横断面表面设计中CNC 机床插补的应用外文文献翻译、中英文翻译

A CNC machine tool interpolator for surfaces of cross-sectional designSotiris L. Omiroua,_, Andreas C. NearchouAbstractA machining strategy for milling a particular set of surfaces, obtained by the technique of cross-sectional design is proposed. Thesurfaces considered are formed by sliding a Bezier curve (profile curve) along another Bezier curve (trajectory curve). The curves arelocated in perpendicular planes. The method employs a three-axis CNC milling machine equipped with suitable ball-end cutter and isbased on the locus-tracing concept. 1. IntroductionIn the automobile, aerospace and appliances industry, a variety of functional or even aesthetic free-form surfaces are engaged by engineers and designers to achieve the desired performance of a product. The machining of such complex geometries is a basic problem in computer-aidedmanufacturing since the available NC machines are constrained, by their software, to linear and circular motions. In this paper we deal with a set of surfaces obtained with this design technique. More particularly we use Bezier curves to define the shapes of both the profile and the trajectory. Bezier curves as free-form curves are a powerful designing tool. They need only a few points to define a large number of shapes, hence their wide use in CAD systems. The principle for generating the considered surfaces is shown in Fig. 1. The curves are located in perpendicular planes. The upper end of the profile curve lies on the trajectory curve which is a plane contour. Fig. 2 shows a sample surface obtained by the above-mentioned technique. This paper, following the present intention of research engineers to take advantage of the hardware capabilities of modern CNC systems, proposes a real-time surface interpolator for machining the specified surfaces onFig. 1. Surface is generated by sliding the profile curve along thetrajectory curve.Fig. 2. Sample surface obtained by cross-sectional designvertical three-axis CNC milling machine. However we keep in mind that whenever feasible, three-axis milling procedures are often preferred due to considerations of cost. For the considered surfaces, inaccessibility issues are directly dependent upon the form of the profile curve. So by controlling the form of theaccuracy are the main advantages of this manufacturing method.Finally, accuracy is obtained by applying the locus-tracing concept for driving the tool along the Beziers offset. The concept is generally applicable in motion generation. In this paper, its application is illustrated in the context of motion generation along Beziers offset. Compared to the customary offset-modeling schemes, an additional advantage besides accuracy, is the fact that we avoid the complexity of using an exact analytic expression or a piecewise-analytic approximation for the offset. 2. Cross-sectional design with Bezier curvesMany commonly seen and useful surfaces are surfaces of cross-sectional design. For example a surface of revolution is produced under this technique. The surface is generated by revolving a given curve about an axis. The given curve is a profile curve while the axis is the axis of revolution. This paper deals with a more complex type of surface which is an extension to the surfaces of revolution. We still need aprofile curve that rotates about the axis of revolution, but the rotation