附錄 A 外文文獻Geometry design model of a precise form-milling cutter based on the machining characteristicsAbstract This paper presents a new approach to design a form milling cutter for precisely obtaining the complex free-form surfaces. In this study, the intersection point of the rake surface, helix flute and clearance flank is appropriately defined due to its significant role in the design and grinding performance. The angle-solid-block analysis is developed to establish the new cutter geometry model. Hence, a new form-milling cutter satisfying the requirements of machining characteristics of workpiece can be designed. In addition, the cutter geometric model can be adopted to map out the measuring strategy with minimum measured points to attain the exact geometric feature of cutter.1 Introduction1.1 Motive of researchA multifacet drill (MFD) with multifacet and multiflanks has been distinguished from its significant characteristics such as lower cutting forces, better heat transfer, more accuracy, longer tool life, higher productivity, etc [1]. A three-axis milling machine with a ball-nosed milling cutter is the familiar method in the free form surface milling process. However, its precision has less refinement than a five-axis milling machine with an end mill. In every milling location, the coincidence between tool axis and surface normal for five-axis milling process has better performance than three-axis process. If the coincidence between tool axis and surface normal is identical, it is referred to as being always normal; the term “quasi-normal” is proposed to describe the relative normal degree. In this case, the consideration of milling characteristics and application to multi-axis degree of freedom has to be taken into account to help design a ball-nosed milling cutter with multifacet and multiflanks. Accordingly, it leads to better geometric precision and geometric compatibility of workpieces, and this milling cutter is called a form milling cutter in this study. Therefore, a geometric design model for precise form milling cutter is the main purpose of this study.This study probes into the modeling research of cutting edge geometry to drive off the problems of surface normal vectors, which are different from point to point based on the concepts of always normal and quasi-normal. Accordingly, the inspection method is developed to verify the theory and performance of modeling.1.2 ReviewIn order to exactly solve the above problems, both the cutter and workpiece’s geometric characteristics have to be verified and unified at first. Glaeser, Wallner and Pottmann [2] made a concept definition for workpiece characteristics, but the corresponding cutter characteristics have not been well defined. The method of always normal (named as axis milling by Baptista and Simoes [3]), proposes that the spindle is normal to the free-form surface on the cutting point in the workpiece. Baptista and Simoes showed some larger scallops were left and then replaced the ball-nosed end mill by an end mill inclined in the feeding direction to reduce the scallop dimension. However, it was not suitable for 3-axis milling.Lee and Chang [4] used the 2-phase approach with 4-th and 5-th axis postures to avoid the global cutter interfer-ence caused by the cutter and its holder. A method by Yoon et al. [5], presented locally optimal cutting positions for cutting directions on the 5-axis sculptured surface machining. The cutter positions can guarantee local gouging avoidance. Both [4] and [5] briefly described the limitation of cutter posture in the 3-axis milling processes. Glaeser [2] offered an idea on selecting a cutter of collision-free 3-axis milling for free-form surfaces, but the cutter design has never been studied. Regarding the suitable tool-path of 3-axis milling on free-form surface, Park and Choi [6] described a direction-parallel method and Park [7, 8] proposed both characteristic contour and z-map methods. Nevertheless, the geometric relationship between the cutting edge and the workpiece was not fully described.Tsai and Wu [9] showed the quadratic surface flanks for drill point design and grinding. Fujii et al. [10] defined the cutting edge of drill geometry as wire frame algorithm. Wu and Shen [1] derived the configuration of the multifacet drill. All of the above cutter geometric models were analyzed by the analytic geometry method. Then, Wang et al. [11] investigated the geometry of a multifacet drill with a new approach - the angle-solid-blocks method. This study analyzed the relationship of the geometry of the drill-tip, flanks and straight cutting edges and then found the relationships of the cutting parameters, which depend upon one another. However, the approach [11] has not yet developed the angles-relation of the cutter geometry for the generation of curved cutting edges, multi-segments edges and flutes. So a new modeling method for the form milling cutter is still required.Bradley and Chan [12] used reverse