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黄河科技学院毕业设计(论文)文献翻译 第 18 页 非正交主轴与工作台型五轴工具机后处理程序开发黄昭堂.佘振华摘要：后处理程序是将刀具位置数据转换成加工操作所需数据的重要接口，其对五轴工具机来说是非常复杂的，因为在五轴工具机中线性轴和旋转轴是同动的。以前大部分的五轴后处理方法研究只局限于正交的工具机构型，本论文针对主轴型与工作台型及工作台/主轴型有非正交旋转轴的五轴工具机开发其后处理算法，这种构型的工具机具有从立式加工转换为卧式加工的优点。本文以齐次坐标转换为基础，利用运动学的前向转换，求得五轴工具机的形状创成函数矩阵，再由逆向转换，解出工具机各轴运动的解析方程。后处理程序中的线性算法是为了保证加工的精确性而开发的。五轴后处理程序接口是利用Borland C+、Builder与OpenGL开发，以产生三种构型的NC码，经由商业实体切削仿真软件VERICUT验证及试加工实验，证实所提出的后处理方法论的可行性。关键词：后处理、五轴加工、形状创成函数、非正交旋转轴1、引言五轴工具机被越来越多地的用户所使用的，特别是用于加工复杂自由曲面。传统的五轴工具机有三个正交的线性轴和旋转轴。这里所说的旋转轴通常是指与相互正交的中心线平行的线性轴。各国的机械工具制造商，如Makino，Ingersol和Deckel Maho，将非正交旋转轴或工作台进行改进使机器具有更好的多功能性和灵活性。“非正交”是指轴旋转体的振荡运动，这类似与一张桌子上的硬币的缓慢旋转。五轴工具机有一个旋转轴的倾斜面1，而不同于平行的直线轴，它提供的优势可使切削刀具在一个半球内指向任意角度2，3。这种机器可以在连续的水平和垂直位置移动。非正交旋转轴为生产航空部件及汽车头部提供了便利。运动经电机主轴传递给空心轴和齿轮4。由于线性和旋转运动同时作用在五轴数控机床上，导致了五轴数控程序比三轴数控程序更加的复杂。后处理程序必须利用刀具位置（CL）将数据从凸轮系统转化为机器控制数据。尽管先进的控制器可以接受实时的数据，而不需要后处理，但他们是相当昂贵的5。该方法主要可以分为三类：图形 6，7 和坐标数值迭代8-10。由坐标变换方法解析方程，产生的数控数据最有效，它已被广泛采用在最近的研究中。然而，几乎所有的这些方法包括后处理方法均采用正交旋转轴五轴工具机。研究解决非正交配置的相对较少。例如，有为主轴倾斜式发展的非正交旋转轴五轴机床后处理程式11。最近，Sorby 12发表了一篇关于封闭形式五轴工具机的非正交旋转工作台论文。然而，该解决方案具有一定的局限性。例如，工件原点的偏移向量和二次主旋转不明确，及角度倾斜45度的非正交轴的固定。本研究开发一种后置的双主轴和工作台五轴工具机。基于齐次坐标变换矩阵的解析方程，确定方程的一般形式；偏移向量定义为从工件的起始位置回转至工作台，偏移向量在非正交轴中是可变的。此外还包括线性化算法的后处理开发，保证加工精度。一个基于后处理是开发和图形界面动态显示的表面模型议案的提出帮助用户输入相关参数正确。此外，生成的NC数据进行验证，使用商业实体切削仿真软件VERICUT 13进行五轴加工实验工具机的非正交旋转工作台的后处理方法确认。2、五轴工具机的配置与类型大多数五轴工具机有两个旋转轴作为常规X轴，X轴和Z轴。五轴机床可分为三种类型：主轴型，工作台型和工作台/主轴型。商业方面用正交配置，如图1所示三种类型。图1（a）为非正交旋转主轴型。图1（b）为非正交旋转工作台型，如Deckel Maho DMU 70改进型 15，其在工作台上具有两个旋转轴，和一个平行与Z轴而与非正交旋转轴存在一定的倾斜角度的旋转轴（C轴）。图1（c）为工作台主轴型，如Deckel Maho 200P 15，其中一个旋转工作台（c）是以在工作台上的非正交旋转轴（B轴）为主轴。由于作者已经提出过主轴型非正交旋转主轴的后处理程序，本研究着重于发展与其他两种配置的后处理。五轴机床可以看作是平动与旋转运动组合的机床。正向运动学方程必须建立数学模型来描述刀具相对于工件的切削运动。基本的坐标变换矩阵，包括Trans矩阵和Rot矩阵 16 。Trans矩阵式可以表示如下：Trans（a，b，c）表示矢量a i+b j+c k一般Rot矩阵用来描述旋转的主轴单元。本坐标系设定；则Rot矩阵可以表示为： 其“C”和“S” 分别为余弦和正弦函数，且 图1五轴工具类型：a.主轴型 b.工作台型 c.工作台主轴型3、后处理程序3.1工作台倾斜型图2描绘了相关的坐标系配置。工件坐标系为而为刀具坐标系。由于这两个旋转轴并不相交，则必存在一条公法线垂直于两轴。公法线分别与C轴和B轴相交于RC和RB点。偏移向量为从原点至RC，而偏移向量为从RC至RB。 图2倾斜型坐标系组成机床的结构有：回转工作台C，回转工作台B，机床床身， X轴方向工作台，Y轴方向工作台，Z轴方向工作台，主轴和刀具。根据刀具与工件的相对位置和方向，将从工件开始至刀具完成的过程称为形式塑造功能，17。这种机床的形式塑造过程，用数学矩阵形式表示如下： 其中Px，Py和Pz分别表示X，Y和Z轴的相对距离。和分别为与C轴和B轴的旋转角度。采用右手螺旋定则判定+C和+B。方程（3）表示的函数矩阵，结合机床参数Px，Py，Px，和。第一步是计算刀具所需的旋转角度，二是根据已知的旋转角中心位置的直线计算所需的位置关系。当刀具位置和刀具的方向向量确定后，CL数据可用矩阵形式表示如下：由于方程（3）和（4）都表示相同的刀具和工件之间的关系，联立这两个矩阵，确定所需的参数。结合两个矩阵得到下列公式： 首先可以确定，的值。代入式（5）得：值得注意的是，在范围内的表达方式如下：如果范围在到0之间，方程应修改为式（8）所示。另一方面，如果同时满足以上两种情况，则以最小的旋转角选择算法。此外，将式（5）对应的第一值第二值联立求解线性方程组得到： 由于方程（9）和（10）分母是相同的，总是正的，C轴转角可以确定如下： 其中arctan2（y，x）是在范围内的函数返回值，表示y和x的夹角16。此外，结合矩阵（6）式两边的相应参数，产生三个未知数Px，Py和Pz。联立方程组，设定程序坐标系为工件坐标系。因此，可以得到所需的NC数据（记为x，y和z），考虑两个偏移向量和，并表示为如下： 3.2工作台/主轴倾斜型工作台/主轴倾斜型有一个旋转主轴和一个旋转轴的工作台。图3分别显示了C轴和B轴上的两个交点RC和RB。交点RC位于C轴上任意点，交点B为非正交旋转B轴和刀具的交点。偏移向量是按从原点到交点RC，有效刀具长度代表交点RB和刀尖中心之间的距离，。由其造型函数矩阵可以得到坐标变换矩阵如下：图 图3工作台/主轴倾斜型坐标系统 等值式（14）式（15）联立得： 结合参数，可以采用同工作台倾斜型的计算过程。但要注意的是， NC参考点，在此假设为交点RB。这个定义是根据主轴倾斜和工作台/主轴倾斜型得来，而且使用的是相同的商业后处理器程序的软件包。完整的NC数据的分析方程可以表示为： 3.3线性问题从理论上讲， CAD / CAM系统生成的CL数据是以假设刀具在连续两个点之间的线性移动为基础。然而，实际的刀具与工件的运动轨迹并不是直线和旋转轴移动同时进行。弯曲的路径偏离线性插值的连续路径点之间的直线路径被称为线性问题。以下算法可以解决这个问题。假设，在图4中 为三个相邻的CL数据点。矢量Pn的矩阵形式可表示为，其中和组成刀尖的中心位置，和组成刀具的方向。 相应的机床数控代码Pn为。由于五轴同时从当前位置Pn移动到随后的位置的Pn+1，每个轴之间移动假定为线性的18。因此，实际的曲线路径的每个点可以表示如下：其中t是一个虚拟的时间坐标。其中CL数据和为正值。例如，工作台倾斜型的式（5）、（6）和工作台/主轴倾斜型的式（16）、（17）。此外，在理想的线性刀具路径下每个点可以决定如下：理想的线性刀具路径实际曲面刀具路径内插刀具路径图4多轴加工线性问题图5后处理程式对话框：a工作台倾斜型 b工作台/主轴倾斜型图6工作台倾斜型生成NC数据对话框之间的距离为偏差。如果最大偏差超过规定的公差，应将插入到原CL数据。理论上，必须采取数值迭代方法计算。实际上，中间点，t=0.5，常被选为候选点10。将中间点插入后，即可以生成相应的数控代码。4、讨论1非正交旋转轴的主要特征是在同一台机床上水平位置和垂直位置之间的连续运动。在当前商业工具机的配置中，可以由以上方程得出非正交旋转轴倾斜45度。可以拿工作台型倾斜是用来作为一个例子，方程（5）表示刀具相对于工件的方向。在初始位置，工作台水平，可以确定。非正交旋转轴假定绕x轴旋转角使矢量及。将以上条件代入式（5）且，产生了如下方程：解得，。因此，当工作台转动角度，非正交旋转轴B轴转动/4时工作台的处于垂直位置。 图7工作台/主轴倾斜型生成NC数据对话框2、非正交坐标系的采用提高了五轴工具机床灵活性。然而，在CL数据方面是有限制的。只有在方程（7）显示的条件满足时方能使用。当非正交轴设置在45度角时，的取值范围在。所以，为负值时，通过CAD / CAM软件生成的CL数据无法进行加工。3、生成的NC数据是一个普遍的形式，它可以运用到正交配置中。工作台倾斜型就是一个例子。如果向量W是在X轴方向，且Wx =1 Wy=Wz= 0就是CA工作台倾斜性的配置。分析方程中的NC数据，例如Y轴的值，与文献8中的一致，可以表示如下：注意，在所列举的例子中，假设两个旋转轴相交且偏移向量用于推导上述方程。4、基于和，刀具解可能通过，且12, 18未知的点。该点发生在且C轴平行于刀具轴时。正如在图4所示，如果当Pn+1是该点时， 在理论上可以是任意的值，因为Pn+2是未知的。Pn+2应进一步确定，以确保的值是在该连续两个点之间的线性变化。 的值可定义为Pn到Pn+2之间的距离。5、在实际的多轴加工中进给速度控制是一个重要的问题。大多数控制器，如FANUC公司和Cincinnati Milacron公司采用字符（FRN）和G93代码来控制进给速度。FRN由工件的进给率的所决定。当两个或两个以上线性轴旋转运动时，路径长度的确定变得非常复杂。