is controlled by a trajectory curve. Now, the profile curve swings about the axis of revolution, guided by the trajectory curve. Both curves, profile and trajectory, are Bezier curves located in perpendicular planes. A Bezier curve of degree n is a polynomial interpolation curve defined by en t 1T points defining the Bezier control polygon. The interpolation basis functions used in Bezier interpolation are the Bernstein polynomials defined for degree n aswhere the binomial coefficients are given byThe parameter t is in the range 0,1 and there are n t 1 polynomials defined for each i from 0 to n. The Beziercurve is therefore defined over the interval 0,1 aswhere bi are the control points defining the Bezier polygon. A recursive algorithm defined by de-Casteljau 3,5,12, calculates for a given control polygon the point that lies on the Bezier curve for any value of t, and can be used to evaluate and draw the Bezier curve simply, without using the Bernstein polynomials. The algorithm advances by creating in each step a polygon of degree one less than the one created in the previous step until there is only one point left, which is the point on the curve. The polygon vertices for each step are defined by linear interpolation of two consecutive vertices of the polygon from the previous stepwith a value of t (the parameter):An interactive drawing tool based on the de-Casteljau algorithm, capable to design and manipulate Bezier curves supports the method proposed in this paper. Since the design process is very often iterative, the designer first lets the computer draw the Bezier curve defined by a given polygon. Next, checks whether the shape is acceptable (or optimal) based on various criteria, and, if necessary, adjusts the location and the number of the polygon vertices. The edit, add, move and delete operations of this drawing tool, presented in Figs. 3(a)(d), respectively, were used to achieve the desired form for a profile curve. Once the forms of the profile and the trajectory curve are definitively accepted, the coordinates of their control points are advanced to the input of the CNC surface interpolator, constituting part of the geometric information required.3. Offset tracing for a Bezier curveAn accurate machining of the considered surfaces requires accurate offset cutter paths along the trajectory and the profile curves. Since both of them are implemented in terms of Bezier curves our interest is focused on the motion generation along Beziers offset. The generation of an accurate motion along Beziers offset is treated as alocus-tracing problem. The formulation of the interpolation algorithm demonstrates the versatility and effectiveness of the locus-tracing concept in this practical case of machining. The algorithm guides the tool-center through repeated application of two analytically implementedconstruction operations, maintaining exact contact (within 1BLU1) along the entire path. In each iteration, the set of candidate steps is represented by the vector expressionassuming a unit of length equal to the step size. The number of possible steps in each point is 8 (Fig. 4). The last inequality excludes the combination of zero values for both dX, dY, which does not constitute a step. The optimal step is one, which maximizes the advance TidP (Fig. 5) along the local tangent Ti while, at the same time, it satisfies a criterion