. For the cutter itself, the CC point may be contacted on the taper envelope, the torus region or the lower cone. For these various contact conditions, the computations of CL points are not the same. It is necessary to determine the contact regions and CC points first, and then from the locations of CC points, the CL points of the tool path can be calculated. For a generalized APT cutter, there are nine types of computation model. In reality, not every cutter profile contains all the three regions. Very often, a cutter usually contains only one or two cutting regions (figure 4).The CLpoint_computation procedure begins the computation of the CC point with the triangular facet first. This is a time-saving strategy because if the CC point is located inside the facet, then the cutter does not touch the vertices or edges of the triangle and, therefore, the latter two more time-consuming steps can be avoided.