Tool path generation for free form surfaces using Bézier curves/surfaces

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Abstract

Presented in this paper is a tool path generation method for multi-axis machining of free-form surfaces using Bézier curves and surfaces. The tool path generation includes two core steps. First is the forward-step function that determines the maximum distance, called forward step, between two cutter contact (CC) points with a given tolerance. The second component is the side step function which determines the maximum distance, called side step, between two adjacent tool paths with a given scallop height. Using the Bézier curves and surfaces, we generate cutter contact (CC) points for free-form surfaces and cutter location (CL) data files for post processing. Several parts are machined using a multi-axis milling machine. As part of the validation process, the tool paths generated from Bézier curves and surfaces are analyzed to compare the machined part and the desired part.

Introduction

A CAD/CAM system is a computer interactive graphic system equipped with software to accomplish certain tasks in design and manufacturing and to integrate the design and manufacturing functions. One of the important tasks performed on CAD/CAM system is NC part programming (Chang, Wysk, & Wang, 1998). A manufactured part is produced by a NC program containing a series of coded instruction called NC code that directly affect accuracy and cost of manufactured part, because accuracy and cost are proportional to the machining time. The coded instruction (specific command and numerical value) makes specific trajectories on the part being processed called tool path. There are two main tasks which are main subjects of this paper in NC part programming. The first task is tool path generation. The second task is defining geometry of the part to be machined. We discuss more in detail about each task in Sections 3 Mathematical representation, 4 Proposed approach. Milling operation is the primary machining process in the manufacture of the part and divided by two stages. The first stage is rough stage, the second stage is finish stage. In rough stage, part is machined in incremental layers and cutter removes most of the material on the surface so as to avoid damage of tool or/and machine. In finish stage, the surface is machined smoothly by approximating surface using line segments to get desired part with predetermined accuracy and shape. The tool paths in finish stage are important, since the tool path directly affect the accuracy and manufacturing time of the manufactured part. In milling operation, a rotating spindle is touched with the surface at cutter contact point (CC) and moves to next CC point linearly, that is curved path is approximated by a straight line segment as shown in Fig. 1(a). The accuracy of this linear approximation controlled by deviation is called tolerance. There is also an un-machined region between adjacent tool paths called scallop or cusp (Fig. 1(b)). After the machining, the grinding is needed to remove scallop. However, the grinding operation to remove scallop between adjacent tool paths is very expensive and time consuming. The large scallops increase amount of machining time to smooth the machined surface. Therefore, appropriate tool path in finish stage is very important to reduce the amount of secondary processes such as grinding and/or polishing. It is also important to generate tool path with less cutter contact point with given tolerance and scallop height that affect accuracy of work part being processed since we assume that the more line segments, the more machining time. Each discrete line segment quantity is called a forward-step denoted as “s” in Fig. 1(a) and the maximum allowable deviation is referred to as the tolerance denoted as “e” in Fig. 1(a). Further, the distance between two adjacent tool-paths called the side-step denoted as “g” in Fig. 1(b). The maximum allowable height of this scallop is called the scallop-height denoted as “h” in Fig. 1(b). The value of “e” and “h” are determined in advance and then “s” and “g” are calculated from the value of “e” and “h”, respectively (Suresh & Yang, 1994).

In this paper, a ball-end mill cutter on a 3-axis milling machine is used to generate tool paths. When we design the NC part program, one of the main tasks is defining geometry of the part. A surface is the image of a sufficiently regular mapping of a set of points in a domain into a 3D space and expressed asr(u,v)=(x(u,v),y(u,v),z(u,v))where u and v are the parameters of the surface. If the surface is defined on a bounded domain, generally u = 1, v = 1, it is called a surface patch.

A composite surface may consist of single patches with predetermined continuity condition between patches. When the domain of a surface is the xy-plane of the given Cartesian coordinate system, the parametric surface equation given by Eq. (1) reduces toZ=f(x,y)It is in an explicit function form and is called a nonparametric surface equation. Now consider an analytic functiong(x,y,z)=0which gives an implicit surface equation. If g(x, y, z) is a linear function it becomes a plane equation; if g(x, y, z) is a polynomial of degree 2 then it gives a quadratic surface. A surface represented by Eq. (3) is called an ‘analytic surface’, and one represented by Eq. (1) is called a ‘sculptured surface’. A surface that consist of analytic surface elements (Eq. (2)) or (Eq. (3)) is called ‘analytic compound surface’, and one that consists of sculptured surface elements (Eq. (1)) is called a ‘parametric compound surface’ (Choi, Lee, Hwang, & Jun, 1988).

In this work, we develop an efficient approach to generate tool paths for NC machining of free-form surfaces. The primary goals of this paper are as follows. First, reduction in the size of the CL (Cutter location) data file and machining time with given tolerance and scallop height. Second, verify real machining error by comparing desired surfaces and machined surfaces. To achieve these, first, we determine reliable and near optimal “s” and “g” to control the accuracy of the manufactured part and the time involved in manufacturing. Second, the machined surface scanned by point cloud method compare to CAD file of desired surface to verify true machining error that includes floating and measuring error as well. As a result of this algorithm, the part can be machined in the least machining time while keeping the given tolerance and scallop height in the tool path.

Section snippets

Literature review

The tool paths generation can be classified into 3 methods, iso-parametric, iso-planning, and iso-scallop height. In iso-parametric method, Loney and Ozsoy, 1987, Broomhead and Edkins, 1986 have studied the tool-path generation using iso-parametric curve. They approximated tool path into surface using iso-parametric curve on the surfaces. The forward step was calculated by “quick and dirty” method which is iterative and lengthy. Although this approach is the most widely used for the tool paths

Bézier curve

Bézier curves can be expressed in terms of Bernstein polynomials, defined byBin(t)=niti(1-t)n-iWith the control points bir, a Bézier curve can be expressed byb(t)=i=0nbi(t)Bin(t)This is to be interpreted as follows: first, the first point of curve is first control point, b0 and the last point is the last control point, bn. Second, the curve must be tangent to the line, b1b0 at b0 and at bnbn  1 at bn as shown in Fig. 2. In the Fig. 2, b0, b1, b2, and b3 are user defined control points. The

CC point and CL point

The cutter location (CL) point is a reference point on the cutter, with which the machine tool moves linearly along the tool path. Fig. 3 shows CC and CL point. The CC point is the point on the cutter that touches the work piece surface. Ideally, it is desired that the CC point lie on the work piece surface and is an addressable point which is a point along the axis at which the tool can be placed so as to minimize the manufacturing error.

The surface normal n at an arbitrary point p(u,v) on the

Implementation and result

The proposed approach was developed and implemented. Several parts for which the CC point was generated using the proposed algorithm, was also machined using a 3-axis milling machine. The algorithm has been coded in MATLAB on a personal computer. We machined free form shaped part using a block of wax and measured tolerance between machined surface and desired surface. After machining, a point cloud method was used to scan the machined surface to compare them. The desired surface was generated

Conclusion

The proposed algorithm for tool path generation is developed and implemented successfully. The implementation of this algorithm shows that the algorithm is very efficient for finish machining. There are some advantages. First, we reduced CL points by which NC code was generated significantly as shown in Table 1, Table 2. For example, 100 X 100 points generated the designed part. However, we generated machined surface by under 200 CL points with predetermined scallop height and tolerance.

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