of proximity to the offset. Implementation of the proximity criterion requires the use of a proximity function which, in the neighborhood of Pi, provides a measure of closeness to the offset.A suitable proximity function is derived from the fixed distance property of the offset where d is the radius of the cutting tool.Notice that for P lying on the offset p 0, while pincreases absolutely as the distance of P from the offsetincreases. Since the choice of step is limited to thoseprescribed by Eq. (4), the fixed distance property cannot beapplied in a rigid manner. Rather, p is used as a proximity measure, from which a differentialcan be developed, giving the effect of each candidate step on the position error. To satisfy the proximity requirement, dP must point towards the offset locus. In algebraic terms, it must drive the value of p towards 0. Specifically, it should give Dpo0 if p40 and Dp40 if po0 or pDpo0.Thus, if Ti is the local tangent vector, step selection is formulated as a constrained optimization problem:maximize Ti dPwhere the sign o4 stands for X when po0 and for o0 when pX0. A more explicit formulation can be obtained by introducing Eqs. (4) and (5b)and an expression of the local tangent vector.For a parametric representation x uetT, y vetT,Ti u0,v0 the problem to be solved ismaximize u0 dX t v0 dYsubject to This is an integer programming problem in which the variables dX, dY take values from the set _1,0,1 with the added restriction that at least one must be non-zero. Once an optimal step dP is determined from (6), the normality condition is enforced by throwing a normal fromthe new point Pit1 Pi t dP to the Bezier curve, to locate the next point pi+1. This point is computed by solving the normality conditionfor p by Newtons method, using pi to start the iterations. For a parametric representation, the normality conditionis solved for t ti+1 to determine the new pointpi+1 u(ti+1), v(ti+1). The new t is obtained as the root of Eq. (7b), using Newtons iterative formula, which in this case takes the formThe normality condition (Eq. (7b) relates the coefficients of the unknowns dX, dY in the respective step selection problem (Eq. (6b) and may be interpreted as a constraint on the signs of these coefficients. Eq. (7b) implies that the coefficients u0, (Xi_ui), v0, (Yi_vi) inproblem (6b) cannot all be of the same sign. Furthermore, if the coefficients of dX (the first two) are of the same sign, the coefficients of dY must have opposite signs and vice versa. It has been shown in Ref. 13 that in bivariate integer programming problems possessing this structure, the optimal solution can be tabulated and may thus beobtained by a simple inspection of the coefficients. A flow chart of the proposed real-time Beziers offset interpolator is shown in Fig. 6.In INTIALIZATION and UPDATE steps, the de-Casteljau algorithm offers a quick solution for computing both, the point bi on the Bezier curve (Eq. (3) and the first and second derivative at the same time. The first and second derivative can be expressed in terms of the intermediate points bi+1, bi+2 and the point bi, all generated by the de-Casteljau algorithm:where n is the degree of the curve.3.1. Feedrate controlFeedrate f can be controlled by regulating the time delay associated with each step. Since successive cutter contact points are computed exactly, the required time delay