在大多数情况下，实际的路径长度可以充分接理论的线性位移19。5执行和核查5.1软件实现在Windows XP环境中使用BorlandC + 、Builder编程语言和OpenGL图形库。采用一个半径为35mm、的半球进行加工说明。 CL数据通过商业CAD / CAM软件与PowerMILL20产生。机床采用工作台倾斜型与工作台/主轴倾斜型的二种形式的工具机，进行了测试。图5（a）所示工作台倾斜型配置后处理器开发软件对话框。用户可以用鼠标的旋转放大机床表面模型。当用户输入相关参数，如偏移向量从C轴中心点开始时，系统会显示数字，以帮助用户输入正确的参数，如图6所示。最后，点击“文件”菜单打开CL数据，生成NC代码。图5（b）和图7显示的是工作台/主轴倾斜型启动和实施环节的对话框，。值得注意的是，设值长度是从压刀尖中心到工作台表面。5.2实体切削仿真实体切削仿真软件VERICUT是用来生成数控加工数据。软件中有可供选择的原材料，刀具的规格尺寸，数控数据，控制器的类型，及物理性能不同的数控加工工具，它可以用数控数据来模拟材料去除过程。工作台倾斜型工具机用产品仿真和成品加工进行验证，如图8所示。相关参数如图6所示。图8 工作台倾斜型的VERICUT软件模拟图9工作台/主轴倾斜类型的VERICUT软件模拟图9所示工作台/主轴倾斜类型的VERICUT软件模拟。如前所述，根据图7，应设置相关参数。B轴的向量为。偏移向量从程序原点到旋转刀具轴。5.3实验验证 生成的五轴联动数控数据要进一步验证。工作台倾斜型五轴加工中心（DECKEL MAHO DMU70改进型）配备Heidenhain iTNC530用于半球形工件加工。这项实验是在下列条件下进行：（1）两个球头直径为10毫米和4毫米的刀具分别用于粗加工和精加工（2）主轴转速5000rmin,进给速度为1000mm/min（3）工作台采用7075铝合金材料制造。应该注意的是，本机床C轴的正方向是刀具沿着Z轴的负方向。C轴的实际数控数值再式（11）中为负值。图10显示了实际的加工过程，揭示正确的后处理程式，能成功生成NC数据。图10 DECKEL MAHO DMU1070改进型机床的实际加工实验 a.粗加工 b.精加工六、结论 非正交工作台和主轴型五轴工具机床的后处理程序有了一定的发展。一般的NC数据是由齐次坐标变换矩阵，正向和逆向运动学的分析来确定的。生成的NC数据对那些旋转轴需要相互交叉和非正交轴的倾斜角度为变量的这类机床是有用的。产生的可变倾斜角能增加派生方程的有效性，从而NC数据可降低正交型的配置。该种算法也可以应用到线性轴和旋转轴非正交的多功能磨/转机床中21，目前这项工作正在进行。致谢 对中华人民共和国理事会NSC95-2221-E-150-101的财政资助深表感谢。同时也对金属工业研究发展中心提供五轴设备，及对在台湾Delcam公司的Bacchus Yu先生提出的有效建议意见表示感谢。参考文献1.Goode KF, Rockford IL (1983) Tool head having nutating spindle,US Patent No. 43700802.Makino (2003) High-Productivity Aerospace Machining Center.MMS Online, URL: /equipment/mcen389.html3. Ingersol Mastercenter (2007), URL: /ind/mastercenterH.htm4. Hagiz G (2006) 5-axis machining, URL: /cnc/5axes.htm5.Affouard A, Duc E, Lartigue C, Langeron J-M, Bourdet P (2004)Avoiding 5-axis singularities using tool path deformation. InternationalJournal of Machine Tools and Manufacture 44:4154256. Fauvel OR, Vaidyanathan J, Norrie DH (1990) An analysis oflinearization errors in multi-axis APT-based programming systems.Journal of Manufacturing Systems 9(4):3533627. Nagasaka M, Takeuchi Y (1996) Generalized post-processor for5-axis control machining based on form shape function. Journalof the Japan Society for Precision Engineering 62(11):160716118. Lee RS, She CH (1997) Developing a postprocessor for threetypes of five-axis machine tools. International Journal of AdvancedManufacturing Technology 13(9):6586659. She CH, Lee RS (2000) A postprocessor based on the kinematicsmodel for general five-axis machine tools. SME Journal ofManufacturing Processes 2(2):13114110. Chou HL (1989) Development of an APT universal postprocessorfor multi-axis CNC milling machine tools. Masters thesis, NorthCarolina State University, USA11. She CH, Chang CC (2007) Development of a five-axis postprocessorsystem with a nutating head. Journal of MaterialsProcessing Technology 187188:606412. Sorby K (2007) Inverse kinematics of five-axis machines nearsingular configurations. International Journal of Machine Toolsand Manufacture 47(2):29930613. VERICUT V5.3 User Manual, URL: 14. Sakamoto S, Inasaki I (1993) Analysis of generating motion forfive-axis machining centers. Transactions of the Japan Society ofMechanical Engineers, Series C. 59(561):1553155915. Deckel Maho, URL: 16. Paul RP (1981) Robot Manipulators: Mathematics, Programmingand Control. MIT press, Cambridge, MA17. Reshetov DN, Portman VT (1988) Accuracy of Machine Tools.ASME press, New York18. Bohez E, Makhanov SS, Sonthipermpoon K (2000) Adaptivenonlinear tool path optimization for five-axis machining. InternationalJournal of Production Research 38(17):4329434319. Cincinnati Milacron (1994) Programming Manual for CincinnatiMilacron Acramatic 950MC Rel 3.0 Computer NumericalControl. Ohio20. Delcam, URL: 21. OKUMA MULTUS, URL: http:/www.okuma.co.jp/english/ORIGINAL ARTICLEPostprocessor development of a five-axis machine toolwith nutating head and table configurationChen-Hua She&Zhao-Tang HuangReceived: 21 August 2006 /Accepted: 7 June 2007#Springer-Verlag London Limited 2007Abstract The postprocessor is an important interface thattransforms cutter location data into machine control data, andin a five-axis machine tool is highly complex because thesimultaneous linear and rotary motions occur. Since mostworks of the five-axis postprocessor method have dealt onlywith the orthogonal machine tools configuration, this studypresents a postprocessor scheme for two types of five-axismachine