engineering in the inspection of a ground cutter with a touch probe and laser scanning. But they still lacked the better method for measuring the geometry of cutting edge.1.3 Guidance of researchThe intersection of rake surface, helix flute and clearance flank is defined as PN and also the connected point of the cutting edge and side cutting edge. It plays an important role in the design and grinding of the milling cutter. The first step of the research is to define the nominal relationship of the cutting tool as shown in Fig. 1. The correlation between PN and the adjoining cutter geometry has to be taken precisely into account, otherwise it results in inaccurate geometry for grinding due to the parameters interference mentioned by Wang et al. [11]. Therefore, PN is taken as the prerequisite key point for the following designing, grinding and measuring methods.Fig. 1 The defined characteristic points along the cutting edge of a precise form-milling cutterOne of the key terms of the precise milling is that form accuracy can be improved if the cutter point on the cutting path is kept as normal to the cutting surfaceas possible.If any cutting position of the cutter allows the workpiece to be quasi-normal in the 3-axis machining process, then some strategies need to be studied. The 4-th axis inclination has an important effect on the geometry of the cutter projection-profile so that the manner of 4-th axis is incorporated directly into the cutter design. For improving the cutter precision of 3-axis precise milling, it is attempted to impose both 4-th and 5-th axes’ movement degrees of freedom and the relative characteristics on the cutter’s geometric design.An inspection method verifies the geometric quality of the cutter with the measured characteristic points through reverse engineering. Additionally, the angular-solid ap-proach [11] for proving the relationships among the cutter’s design parameters is required. Then it has been realized that once the minimum of measured characteristic points have been defined, where the accuracy of the design parameters of N-segments-edge form the milling cutter, can be verified by the re-modeling process of reverse engineering.A new approach is developed in this work based on the specification of machining characteristics. In this research, some workpiece’s geometries are taken as practice machin-ing characteristics. A precise form milling cutter’s CAD solid geometry model is designed according to the geometrical characteristics as described above. The cutter geometric models are then constructed by the rapid prototype (RP) method called 3D printing. Finally, the accuracy is verified by the least amount of measured characteristic points satisfying the requirement and fitting in the relationship among design parameters. Hereby, a cutter with more complicated features is designed, based on the research so that the geometry’s reliability is advanced concretely and feasibly.Furthermore, for the total solution oriented objects, there must be an effective approach to develop a model for design, grinding and inspection.2 Normal algorithmBased on the machining characteristics of the workpiece, the major normal vector interval is defined, and a multi-segment cutting edge by the utilization of few characteristic points, such as a threshold point or control point in the interval, is accordingly designed. The characteristic point has to be always normal and regarded as the basis information for forming the cutting edge, according to the adoption of this study. However, the projection-profile of each adjoining characteristic point has to fit in with the condition of quasi-normal and becomes the major issue of this section.2.1 Classification of always normal2.1.1 Hermite curveFaux and Pratt [13] demonstrated the advantage of piecewise Hermite interpolation to easily get the first-derivative continuity without causing severe oscillation problems. They also mentioned the main disadvantage that the gradient values is hard to find. While in this paper, each characteristic point can actually define the normal condition on the 2D (two dimensional) projection-profile of the cutter and the tangent vector which is perpendicular to the normal vector.2.1.2 ArcThe arc, similar to Hermite curve, also has the first-derivative continuity, but it causes a severe oscillation problem. Its main advantage is suitable for defining the multi-arc edge milling cutter [14]. The normal vector is derived by linking the arc center to the target cutting point.2.2 General equations available for quasi normal2.2.1 Brief expressions of projection-profile and cutting edgeThe milling cutter can be formed by grinding or turning as a revolved-body [15]. Then, flutes, rake surfaces, clearance surfaces and the other surfaces of the cutter are ground sequentially. The continuous projection-profile of the cutting edge is divided into some segments as piecewise continuously in the next step - called N-segments edge. The definition