for the ith step is4. Tool path planningA convenient tool center path, corresponding to one pass in the machining process, consists of a series of small arcs of prescribed length along the profiles offset, followed by offset motion along the entire length of the trajectory curve, until the end of the profile curve is reached(Fig. 7).Based on the real-time approach for motion generationalong Beziers offset (Section 3), the interpolation programgenerates the necessary steps for this motion, using as datainput the following:_ the coordinates of the Beziers control points definingthe profile curve in XZ-plane,_ the coordinates of the Beziers control points definingthe trajectory curve in XY-plane,_ the tool-radius,_ the step size,_ the distance between scallops (t),_ the federate.The programmed distance between scallops (t) is used to determine when to switch from motion generation along the profile curve to the motion generation along the trajectory curve. It must be noted that the control points of the trajectory curve define that curve only at the top plane.At this level the tool executes the first offset motion along the trajectory curve with offset distance the programmed tool-radius. However, as the tool advances along the profile curve, any movement of the tool along the X-axis, inevitably induces changes in the form of the trajectorycurve in the current plane section. It is evident that these trajectories in the following sections are offset curves of the initial trajectory curve but in different Z-levels (Fig. 8).5. Test resultsA representative example for machining a surface ofcross-sectional design is illustrated in Figs. Figs. 6and 7 show the selected forms for a profile and a trajectory curve, respectively. Both curves are defined asFig.6 Profile curve defined with eight control pointsFig. 7. Trajectory curve defined with 16 control points.Fig. 8. Generated tool paths for a surface of cross-sectional designBezier curves. The command block used in the part program is shown below:G62 P01 0; 0; 16; 11:5; 11; 21; 33; 33P02 0;_39:5;_22:5;_21;_45,_ 41;_31;_47P03 0;_3;_98;_110;_159;_196,_ 247;_298;_298;_247;_196,_ 159;_110;_98;_3; 0P04 0; 102; 102;_86;_86; 134; 52,122;_122;_52;_134; 86; 86;_102,_ 102; 0P05 4 P06 5 P07 100.The programming parameters are in mm and the BLU isset equal to 0.5 mm.6. Concluding remarksA manufacturing method for machining a particular set of surfaces, obtained by the technique of cross-sectional design is presented. The generatrix curves of these surfaces (profile and trajectory) are implemented in terms of Bezier curves, a powerful design tool widely used in computeraided design (CAD) systems. Simulation results have shown the effectiveness of the locus-tracing concept in generating the Beziers offset and its capability in automatic error control.References1 Koren Y. Computer control of manufacturing systems. New York:McGraw-Hill; 1983.2 Bezier P. Numerical control: mathematics and applications. NewYork: Wiley; 1972.3 Casteljau P de F. Shape mathematics and CAD. London: KoganPage; 1986.4 Choi BK. Surface modelling for CAD/CAM. Amsterdam: ElsevierScience; 1991.5 Farin G. Curves and surfaces for computer aided geometric design,4th ed. Boston: Academic Press; 1997.6 Papaioannou S, Omirou S. Motion generation as a locus tracingproblem. Proceedings of the Patras/Greece third internationalsymposium on advanced electromechanical motion systems, vol. II.1999. p. 10139.7 Omirou S. A CNC interpolation algorithm for boundary machining.Robot Comput Integr Manuf 2004;20(3):25564.