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徐州工程学院毕业设计(论文)附录英文原文A GENERIC KINEMATIC ERROR MODEL FOR MACHINE TOOLSYizhen Lin, Yin-Lin ShenDepartment of Mechanical and Aerospace EngineeringThe George Washington University, Washington, DC 20052ABSTRACT: A generic kinematic error model is proposed to characterize the geometric error of machine tools. Firstly, modeling was made on a moving bridge gantry machine, a moving table machine and a horizontal spindle machine respectively by means of the conventional homogeneous coordinate transformation. Then these models were generalized to derive the generic error model which is able to accommodate the different configurations and axis definitions in various kinds of 3-axis machine tools. Finally, the generic kinematic model is implemented in a virtual CNC computer program, which has rigorous procedures to interpret machine tool metrology data into 21 parametric errors. The effectiveness of the generic error model is tested by using the measurement data from a horizontal spindle machining center. The result of the diagonal displacement test is presented and compared with the model prediction. It is shown that the generic kinematic model is efficient and easy to implement, which can substantially reduce the modeling and implementation efforts.INTRODUCTIONGlobal competition has imposed more and more stringent requirements on the levels of accuracy and productivity in the manufacturing industry.1 Since the accuracy of the manufactured workpieces is closely related to the accuracy of machine tools,2 a lot of research work has been carried out to enhance the machine tool accuracy and reduce the operational cost. Machine tools performance evaluation and real-time error compensation have provided an effective way to build up a highly precise manufacturing system.3-8Currently, extensive research has been conducted to model the geometric and thermal errors of machine tools.311 These research works have proposed effective approaches in modeling the volumetric error of machine tools. However, these models are mostly developed for specific machines instead of generic machine tools They could not provide a universal and ready-to-implement model for various kinds of different machine tools. Here, the main challenge is how to develop a generic machine error model12 which could accommodate different machine configurations and axis definitions in the shop floor. For example, homogeneous coordinate transformation,13 the most extensively used technique in modeling the geometric error of machine tools, only provides a general approach and proves to be less efficient for each new machine configuration and axis definition, people have to go through the same modeling procedures.To this aim, we developed a generic error model for machine tools which can be used to characterize various kinds of 3-axis machine tools quickly and efficiently. The generic error model has been implemented in a virtual CNC computer program. The test results show that the generic model can predict the geometric error of machine tools well.MACHINE ERRORSAmong the errors attributed to machine tools in the manufacturing systems, quasistatic errors, including geometric and thermal errors, are the major contributors to the positioning inaccuracies of machine tools. These errors, estimated to account for 70 percent of the errors of machines, have been observed to be as high as 70 to 120 m for production class machine.11 For these errors, a variety of machine tool performance test systems have been developed.14 Among them, the parametric representation describes the machine error characteristics in a kinematic