of N-segments edge profile is shown in Fig. 2 - a 2D half projection-profile of cutter in XProf, YProf coordi-nate system (local coordinate system). The more N-seg-ments, the larger the database of cutter model, and the better the accuracy of the geometry.2.2.2 Quasi normal for the cutter geometry with N-segments edgeThe general design model can be derived from the relationship between the cutter design parameters and the projection-profile of N-segments edge. The fundamental geometry of the N-segments edge described in Fig. 2 can be illustrated in the form of Ferguson’s parametric cubic curves.Each segment of cutting edge can be expressed as [13, 16]: )(1)2)(032)(132)(0( 233trttrtrtrt ???????Fig. 2 Definition of the half projection-profile in the x-y profile coordinate system (local coordinate system). n: normal vector. Pprof i: characteristic point i on half projection profile3 The geometry characteristics of both workpiece and cutterTo choose or design a suitable cutter by both analyzing and surveying the geometric characteristic of the workpiece is much more beneficial for precisely milling the free-form surfaces.3.1 Definition of the geometrical characteristics between workpiece and cutterThere are some assessments and definitions obtained by the following procedures. By studying on the definition for the geometry characteristics of workpiece, a proper shape of cutting edge will be developed for promoting the precision in 3-axis milling up to the 4-axis or 5-axis milling has, and the cutting edge will be designed as properly piecewise division.For a case the 3-axis milling on free-form surfaces, Fig. 3a shows the existing method without the spindle tilt axis. Regarding the conditions in Fig. 3a, it is necessary to define as a milling characteristic for designing a proper cutter. For a cutter without the traditional concept, as shown in Fig. 3b, its projection-profile can be determined by scaling up or down from the workpiece’s projection-profiles, which are suitable in the finish milling.3.2 Main approach of designing the high-accuracy form milling cutter3.2.1 Definition of the characteristic points of cutter projection-profileThe sketch map in Fig. 4 shows the position on the cutting location of the 3-axis milling with normal vectors of workpiece and cutter. This is an improved type z-constant contour machining from the methods of Park [7, 8]. By this machining approach, a suitable projection-profile of cutter is regarded as the design goal.The basic position-calibration points including the cutter tip (coordinate origin) and the point PN, and the character-istic points on the projection-profile of the cutter, greatly influence the cutting efficiency. The normal vectors or tangent vectors are defined on the cutter’s characteristic points.Fig. 3 Schematic illustration of the 3-axis milling on the freeform surface: (a) the existing method; (b) the cutter projection-profiles designed based on the characteristics of workpiece geometry3.2.2 Projection-profile of cutter mapped by the workpiece’s characteristic pointsA significant goal of this research is dedicated to satisfying the requirements - the least characteristic points of cutter and the minimum workpiece’s geometry error. An available mathematical model is herein developed for finding the interval coordinate value between any two neighbouring characteristic points. And also be applicable for verifying the inspection stratagem.The characteristic points of the workpiece can be mapped to the 2D projection-profile of cutter. For always normal, any two adjacent characteristic points will be linked by a Hermite curve.3.2.3 Determination of tangent vector on the characteristic pointThe projection-profile of cutter is expressed as a piecewise continuous type of Hermit curves in the x-z 2D coordinatesystem. From the numerical method and the assumption of dx≈0, the 2D coordinate of the characteristic point on projection-profile has a general form of (x, z(x)). The point (x+dx, z(x+dx)) is near to the characteristic point at the designed Hermite curve. The derivative is set as x′=dx, . )((' xzdz???Therefore, the tangent vector on the i-th characteristic point is(2)LzxtzLxt iitiiii ??????2'''2''' )(,)(where, L is the average chord length among the three characteristic points),(),,(11??iiiii zzx,as given by??21212121 )()()(2 iiiiiiii zxx ??????）