tools, each with a nutating head and a table whoserotational axes are in an inclined plane. The benefit of such aconfiguration is that it allows switching from vertical tohorizontal machining by a single machine. The generalanalytical equations of NC data are obtained from the forwardand inverse kinematics and the homogeneous coordinatetransformation matrix. The linearization algorithm for thepostprocessor is developed to ensure the machining accuracy.Thepresentedalgorithmisimplementedusingawindow-basedfive-axis postprocessor with nutating axes, and programmed inBorland C+ Builder and OpenGL. A simulation is performedusing solid cutting software and a trial-cut experiment wasconducted on a five-axis machine tool with a nutating table toelucidate the accuracy of the proposed scheme.Keywords Postprocessor.Five-axismachining.Form-shapingfunction.Nutatingaxis1 IntroductionFive-axis machining is becoming increasingly used bymachine tool users, especially in machining complexfreeform surfaces. The conventional five-axis machine toolhas three orthogonal linear axes and two rotary axes. Therotary axes are typically orthogonal to each other and thecentre line of the rotary axis is parallel to the direction ofthe linear axis. Various machine tool builders, such asMakino, Ingersol and Deckel Maho, incorporate a nutatinghead or a nutating table in the machine tools to improvetheir versatility and flexibility. The word of “nutating”means oscillatory motion about the axis of a rotating body,which is similar to the slow spinning of a coin on a table. Afive-axis machine tool with a nutating unit has a rotationalaxis in an inclined plane 1, and not parallel to the linearaxis, providing the advantage that allows the cutting tool toorient itself toward any angle within a hemisphere 2, 3.Such machines can move continuously between thehorizontal and vertical positions in a single setup on thesame machine. The nutating head provides very useful inmanufacturing aerospace parts because it has no motors onthe head, and is more rigid. The motors for the spindle areon the machine and the motion is transferred to them byhollow shafts and gears 4.Because the linear and rotary axes move simultaneouslyon a five-axis machine, the derivation of the five-axisprogram is more complex than that of the three-axisprogram. A postprocessor must be utilized to translate thecutter location (CL) data from the CAM system into themachine control data. Although the advanced controllerscan accept the CL data to machine the workpiece in real-time without the need of postprocessor 5, they arerelatively expensive and not commonly used in mostindustries. The methods of developing multi-axis postpro-Int J Adv Manuf TechnolDOI 10.1007/s00170-007-1126-5C.-H. She (*)Department of Mechanical and Computer Aided Engineering,National Formosa University,64 Wen-Hua Road, Huwei,Yunlin 632, Taiwan, Republic of Chinae-mail: chshe.twZ.-T. HuangDepartment of Mechanical and Automation Engineering,Da-Yeh University,112 Shan-Jiau Road, Da-Tsuen,Chang-Hua 515, Taiwan, Republic of Chinacessors can be mainly divided into three categories -graphical 6, numerical iterative 7 and coordinatetransformational 810. Since the coordinate transforma-tion method yields the analytical equation of NC data mostefficiently, it has been adopted extensively in recent work.However, almost all of these approaches involve postpro-cessor methods for five-axis machine tools with orthogonalrotary axes. Relatively few studies have addressed non-orthogonal configuration. For example, the authors havedeveloped the postprocessor for the spindle-tilting typefive-axis machine tool with a nutating head 11. Recently,Sorby 12 has presented a closed-form solution for a table-tilting type five-axis machine tool with a nutating table.However, this solution exhibits some limitations. Forexample, the offset vectors such as from the workpieceorigin to the rotary table and from the secondary rotary tothe primary rotary are not defined, and the angle ofinclination of the nonorthogonal axis is fixed at 45 degrees.This study develops a postprocessor for two five-axismachine tools each with a nutating head and table config-uration. Based on the homogeneous coordinate transforma-tion matrix, the general analytical