model that provides the position and orientation errors of the cutting tool in terms of the axis position, tool length, and machine axis characteristics (positioning accuracy, straightness, axis rotations and squareness). It is the most convenient format for characterizing the machine tool errors and has been shown to be very flexible and robust in the performance evaluation.15The parametric errors are errors in the relative position and orientation between two successive axes in the kinematic chain from the workpiece to the tool. It has been well known that 21 parametric errors are enough to represent all the geometric error sources of a generic 3-axis machine. They are named as xTy, zRx, Sxy, etc., where R means rotation, T means translation and S means squareness.The left subscript means the moving slide and the right subscript means the error direction.15The kinematic model relates errors in relative position and orientation of the tool to 21 parametric errors. In deriving the kinematic error model, we make the assumptions of rigid body kinematics and small error motions. Donmez9 developed a general methodology to derive the kinematic error model by using the homogeneous coordinate transformation, which represent the error motion as9，13 (1)KINEMATIC MODELSBy means of the homogeneous coordinate transformation, we can derive kinematic error models for several specific machine types. Figure 1 shows a bridge type moving gantry machine which can be classified as FXYZ system, where F means the machine fixed base, X axis is the first slide stacked on the fixed base, Y axis is stacked on X axis and Z axis stacked on Y axis.From Equation (1), we have (2) (3) (4)Here Hx, Hy and Hz are the transformation matrices for each axis. xRx, xTx, xRy, Sxz are the 21 parametric errors. The squareness error is interpreted as an angular error in the derivation.15 The positive direction of the squareness error is defined by the corresponding angular errors.Figure 1.Bridge type moving gantry machineAlso we have the tool link offset vector: (5)According to the machine kinematic chain, (6)Apply Equations (2)-(5) to Equation (6), we have the kinematic equations for the FXYZ machine: (7) (8) (9)We further derive the model for machines with a moving table. A typical machine with a moving table (X axis) is shown in Figure 2. It can be classified into the XFYZ group. For XFYZ machine type, (10)The homogenous coordinate transformation also holds true for each axis so that Equations (2)-(5) are still valid here. Apply them to Equation (10), we have (11)Figure 2.Machine with a moving table-X axis Figure 3.Machine with moving tables(X,Y) (12) (13)For the machines with two moving tables (X, Y axis), they can be classified into the XYFZ group,asshown in Figure 3. For XYFZ machine type,Therefore, we have (14)Apply Equations (2)-(5) to Equation (14), we have (15) (16) (17)We have discussed the kinematic models of FXYZ, XFYZ and XYFZ machines. All of them are vertical spindle machines. It is therefore of interest to study the case of the horizontal spindle machine. By convention, the spindle is defined as the Z axis.16 Figure 4 shows the kinematic chain of a FXZY-type horizontal spindle machine. Because Z axis, the spindle, is stacked on X slide now,Equations (3)-(4) will become (18)Figrue 4.Machine with horizontal spindle (19)Also, by the kinematic chain, (20)Applying Equations (2), (18) and (19) to Equation (20), we have (21) (22) (23)GENERIC KINEMATIC ERROR MODELAlthough the kinematic equations we have derived for different machines are different in mathematical forms, they hold the same structure in formulation because they have the similar physical kinematic chains. Therefore it is possible for us to generalize these specific models to develop a generic error model for 3-axis machine tools. In general, we have the following model: (24) (25) (26)I, II and III are the first, second and third physical axis of machine. I is the first axis directly related to the workpiece. III is the axis directly related to the tool link. II is the axis in between. 