（Fig. 4 Modified z-constant contour machiningSubstituting Eq. (2) into the general Eq. (1) yields a simplified 2D projection-profile, and the tangent vectors of the two ends of a Hermite curve are obtained as .),1(,0),(11 iiiiiiiii zxzx?????As mentioned above, the equations of the N-segments Hermite projection-profile are necessary for precisely designing a quasi-normal form-milling cutter. Therefore, the definitions are extremely important in the processes of designing, grinding, inspection and tool path simulation.4 Design processes and computer-aided modeling of cutter geometryThe flow chart, as shown in the left part of Fig. 8, illustrates the continuous processes from the projection-profile defi-nition of the cutter to the NC path decision for grinding the clearance surfaces. It is less easy for searching the gradient values of 3D Hermite curves. The gradient values of the Hermite curves, including 3D cutting edge and NC grinding path, must be able to be calculated only by the 2D projection-profile definition. Therefore, a series of program-ming processes needs to be developed.4.1 Cutting edge geometryFor describing the accuracy of the cutting edge, the coordinate value needs to be obtained from any position of the cutting edge. From the given characteristic point Pi(xi, 0, zi) of the projection-profile, a relationship mapped on the cutting edge for developing characteristic point Qi(Xi, Yi, Zi) is expressed in the following equations:Fig. 5 Combination of angle-solid-block OPNO1Q and tool solid model for performing coordinate transformation into a inspection gestureFig. 6 Flow chart of the design processes and the geometry constructionBased on the developed equations of projection-profile and rake surface (Eq. (4)), it is possible to derive the Hermite equations of the cutting edge.A more close neighboring point approaching to Qi on the cutting edge is defined as Q′i(Q′ix, Q′iy, Q′iz), and the tangent vector of the i-th characteristic point is expressed as（11）???????LdZizi'iyiixi ,-YLdQ-X’，where dL is the length of and i is chord length.'i 1iQ??Accordingly, the coordinate values and tangent vectors, described by Eq. (11), of the two adjacent characteristic points are substituded into the general Eq. (1) of Hermite curve. The geometry equations of the piecewise continuous cutting edge can be derived exactly and piecewisely.4.2 Grinding path for cutting edge with quasi normalThe matrix form is suitable for representing the coordinate transformation. The relationships between different forms of coordinate systems XYZ (globe) and XgYgZg (grinding), need to be developed. The coordinate systems XgYgZg is the same as XMYMZM shown in Fig. 7. Thus, the key matrix M is the product of two coordinate transformation matrixes, RZ (?γS) and RX (?γ′).The transformation matrixes are useful in the jug-calibration process for grinding the clearance surfaces.The askew plane OPNQ on the angle-solid-block OO1PN Q, as shown in Fig. 6a, can be set for being coincident with orthogonal plane y=0 after the angle-solid-block OO1PNQ is rotated twice. The angle-solid-block ，as shown in Fig. 6b, is QO1??obtained by rotating the angle-solid-block an angle ?γs about Z-axis. The Nangle-solid-block an angle ?γ′about X-axis. In the Fig. 6c, the Y-coordinate QPON??1values of point are equal to zero. andFrom the above verification illustrated in Fig. 6, any point or vector (e.g., a characteristic point and tangent vector on cutting edge) on the rake surface can be transformed to locate on the plane y=0 according to Eq. (12). Therefore, a group of new 2D Hermite curves (y=0) for NC grinding paths of clearance surfaces can be successfully designed.4.3 Computer-aided modelingThe flow chart, as shown in the right part of Fig.6, shows the feature-based solid modeling processes, taking into the grinding sequence of the cutter. The main processes including the grinding sequence are described as follows.4.3.1 Parameters selectionThe cutter design parameters, as shown in a template of Fig. 7, are classified into external and internal parameters.1. External parameters for constructing N-segments cutting edgeA N-segments cutter, consisting of characteristic points (P0(x0, z0), P1(x1, z1), P2(x2, z2),,PN,(XN, ZN)) andnormal vectors and diameter D, is similar to the main projection-profile as shown in Fig. 2.2. Internal geometry parametersThe parameters are classified as design and grinding parameters (as shown in the local left part of Fig. 7) for constructing various features of cutter model.Fig. 7 Geometry parameters of a quasi-normal form-milling cutter with four characteristic points (N=3)4.3.2 Angle-solid-blocks for computer-aided modelingBecause the internal parameters cannot be straightly measured, the internal parameters for CAD modeling need to be obtained by using the angle-solid-blocks approach.1. Angle-solid-block of rake surfaceThe tetrahedron body OO″ PNQ is shown in Fig. 5. 2. Angle-solid-block of fluteThe cross section profile of the flute on plane z=?h is defined based on the rake surface. Figure 10a shows the angle-solid-block of flute. The grinding parameter τ (contain DD1PN angle DD1P2 of plane and) as a function of γ, LC,γs D, β, β0 can be determined from the angle-solid-block.3. Angle-solid-block of clearance surfaceThe angle-solid-block based on the point PN, as shown in Fig. 10b, presents the relationship between the clearance angle α0 (for computer-aided modeling) and the measured-clearance angle α.Fig. 8 Analysis diagram of angle-solid-blocks for creating primitives of the cutter model: (a) construction of the flute angle-solid-block based