equations of NC data areobtained from the forward/inverse kinematics and themachine tools form-shaping function matrix. The deter-mined equation is in general form because the rotary axes areassumed not to intersect each other; the angle of inclinationof the nonorthogonal axis is variable, and the offset vectorfrom the origin of the workpiece to the rotary table isdefined. Moreover, the linearization algorithm of the post-processor is developed to ensure the machining accuracy.A window-based postprocessor is developed and a graph-ical interface that dynamically displays the surface model andthemotions ofallofthe axesoftheconfiguredmachinetoolispresented to help the user to input relevant parameterscorrectly. Additionally, the generated NC data are verifiedusingthecommercialsolidcuttingsoftwareVERICUT 13and a machining experiment is conducted on a five-axismachine tool with a nutating table to confirm the effective-ness of the proposed postprocessor methodology.2 Configuration and modeling of five-axis machine toolMost five-axis machine tools have two rotary axes as wellas the conventional X, Y and Z axes. Following Sakamotoand Inasaki 14, the configurations of five-axis machinetools can be categorized into three types: spindle-tilting,table-tilting and table/spindle-tilting. Commercial machinetools with the nonorthogonal configuration, as shown inFig. 1, are also of three types. Figure 1 (a) shows thespindle-tilting type with a nutating head, such as theMakino MAG3 2, which is designed with a rotary axis(C-axis) behind a nutating head that rotates about the B-axis. Figure 1 (b) displays the table-tilting type with anutating table, such as the Deckel Maho DMU 70 eVolution15, which has two rotary axes on the table, and one rotaryaxis (C-axis) is parallel to the Z-axis while the non-orthogonal rotary axis is inclined at an angle to the C-axis.Figure 1 (c) presents the table/spindle-tilting type with anutating head, such as the Deckel Maho 200P 15,inwhichone rotary table (C-axis) is on the table and the nutatingrotary head (B-axis) is on the spindle. Since the authors havealready presented the spindle-tilting postprocessor with anutating head 11, this study focuses on developing thepostprocessors with the other two configurations.A five-axis machine tools can be regarded as amechanism with serially connected links with revolute orprismatic joints. Forward kinematic equations must beestablished to describe mathematically the motion of thecutting tool in relation to the workpiece. The fundamentalcoordinate transformation matrices, including the transla-tion matrix Trans and the rotation matrix Rot 16, areintroduced. The translation matrix Trans can be expressedas follows:Transa;b;c 100a010b001c0001266437751where Trans(a, b, c) implies a translation given by thevector a i+b j+c k.The general rotation transformation matrix should beused to describe the rotation of the nutating unit. Thecoordinate system is assumed to rotate through an angle offwaround any arbitrary vector W=Wxi+Wyj+Wzk; therotational transformation matrix can be expressed as:Rot W;w W2xVw CwWxWyVw? WzSwWxWzVw WySw0WxWyVw WzSwW2yVw CwWyWzVw? WxSw0WxWzVw? WySwWyWzVw WxSwW2zVw Cw00001266437752Int J Adv Manuf Technolwhere “C” and “S” are cosine and sine functions,respectively, and Vfw=1Cfw.3 Postprocessor3.1 Table-tilting type with a nutating tableFigure 2 depicts relevant coordinate systems for thisconfiguration. The coordinate system for the workpiece isOwXwYwZwwhile the system OtXtYtZtis attached to thecutting tool. Since the two rotary axes are assumed not tointersect each other, a common normal line is mutuallyperpendicular to both axes. The common normal line inter-sects with the C-axis and B-axis at two points, RC and RB.The offset vector Lxi+Lyj + Lzk is determined from theorigin Owto the pivot point RC, whereas the offset vectorMxi+Myj+Mzk is calculated from the pivot point RC tothe pivot point RB.Since the structural elements of the machine toolcomprise the C rotary table, the B nutating rotary table,the machine bed, the X linear table, the Y linear table, the Zlinear table, the spindle head and the cutting tool. Therelative position and orientation of the cutting tool withrespect to the workpiece can be determined sequentiallystarting from the workpiece and ending at the cutting tooland is referred to as the form-shaping function 17. TheCBXYZaXYZCBcBCYXZbFig. 