123 is a multiplier,which will have:(1). 123=1,when I, II, III form a right hand coordinate system; (2). 123= -1, when I, II, III cannot form a right hand coordinate system.By simply assigning I, II, III to X, Y, Z and setting xyz=1 because XYZ form a right hand coordinate system in Figure 1, Equations (24)-(26) will change to Equations (7)-(9). Assigning I,II, III to X, Z,Y and setting xzy = -1 because XZY form a left hand coordinate system in Figure 4, Equations (24)-(26) will change to Equations (21)-(23). For the other different axis naming conventions in the shop floor, by assigning the generic axes I, II, III to their respectively named axes, such as Y, X, Z, the specific error model can be obtained easily. It can be seen that the generic error model can handle different axis definitions well.After assigning the axes to the generic model, we need to make the relevant change for moving table machines. As shown in Equations (27)-(29), we decompose the structure of the formulation in Equations (24)-(26) into three parts zone-I, zone-II and zone-III respectively.Equations (27)-(29) are the model for machines without a moving table. For machines with one moving table (such as XFYZ, YFXZ, etc.), the following changes will be made:(1-1). zone-II and zone-III stay the same.(1-2). All the terms in zone-I change signs.(1-3). If Ip (excluding the one inside (Ip+I), where rule 1-4 applies) appears in zone-I, Ip should be changed to Ip-I.(1-4). If (Ip+I) appears in zone-I, (Ip+I) should be changed to (Ip-I). (27) (28) (29) On basis of this, if one further considers machines with two moving tables (XYFZ or YXFZ etc.),the rules will be(2-1). zone-III remains the same.(2-2). All terms in zone-II change signs.(2-3). For any IIp (excluding the one inside (II+IIp), where rule 2-4 applies) appears in zone-I or zone-II, IIp should be changed to IIp-II.(2-4). For any (II+IIp) appears in zone-I or zone-II, (II+IIp) should be changed to (IIp-II).These rules can be easily verified by comparing Equations (7)-(9) (FXYZ machine) with Equations (11)-(13) (XFYZ machine), then with Equations (15)-(17) (XYFZ machine). It can be seen that the generic error model also handles the moving table(s) machine well.IMPLEMENTATION OF GENERIC ERROR MODELA virtual CNC computer program is developed to implement the generic error model. The program is capable of predicting the effects of machine error motions in the machine gauge point for the given XYZ nominal commanded movement of machines.Figure 5 shows the inputs/outputs and functionality of the virtual CNC computer program. The program inputs include: (1). Machine type and axis assignment; (2). Machine tool metrology data, which consist of laser measurement data of machine axes, including positioning error, straightness errors, roll, pitch, yaw and the squareness measurement; (3). The commanded XYZ motion of the gauge point and moving directions of axes (to account for backlash). The program outputs will predict the actual XYZ position of the machine gauge point and IJK orientation of the cutting tool.In the virtual CNC program, we use the machine tool metrology data to generate a lookup table for each of the 18 translation and angular errors for the 3-axis machine. The program also keeps three squareness numbers. The procedures to decode 21 parametric errors from the laser system measurement data are as follows:15Figrue 5.Virtual CNC computer program implementing generic error model(1). Construct an error lookup table of 6 parametric errors (linear displacement, 2 straightness,roll, pitch and yaw) for each axis. Initialize all the entries in the lookup table to zero.