横断面表面设计中CNC 机床插补的应用Sotiris L. Omiroua,_, Andreas C. Nearchou摘要在这里提到的是通过典型设计技术而确立的磨特殊表面的加工方法。这个提到的表面贝塞尔曲线 (轨道曲线) 滑一个贝塞尔曲线曲线 (描绘曲线) 被形成.曲线是在垂直的飞机中位于。 方法雇用被装备适当的球-结束的裁剪者的一个三轴的 CNC 铣床而且是基于追踪场所的观念。 1. 介绍在汽车、航空宇宙和器械业中，多种功能或甚至美学自由形态的表面被达成产品的被需要的表现工程师和设计者预订。 如此复杂几何学的机制在计算机中是一个基本的问题-援助自从可得的收据控制机器之后制造被强迫, 藉着他们的软件,对线性和圆形的运动 . 在这纸中，我们处理与这设计技术一起获得的一系列表面。 更特别，我们使用贝塞尔曲线曲线定义描绘和轨道的形状。 贝塞尔曲线曲线当做自由形态的曲线是一有力的设计工具。 他们需要只有一些点定义很多的形状, 因此在 CAD 系统的他们的宽使用。 产生考虑过的表面的原则在图 1 被显示。 曲线位于垂直的飞机。 在一个平的等高线的轨道曲线上的描绘曲线谎言的上端。 被上述技术获得的图 2 表演一个样品表面。 这纸,跟随研究工程师的现在意图利用现代 CNC 系统的硬件能力, 为机制计画一个即时的表面内插器指定的表面在 图 1. 表面藉由向前滑描绘曲线被产生那轨道弯。图 2. 被代表性设计获得的样品表面垂直的三轴 CNC 铣床。 每当由于费用的考量能实行又三轴的磨程序时常是优先的，然而我们记住。 对于考虑过的表面，难接近议题在描绘曲线的形式之上直接依赖。 因此藉由控制 theaccuracy 的形式是这制造业的 method.Finally 的主要利益, 准确性藉由应用沿着贝塞尔曲线的抵销驾驶工具的追踪场所的观念被获得。 观念在运动世代中通常可适用。 在这纸中，它的申请被在运动世代的上下文沿着贝塞尔曲线的抵销举例。 被习惯的模型抵销的方案,除了准确性以外的一个另外的利益, 相较的是事实我们避免使用精确的分析表达的复杂或一分段地-为抵销的分析近似值。 2. 代表性的设计用贝塞尔曲线弯普遍见到的多数和有用的表面是代表性设计的表面。 举例来说，一个革命的表面在这技术之下被生产。 表面被产生被回转的关于一个轴的一个给定的曲线。 当轴是革命的轴的时候，给定的曲线是一个描绘曲线。 这纸处理对革命的表面的延长的一个更多的复杂类型的表面。 我们仍然需要一描绘有关革命的轴替换的曲线轮廓，但是旋转被一个轨道曲线控制。 现在, 被轨道曲线指导的关于革命的轴的描绘曲线摇摆。 曲线、描绘和轨道, 是贝塞尔曲线在垂直的飞机中位于的曲线。 一个程度 n 的贝塞尔曲线曲线是被按 t 1T 点定义贝塞尔曲线控制多角形被定义的一个多名的窜改曲线。 窜改基础被用于贝塞尔曲线窜改的功能是伯恩斯坦程度 n 定义的多项式当做哪里二项的系数有被 叁数 t 在范围 0,1 中，而且有从 0 为每 i 被定义到 n 的 n t 1 多项式。 贝塞尔曲线 曲线因此被定义在间隔 0,1 之上当做 在 bi 是控制点定义贝塞尔曲线多角形的地方。 一个回归的运算法则根据 de-Casteljau 3,5,12 定义,为给定的控制多角形计算为 t 的任何价值在贝塞尔曲线曲线上, 而且能用来只是评估并且拉贝塞尔曲线曲线的点, 不使用伯恩斯坦多项式。 运算法则藉由在每个步骤一个程度的多角形中创造前进一较少的超过那一在早先的步骤产生直到剩下只有一点, 哪一个在曲线上是重点。 给每个步骤的多角形顶点被来自早先的步骤的多角形的二个连续顶点的线窜改定义藉由 t(叁数) 的价值: 基于 de-Casteljau 的运算法则的一个交谈式图画工具, 有能力的设计而且操纵贝塞尔曲线曲线支援在这纸中被计划的方法。 因为设计程序是时常反复的，设计者首先让计算机拉被一个给定多角形定义的贝塞尔曲线曲线。 下一个, 检查是否形状是可接受的 (或者最佳的) 基于各种不同的标准,如果必需的, 调整多角形顶点的位置和数字。 编辑,增加, 移动而且划除这个图画工具的操作,在无花果树呈现。 3(一)-(d)，分别地，用来为描绘达成被需要的形式弯。一经描绘的表格和轨道曲线决定性地被接受，他们的控制观点的坐标被前进到被 CNC 表面输入的内插器, 构成几何学数据的一部份必需的。3. 抵销追踪为贝塞尔曲线曲线考虑过表面的一个正确机制沿着轨道和描绘曲线需要正确的抵销裁剪者路径。 自从他们两个都之后根据贝塞尔曲线被实现弯我们的兴趣沿着贝塞尔曲线的抵销把重心集中在运动世代。 正确运动的世代沿着贝塞尔曲线的抵销被当做一 追踪场所的问题。 窜改运算法则的形成示范机制的这个实际情形的多种变化和追踪场所观念的效力。 运算法则指导工具-中央的经过二个的重复申请分析地实现工程行动, 沿着整个的路径维持精确的连络。 在每个重复中, 候选人步骤的组是 根据矢量表达表现 假定一单位长度对步骤大小等于。 每点的可能步骤的数字是 8(图 4) 。 最后一个不平等为两者的 dX ， dY,不构成一个步骤排除零价值的组合。 最佳的步骤是一, 沿着当地的接触 Ti 取进步 TidP(图 5) 最大值当,的时候同时，它使对抵销的一个接近的标准满意。 接近标准的落实需要，在 Pi 的邻近地区中，提供对接近的衡量给抵销的接近功能的使用。一个适当的接近功能起源于固定人 抵销的距离特性 在 d 是刀具的半径的地方。为躺在抵销 p 0 上的 P 注意那, 当 p的时候增加完全地当做来自抵销的 P 的距离增加。 自从步骤的选择之后被限制于那些根据情绪商数规定。 (4), 固定的距离特性不能够是以硬的样子应用。 然而， p 被当作接近尺寸使用, 从哪一个一差别的 能被发展,给每位候选人步骤对位置错误的效果。 为了要使接近需求满意， dP 一定向抵销场所指出。 以代数的角度, 它一定驾驶 p 的价值向 0. 明确地，它应该给 Dpo0 如果 p 40 和 Dp40 如果 po 0 或 pDpo 0.因此，如果 Ti 是当地的接触矢量，步骤选择当做一个强迫的最佳化问题被制定:取 Ti dP 最大值 哪里告示 o 4 代表 X 当 po 0 和为 o 0 当 pX 0. 一个比较明白的形成能藉由介绍情绪商数被获得。 (4) 而且 (5 b) 和当地接触的矢量的表达。对于参数的表现 x uetT, y vetT,Tiu 0,v 0 要解决的问题是取 u 0 dX t v 0 dY 最大值服从的到这是一个完整的事物规画问题在哪一个变数 dX, dY 采取来自组的价值 _1,0,1 用附加的限制哪一至少一一定是非零。 一经最佳的步骤 dP 被决定从 (6), 常态情况藉由丢常态被运行从对贝塞尔曲线曲线的新点深坑 1 Pi t dP, 位于下点 pi+1. 这点藉由解决常态情况被计算 因为牛顿的方法的 p, 使用 pi 开始重复。 对于参数的表现, 常态情况 被解决让 t ti+1 决定新的点pi+1u(ti+1),v(ti+1). 新的 t 当做情绪商数的根被获得。 (7 b), 使用牛顿的反复公式,在这情况采取形式 常态为条件 (情绪商数。 (7 b) 和未知者 dX 的系数有关联, 分别的步骤选择问题的 dY(情绪商数。 (6 b) 而且可能当做一个限制被解释在这些系数的告示上。 情绪商数。 (7 b) 暗示那系数 u 0,(Xi_ui), v 0,(Yi_vi) 在问题 (6 b) 不能够全部是有相同的告示。 此外，如果 dX(第一个二) 的系数是有相同的告示， dY 的系数一定有相反的告示和反之亦然。 它已经在裁判员被显示。 13 哪一在持有这结构的双变量的完整事物规画问题，最