1 Configuration for five-axis machine tool with nutating head andtable. a spindle-tilting type with a nutating head. b table-tilting typewith a nutating table. c table/spindle-tilting type with a nutating headtOtXtYtZBRBwOwXwYwZC xyzLLL+ijkRCOffset vectorxyzMMM+ijkOffset vectorFig. 2 Coordinate systems of table-tilting type configurationInt J Adv Manuf Technolform-shaping function of this machine tool can bemathematically expressed in matrix form as follows:Trans Lx;Ly;Lz?Rot z;?zTrans Mx;My;Mz?Rot W;?wTrans Px;Py;Pz?0 00 01 00 1266437753wherePx, Pyand Pzdenote the relative translation distances ofthe X, Y and Z linear tables, respectively. The termsfzandfwrepresent the angles of rotation for the C-axis and the B-axis, respectively. The positive rotation is in the direction of anadvancing right-hand screw about the +C and +B axes.Equation (3) specifies the form-shaping function matrix of thismachine tool and the joint parameters Px, Py, Pz,fzandfwshould be determined by the inverse kinematics. The firststep is to calculate the required rotary angles to yield the toolorientation, and the second is to calculate the required positionin relation to the linear axis to determine the position of thecentre of the tool tip using the known rotary angles.Whenthe CLdataincludingthe positionofthecentreof thetool tip Qxi Qyj Qzk and the tool orientation Kxi Kyj Kzk are known, the CL data can be expressed in thematrix form as follows:KQ01?KxKyKz0QxQyQz1266437754Since both Eq. (3) and Eq. (4) represent the samerelationship between the cutting tool and the workpiece, thedesired joint parameters can be determined by equatingthese two matrices. Equating the CL data matrix and theform-shaping function matrix, and taking the correspondingelements of the two matrices yield the following equations:KxKyKz026643775CzWxWz1 ? Cw ? WySw? SzWyWz1 ? Cw WxSw?SzWxWz1 ? Cw ? WySw? CzWyWz1 ? Cw WxSw?W2z1 ? Cw Cw0266437755QxQyQz126643775PxCzW2z1 ? Cw Cw? SzWxWy1 ? Cw ? WzSw?PyCzWxWy1 ? Cw WzSw? SzW2y1 ? Cw CwhinoPzCzWxWz1 ? Cw ? WySw? SzWyWz1 ? Cw WxSw?Lx CzMx SzMyPx?SzW2x1 ? Cw Cw? CzWxWy1 ? Cw ? WzSw?Py?SzWxWy1 ? Cw WzSw? CzW2y1 ? Cw CwhinoPz?SzWxWz1 ? Cw ? WySw? CzWyWz1 ? Cw WxSw?Ly? SzMx CzMyPxWxWz1 ? Cw WySw? PyWyWz1 ? Cw ? WxSw?PzW2z1 ? Cw Cw? Lz Mz126666666666666666666666664377777777777777777777777756Int J Adv Manuf TechnolThe joint anglesfzandfwshould be determined first.Equating the corresponding third element in Eq. (5) yieldsthe following B-axis angle:B w arccosKz? W2z1 ? W2z?0 ? w? 7Notably, there is another possible solution for B-axisangle in the range of ? ? w? 0, which can be obtainedas follows:B w ?arccosKz? W2z1 ? W2z?8If the operating range of the nutating table is in the rangebetween and 0, the solution should be modified asshown in Eq. (8). On the other hand, if the operating rangeof the nutating table meets the two possible solutions, theshortest rotational angle movement of the nutating table isusually chosen in the algorithm.Furthermore, equating the corresponding first andsecond elements in Eq. (5) and solving the simultaneouslinear equations for Sfzand Cfz, yield:Sz?KyWxWz1 ? Cw ? WySw? KxWyWz1 ? Cw WxSw?WxWz1 ? Cw ? WySw?2 WyWz1 ? Cw WxSw?29CzKxWxWz1 ? Cw ? WySw? KyWyWz1 ? Cw WxSw?WxWz1 ? Cw ? WySw?2 WyWz1 ? Cw WxSw?210Since the denominators in Eqs. (9) and (10) are the sameand always positive, the C- axis angle can be determined asfollows:C z arctan2?KyWxWz1 ? Cw ? WySw? KxWyWz1 ? Cw WxSw?;KxWxWz1 ? Cw ? WySw?KyWyWz1 ? Cw WxSw? ? z? 11where arctan2(y,x) is the function that returns angles in therange ?p ? q ? p by examining the signs of both y and x16.In addition, comparing the corresponding elements ofthe matrix on both sides of Eq. (6) yields three simulta-neous equations in three unknowns Px, Pyand Pz. Theprogram coordinate system is assumed to coincide with theworkpiece coordinate system. Accordingly, the expressionsof the desired NC data (denoted as X, Y and Z) can beobtained by considering the two offset vectors Lxi+Lyj+Lzk and Mxi+Myj+Mzk, and are expressed as follows:X Px Lx Mx Qx? CzMx SzMy Lx?CzW2x1 ? Cw Cw? SzWxWy1 ? Cw ? WzSw? Qy? ?SzMx CzMy Ly?SzW2x1 ? Cw Cw? CzWxWy1 ? Cw ? WzSw? Qz? Mz Lz? WxWz1 ? Cw WySw? Lx Mx12Int J Adv Manuf TechnolY Py Ly My Qx? CzMx SzMy Lx?CzWxWy1 ? Cw WzSw? SzW2y1 ? Cw Cwhino Qy? ?SzMx CzMy Ly?SzWxWy1 ? Cw WzSw? CzW2y1 ? Cw Cwhino Qz? Mz Lz? WyWz1 ? Cw ? WxSw? Ly My13Z Pz Lz Mz Qx? CzMx SzMy Lx?CzWxWz1 ? Cw ? WySw? SzWyWz1 ? Cw WxSw? Qy? ?SzMx CzMy Ly?SzWxWz1 ? Cw ? WySw? CzWyWz1 ? Cw WxSw? Qz? Mz Lz? W2z1 ? Cw Cw? Lz Mz143.2 Table/spindle-tilting type with a nutating headThe table/spindle-tilting type configuration has one rotaryaxis on the table and one nutating rotary axis on the spindle.Figure 3 illustrates two pivot points RC and RB on the C andB axes, respectively. The pivot point RC is located arbitrarilyon the C-axis and the pivot point RB is chosen to be theintersection of the nutating rotary B-axis and the axis of thecutting tool. The offset vector Lxi +Lyj +Lzk is calculatedfrom the origin Owto the pivot point RC and the effectivetool length, Lt, represents the distance between the pivotpoint RB and the cutter tip centre. The form-shapingfunction matrix of this configuration can be obtained bythe coordinate transformation matrices:Trans Lx;Ly;Lz?Rot z;?zTrans Px;Py;Pz?Rot W;w00001 ? Lt012664377515Equating Eq. (4) and Eq. (15) leads to the followingequations:KxKyKz026643775CzWxWz1 ? Cw WySw? SzWyWz1 ? Cw ? WxSw?SzWxWz1 ? Cw WySw? CzWyWz1 ? Cw ? WxSw?W2z1 ? Cw Cw02664377516QxQyQz126643775?CzWxWz1 ? Cw WySw? SzWyWz1 ? Cw ? WxSw?LtCzPx SzPy Lx? ?SzWxWz1 ? Cw WySw? CzWyWz1 ? Cw ? WxSw?Lt?SzPx CzPy Ly? W2z1 ? Cw Cw?Lt Pz Lz1266666666666643777777777777517Int J Adv Manuf TechnolThe joint parameters can be evaluated using the sameprocedure similar to the table-tilting configuration. Notably,the reference driving point of NC data in this configuration isassumedtobethepivotpointRB.Thisdefinitionisadoptedtoboth the spindle-tilting and table/spindle-tilting type config-urations, and is consistent with those used in most of thecommercial post-processor software packages. The completeanalytical equations for NC data can be expressed as:B w arccosKz? W2z1 ? W2z?0 ? w? 18C z arctan2 ?KyWxWz1 ? Cw WySw?KxWyWz1 ? Cw ? WxSw?;? KyWxSw? WyWz1 ? Cw?KxWySw WxWz1 ? Cw? ? z? 19X Lx Px LtWxWz1 ? Cw LtWySw SzLy? Qy? CzLx? Qx Lx20Y Ly Py LtWyWz1 ? Cw ? LtWxSw? CzLy? Qy? SzLx? Qx Ly21Z Lz Pz LtW2z1 ? Cw LtCw Qz223.3 Linearization problemTheoretically, the CAD/CAM system generates the CL databased on the assumption that the cutting tool moves linearlybetween two successive points. However, the actual toolmotion trajectory with respect to the workpiece is not linearand becomes curved since the linear and rotary axes movesimultaneously. The curved path deviates from the linearlyinterpolated straight line path between successive pathpoints, and this problem is known as the linearizationproblem. An algorithm must be developed to solve thisproblem.Assume that Pn, Pn+1and Pn+2are three continuousadjacent points in CL data, plotted in Fig. 4. The vector ofPnin matrix form can be expressed as Qn,xQn,yQn,zKn,xKn,yKn,z, where Qn,x, Qn,yand Qn,zare the components ofthe position of the center of the tool tip, and Kn,x, Kn,yandKn,zare the components of the tool orientation. Thecorresponding machine NC code of Pnis Mn=XnYnZnBnCn. As the five axes move simultaneously from thecurrent position Pnto the subsequent position Pn+1, eachaxis is assumed to move linearly between the specifiedpoints 18. Therefore, each point in the actual curved pathcan be expressed as follows:Mm;t Mn t Mn1? Mn23where t is a dummy time coordinate0 ? t ? 1. Thecorresponding CL data Pm,tfor Mm,tcan be determined bythe forward kinematics equations, e.g. Eqs. (5) and (6) forthe table-tilting type and Eqs. (16) and (17) for the table/spindle-tilting type. Moreover, each point in the ideal lineartool path can be determined as follows:Pn;t Pn t Pn1? Pn24wOwXwYwZtOtXtYtZCB xyzLLL+ijkRCRBOffset vectortLFig. 3 Coordinate systems of table/spindle-tilting type configurationInt J Adv Manuf Technol,nnP M,n tn tPM,m tm tMP11,nn+PM1,1,ntnt+PM1,1,mtmt+MP,n+2n+2PMInterpolated tool pathActual curved tool pathIdeal linear tool path,n tdFig. 4 Linearization problem inmulti-axis machining abFig. 