(2). Read in the measurement data.(3).Compensate the thermal expansion for the positioning error. If the metrology data have been compensated, advance to STEP 4.(4). Shift the coordinates from the measurement coordinate system to the machine coordinate system.(5). Extrapolate the measurement data to cover the whole range of axis in the machine working zone.(6). Abbe Offset compensation for translation errors. Abbe Offset is the instantaneous value of the perpendicular distance between the displacement measuring system of a machine(scales) and the measurement line where displacement in that coordinate is being measured.14 Because of the Abbe Offset translation errors are often compounded by the effects of angular errors.(7). For straightness data, calculate the best fit line through the compensated data and store the residuals.(8). Calculate the squareness errors using the best fit lines obtained in STEP 7.TEST ON A HORIZONTAL MACHINING CENTER AND DISCUSSIONWe use the measurement data obtained by a 5-D laser system17 from a horizontal spindle machining center to verify our generic model. As shown in Figure 6, the horizontal spindle machine can be classified as the XFZY machine. Because the first axis is a moving table, applying the rule of the moving table to Equations (27)-(29), we have (30) (31) (32)Figure 6. Kinematic china of a horizontal spindle machining centerFinally, we substitute the general axes with the defined axes. In the XFZY machine, I = X, II = Z, III= Y, 123= 1 (X, Z, Y form a right hand coordinate system). Therefore, the specific error model for the horizontal spindle machine center would be (33) (34) (35)We also try to derive the specific error model by the homogeneous coordinate transformation. (36)Apply Equations (2), (18) and (19) to Equation (36), we can verify that the specific model obtained from our generic model is exactly the same as that obtained by the homogeneous coordinate transformation. It can also be seen that the generic model is more direct and needs less calculation and modeling efforts. People without profound knowledge in the kinematics and the homogeneous coordinate transformation are still able to derive the machine error model from the generic model.To further test the effectiveness of the generic model and the virtual CNC program, the diagonal measurement data from the machining center are used. The diagonal measurement14 is a simple linear measurement occurring along a diagonal of the machine working volume, which shows the combined effect of error motions of three axes. Figure 7 shows the diagonal test for the horizontal machining center which measured the linear displacement errors at 11 evenly distributed diagonal points back and forth. The prediction from the virtual CNC program was also shown. It can be seen that the virtual CNC program predicts the errors in the diagonal test well (within a few microns).Figure 7.Diagonal test and model predictionCONCLUSIONThe generic kinematic error model can characterize the geometric errors of various kinds of the 3-axis machine tools. It can handle different machine configurations and axis definitions. Compared with the homogeneous coordinate transformation approach, the generic kinematic model is more efficient, easier