5 Initiating dialog for the developed postprocessor. a table-tilting type. b table/spindle-tilting typeInt J Adv Manuf TechnolThe distance between Pm,tand Pn,tdenoted as dn,tformsa chordal deviation. If the maximum deviation (dn,t)maxexceeds the prescribed tolerance, then the additionalinterpolated CL data Pn,tshould be inserted into the originalCL data. Theoretically, the numerical iterative method forcalculating (dn,t)maxmust be adopted. Practically, the middlepoint, t = 0.5, is often selected as the candidate point 10.After the intermediate point Pn,thas been inserted, thecorresponding machine NC code can be generated.4 Discussion1.The main characteristic of the nutating rotary axisconfiguration is the continuous motion between thehorizontal and vertical positions in a single setup onthe same machine. In the current configuration of thecommercial machine tool, the angle of inclination ofthe nutating rotary axis is 45 degrees. This fact can beexplained by the equations derived above. The table-tilting type is used as an example. Equation (5)represents the tool orientation in relation to theworkpiece. The tool orientation relative to the work-piece in the initial position, where the table surface ishorizontal, can be determined by substitutingfz=fw=0 into Eq. (5), and is given by 0 0 1 0T. The nutatingrotary axis is assumed to rotate around X-axis by anangle so that the components of the vector W are Wx=0, Wy=S and Wz=C. Substituting the aboveconditions into Eq. (5), and settingfz=0 and KxKyKz0T=0 1 0 0Tfor the table surface in the verticalposition yields the following equation:0?10026643775SSw?SC 1 ? CwC2 1 ? CwCw02664377525The solutions to Eq. (25) for andfware =/4 andfw=. Therefore, the table surface can be set in the verticalposition when the table is rotated through an angle aboutthe nutating B rotary axis at an angle of inclination of /4.2.The nutating units on the five-axis machine tools canenhance the flexibility of the machining strategy. How-ever, the CL data considered are limited. Equation (7)Fig. 6 Implementation dialog for generating NC data for table-tilting type configurationInt J Adv Manuf Technolshows that the conditionKz?W2z1?W2z? ? 1 should be satisfied.When the nutating axis is set at an angle of 45 degrees,i.e. =/4 and Wz=C45, Kzis in the range 0 Kz 1.Consequently, the CL data generated by the CAD/CAMsystem can not be machined in this configuration if Kzisa negative value.3.The derived analytical equation of NC data is a generalform, which can be reduced to the orthogonal config-uration. The table-tilting type is chosen as an example.If the vector W is in the X-axis direction where Wx=1and Wy=Wz=0, the configuration reduces to the CAtable-tilting type. The analytical equation of the NC data,specifying for example Y-axis values, agrees with thosepresented elsewhere 8 and can be expressed as follows:Y Qx? LxSzCw Qy? Ly?CzCw? Qz? LzSw Ly26Notably, in the cited work, the two rotary axes areassumed continuously to intersect each other, and the offsetvector M 0i 0j 0k is used to derive the aboveequation.4.Based on the solutions forfzandfw, the cutting toolmay traverse the singular point wherefw=0, Qx=Qy=0,Qz=1 andfzis undefined 12, 18. This singularposition occurs whenfw=0 and the C-axis axis isparallel to the cutting tool axis. As displayed in Fig. 4, ifthe current position Pn+1is the singular point, any valueoffzis theoretically acceptable sincefzis undefined.The next point Pn+2should be read further to ensure thatfzvaries linearly between two successive points. Thevalue offzat position Pn+1can be determined byconsidering a linear change between Pnand Pn+2.5.Feedrate control is an important issue in practical multi-axis machining. Most controllers, such as FANUC andCincinnati Milacron, apply the feed rate number (FRN)and G93 code to control the feedrate. FRN isdetermined by the feedrate on the workpiece dividedFig. 7 Implementation dialog for generating NC data for table/spindle-tilting type configurationInt J Adv Manuf Technolby the span length of the resulting path. Whenrotational movements are combined with two or morelinear axes movements, determinations for the pathlength become very complex. In most cases, anadequate approximation of the true path length can bedetermined by using only the linear displacement 19.5 Implementation and verification5.1 Software implementationThe aforementioned postprocessor methodology was imple-mented in the Windows-XP environment using the BorlandFig. 8 Snapshot of VERICUTsimulation for table-tilting typeconfigurationFig. 9 Snapshot of VERICUTsimulation for table/spindle-tilt-ing type configurationInt J Adv Manuf TechnolC+ Builder programming language and the OpenGLgraphics library. A semi-sphere with a radius of 35 mm isused to illustrate machining. The CL data are generated bythe commercial CAD/CAM system, PowerMILL 20. Twotypical configurations of the machine tool, the table-tiltingand the table/spindle-tilting, were tested. Figure 5 (a) showsthe initiating dialogue of the developed postprocessorsoftware for the table-tilting type configuration. The usercan use the mouse button to rotate and zoom in on thesurface model of the machine tool, and the “Animate”button to animate dynamically the machine tool. When theuser enters the relevant parameters, such as the offset vectorfrom the program origin toward the centre of the C-axisrotary table, the system displays a figure to help the user toinput the parameters correctly, as shown in Fig. 6. Finally,the target CL data are opened by clicking the “File” menu,after which the NC data are generated accordingly. Figures 5(b) and 7 show screenshots of the initiating and implemen-tation dialogues,respectively, for the table/spindle-tiltingtype. Notably, the set length from the gauge plane to thetool tip centre and the distance from the gauge plane to thepivot point should be considered in this configuration, sincethe spindle head moves about the rotary axis.5.2 Solid cutting simulationA solid cutting simulation software package, VERICUT isused to confirm the generated NC data. Given the rawmaterial size, the specifications of the cutting tool, NC data,the type of controller, and the kinematics and physicalproperties of an NC machine tool, it can interactivelysimulate the material removal process of NC data. A table-tilting type machine tool with a B-axis nutating rotary tableis built in the simulation environment and the finished partis verified, as shown in Fig. 8. The relevant parametersshown in Fig. 6 should be reflected in the dialogue of theVERICUT hierarchical component tree. For example, theB-axis rotates about the X-axis by 45, so the nutating tableis in an inclined plane and the vector of the B-axis isW 0i ? S450j C450k. The two rotary axes are as-sumed not to intersect each other and the offset vector fromthe C rotary axis to the B rotary axis is ?5i ? 10j ? 15k.Moreover, the raw material should be placed in the desiredposition so that the offset vector from the program origin tothe C rotary axis is 30i 20j 10k. When the machinesetup has been completed, the system performs a realistic3D simulation of the configured machine tool.Figure 9 illustrates the “as-machined” workpiece cut ona table/spindle-tilting type machine tool with a B-axisnutating rotary head and a C-axis rotary table. As before,the relevant parameters should be set according to Fig. 7.The B-axis vector is W 0i S45?j C45?k. The offsetvector from the program origin to the rotary C -axis is?50i ? 60j ? 70k. The distance from the gauge plane to thepivot point of the B- axis is 100 mm and the set length fromthe tool tip to the gauge plane in the tool library is 120 mm.The results shown in Figs. 8 and 9 confirm the effectivenessand feasibility of the proposed postprocessor algorithm.5.3 Experimental verificationThe generated five-axis NC data are further verified byperforming an experimental trial-cut. A table-tilting five-axis machining centre (Deckel Maho DMU 70 eVolution)equipped with a Heidenhain iTNC530 control is used tomachine the semi-spherical workpiece. The experiment isconducted under the following conditions: (1) two ball-endmills with diameters of 10 mm and 4 mm are used forroughing and finishing, respectively; (2) the spindle speedis 5000 rpm and the feedrate is 1000 mm/min; (3) theworkpiece material is made of 7075 aluminum alloy.Notably, the positive direction of the C-axis of this machineFig. 10 Practical machining experiment on the Deckel Maho DMU70 eVolution. a roughing. b finishingInt J Adv Manuf Technoltool is along the negative Z direction. The actual NC valueof the C-axis is the negative valu
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