to implement, substantially reducing the modeling and implementation efforts.The virtual CNC program can implement the generic model and simulate the geometric errors of machine tools. It has rigorous procedures to decode 21 parametric errors from the machine tool metrology data and uses them in the generic model to predict the machine error motion and the tool orientation. The diagonal test result shows that the virtual CNC program can predict the machine errors and help reduce machine errors to a few microns.The generic model will be tested with more data. Further research work on the generic model for machines with more axes is being carried out.ACKNOWLEDGEMENTThis work was supported in part by the National Science Foundation under Grant No. DMII-9624966. The support is greatly appreciated. The authors would like to thank Dr. Johannes Soons of the National Institute of Standards and Technology, Mr. Richard Yang of Automated Precision,Inc., and Mr. Sungho Moon of the George Washington University for useful discussions.REFERENCES1. Mou, J., A Systematic Approach to Enhance Machine Tool Accuracy for Precision Manufacturing, International Journal of Machine Tools & Manufacture, Vol. 37, No.5, 669-685, 1995.2. Mou, J. and Liu, C. R., An Adaptive Methodology for Machine Tool Error Correction, Journal of Engineering for Industry, Vol. 117, 389-399, 1995.3. Zhang, G., Veale, R., Charlton, T., Borchardt, B. and Hocken, R., Error Compensation of Coordinate Measuring Machines, Annals of the CIRP, Vol. 34/1, 444-448, 1985.4. Ni, J., CNC Machine Accuracy Enhancement Through Real-time Error Compensation, Journal of Manufacturing Science and Engineering, Vol. 119, 717-725, 1997.5. Chen, J. S. and Ling, C. C., Improving the Machine Accuracy Through Machine Tool Metrology and Error Correction, International Journal of Advanced Manufacturing Technology,Vol. 11, 198-205, 1996.6. Chen, J. S., Yuan, J. X., Ni, J. and Wu, S. M., Real-time Compensation for Time-variant Volumetric Errors on Machining Center, Journal of Engineering for Industry, Vol. 115, 472-479, 1993.7. Ni, J. and Wu, S. M., An On-Line Measurement Technique for Machine Volumetric Error Compensation, Journal of Engineering for Industry, Vol. 115, 85-92, 1993.8. Kiridena, V. S. B. and Ferreira, P. M., Computational Approaches to Compensating Quasistatic Errors of Three-Axis Machining Centers, International Journal of Machine Tools & Manufacture, Vol. 34, No. 1, 127-145, 1991.9. Donmez, M., A General Methodology for Machine Tool Accuracy Enhancement: Theory, Application and Implementation, Ph.D dissertation, Purdue University, West Lafayette, IN, USA, 1985.10. Ferreira, P. M. and Liu, C. R., A Method for Estimating and Compensating Quasistatic Errors of Machine Tools , Journal of Engineering for Industry, Vol. 115, 149-159, 1993.11. Kiridena, V. S. B. and Ferreira, P. M., Kinematic Modeling of Quasistatic Errors of Three-Axis Machining Centers, International Journal of Machine Tools & Manufacture, Vol. 34, No.1, 85-100, 1991.12. National Institute of Standards and Technology, Web page of project: Machine Tool Performance Models and Machine Data Repository, Gaithersburg, Maryland, USA, 1997.13. King, M. S., Modeling and Compensation of Quasistatic Errors in Machine Tools, Ph.D dissertation, University of Kansas, Lawrence, Kansas, USA, 1995.14. ASME B5.54-1992, Methods for Performance Evaluation of Computer Numerically Controlled Machining Centers, 1992.15. Soons, J., Private Communication, National Institute of Standards and Technology,Gaithersburg, Maryland, USA, 1998.16. ISO841:1994(E), Industrial Automation Systems Physical Device Control Coordinate System and Motion Nomenclature, 1994.17. Automated Precision, Inc., User Manual for the 5/6-D Laser Measurement System, Gaithersburg, Maryland, USA, 1998.中文译文一个通用的运动误差模型机床Yizhen Lin, Yin-Lin Shen机械和航空航天工程部乔治华盛顿大学，华盛顿哥伦比亚特区20052摘要：一个通用的运动误差模型，提出了表征几何误差的机床。首先，建模方面取得了移动桥门式机，移动表机和卧式机床主轴分别为手段的传统的齐次坐标变换。然后这些模型被推断获得普通错误模型，在各种3轴机床上能容纳不同的配置和轴定义。最后，通用运动学模型在一台虚拟数控电脑上被实施，已严格的程序来解释机械工具计量数据转化为21参数错误。通过使用测量数据普通错误模型的有效率来测试一个卧式加工中心。对角位移测试的结果提出并且与式样预言比较。显示普通运动学模型是高效率和容易实施，可能极大减少建模和执行工作。导言在制造业中，水平精度和生产率在全球竞争中有了越来越严格的要求1。由于精确的制造工件是密切相关的准确性机床2，大量的研究工作被进行以提高机床的准确性和减少运作成本。为建立一个高度精确的制造系统38，机床的性能评估和实时误差补偿提供了一个有效的方法。目前，广泛的研究开展模型的几何和热误差的机床。这些研究工作在建模容积误差机床311上提出了有效的方法。然而，这些模式大多是用于专用机床而不是通用机床，他们无法提供一个普遍适用，并准备到实施模型的各种不同的机床。在这里，主要的挑战是在车间如何发展一个可容纳不同的机床配置和轴的定义的通用的机床误差模型12。举例来说，齐次坐标变换13，最广泛使用的技术在建模的几何误差机床，不仅提供了一般方法而且被证明是效率较低为每一个新机的配置和轴的定义的，人们必须通过相同的建模程序.为此目的，我们开发了一个通用的误差模型，机床可以用来表征各种3 轴机床的迅速和高效率。通用误差模型已在一个虚拟数控电脑程式上实施。测试结果表明，该通用模型可以预测的机床的几何误差。机床误差其中的错误归因于机床在制造业中的系统，准静态误差，包括几何和热误差，是机床安置不确定性的主要贡献者。这些错误，估计占机床误差的70 ，已观察到高达70至120 m的生产一流的机床11。对于这些错误，各种机床的性能测试系统已被开发14。其中，参数的代表介绍了机床误差特性，在运动学模型提供了位置和方向的错误，刀具在条款的轴心地位，工具的长度和机器轴特性（定位精度，直线度，轴的轮作方式和矩形） 。这是描绘机床误差最方便的格式，并已证明性能评估是非常灵活和稳健的15。从工件到工具，参数错误是运动链的相对位置和方向的连续两个轴线之间的错误。它已众所周知，21参数错误是代表一个通用的3轴机床所有的几何误差来源是足够的。他们命名为xTy ， zRx ， Sxy等，其中R是指旋转， T代表翻译和S是指矩形。左下标意味移动的幻灯片和正确的下标意味错误方向15。该运动学模型与21参数错误关系在工具的相对位置和取向的错误。在所产生的运动误差模型，我们作出的假设刚体的运动学和小错误行动。Donmez9制定了一般的方法推导出运动误差模型用齐次坐标变换，代表错误行动9，13 (1)运动学模型方式均匀的坐标变换，我们可以得出几个具体类型机床的运动学误差模型。图1显示了一桥式移动机床，可以被列为FXYZ系统，其中F是指机床固定基地， X轴是第一幻灯片堆放在固定基地， Y轴是堆放在X轴和Z轴堆放在Y轴。从方程（ 1 ） ，我们有 (2) (3) (4)图1 桥型移动龙门机这里的Hx ,Hy和Hz是变换矩阵为每个轴。 xRx ， xTx ， xRy ， ， Sxz是21参数错误。该矩形的错误解读为在派生的1个角误差15。矩形错误的正向是由相应的角的错误定义的。我们也有工具的链接，抵销了向量： (5) 根据机器的运动链， (6)适用于方程（ 2 ） （ 5 ）方程（ 6 ） ，我们有运动学方程为FXYZ机： (7) (8) (9)我们进一步推导出模型机与运动就座。一个典型的机器与运动表（ X轴）是如图2所示。它可分为XFXY组。为XFYZ机的类型， (10)均相坐标变换还拥有真正的为每个轴，使方程（ 2 ） -（ 5 ）仍然有效，在这里。它们应用到方程（ 10 ） ，我们有 (11)图2 机与移动表-X轴 图3 机与移动表（X ，Y ） (12) (13)为机器与双动桌（X ， Y轴） ，他们可分为XFYZ组，如图3所示。为XFYZ机类型， 因此，我们有 (14)适用于方程（ 2 ） -（ 5 ）方程（ 1 4） ，我们有 (15) (16) (17)我们讨论了运动学模型的FXYZ ， XFYZ和XYFZ机器。所有这些都是垂直主轴的机床。因此，这是有兴趣的研究案件的水平主轴机。按照惯例，主轴定义为Z轴16。图4显示运动链的一个FXZY型卧式机床主轴。因为Z轴是主轴，是堆叠在X幻灯片现在，方程（ 3 ） -（ 4 ）将成为 (18)图4 卧式机床主轴 (19)此外，由运动链， (20)运用方程（ 2 ） （ 18 ）和（ 19 ）方程（ 20 ） ，我们有 (21) (22) (23)通用运动学误差模型虽然运动学方程，我们为不同的机器获得了不同的数学形式，他们持有相同的结构，在制定，因为他们拥有类似的物理运动链。因此，这是使我们有可能一概而论，这些特定的模式，以开发一个通用的误差模型为3 -轴机床。总的来说，我们有以下型号： (24) (25) (26)I ， II和III是第一，第二和第三的物理轴机床。 I是第一轴，直接关系到工件。是轴，直接关系到工具的链接。是中轴线之间。123是一个乘数，这将有：（1）.1231，当我第I ，II，III形成右手坐标系； （ 2 ）. 123-1，当我第I ，II，III不能形成一个右手坐标系统。只要分配I ，II，III向至X ， Y ， Z和设置xyz1 ，因为XYZ形成一个右手坐标系统，如图1 ，方程（ 24 ） -（ 2 6）将变更为方程（ 7 ） - （9 ） 。指派I ，II，III到X ,Z,Y和设置xyz1因为XYZ形成一个左手坐标系统在图4 ，方程（ 24 ） -（ 26）将变更为方程（ 2 1） - （2 3）。至于其他不同的轴的命名惯例，在车间，指派通用轴，I ，II，III，以他们分别命名为轴线，例如的Y ，X的Z的具体误差模型可以得到很容易。可以看出，一般误差模型可以处理不同轴的定义。后指派轴线向通用模型，我们需要作出有关改变移动表机器。显示，在方程（ 27 ） -（ 2 9） ，我们分解结构的提法，方程（ 2 4） - （2 6 ）分别分为三个部分区I，区II和区。 (27) (28) (29) 方程（ 27 ） -（ 2 9）是模型机，没有一个移动表。为机器一动表（如XFYZ ， YFXZ等） ，以下的变化会作出：（1-1）.区二和区三保持不变。（1-2）.所有条款在区改变的迹象。 （1-3）.如果IP（除去一个在内（即IP +I），其中的规则适用于14）出现在区，IP 应改为的IP-I。（1-4）.如果（即IP +I）出现在区，（即IP +I）应改为（IP-I） 。在此基础上，如果进一步认为机器与双动表（ XYFZ或YXFZ等） ， 该规则将是（2-1）.区维持不变。（2-2）.所有条款在区改变的迹象。 （2-3）.任何P（除去一个在内（P），其中的规则适用于24）出现在区或区，P应改为P。（2-4）.任何（P）出现在区或区， （P）应改为（P） 。这些规则可以很容易地验证了比较方程（ 7 ） -（ 9 ） （ FXYZ机）与方程（ 1 1） - （1 3 ）（XFYZ机） ，然后与方程（1 5 ）- （ 17 ） （XYFZ机）。可以看出，一般误差模型，还处理的感人表（s）的机器。通用误差模型的实施一个虚拟数控电脑程序被开发实施通用误差模型。该计划是能够预测机床误差运动在机床中收集点，以便给予的XYZ的名义指挥运动机床。图5虚拟数控电脑程式执行一般性错误示范图5显示输入/输出和功能的虚拟数控电脑程式。该计划的投入，包括：（ 1 ）.机器型号和轴转让；（ 2 ）.机床计量数据，其中包括激光测量数据的机轴，包括定位误差，直线度误差，轧辊，音高，方向舵和矩形测量；（ 3 ）.该指挥的XYZ的议案，压力计点和移动方向的轴（以帐户为反弹） 。该计划的产出预测实际的XYZ的立场，机器计点和IJK方向刀具。在虚拟数控程序，我们会使用机床计量数据生成一个查找表为每18个翻译和角误差的3轴机床。该计划亦备有三个矩形号码。程序解码21参数错误，从激光系统的测量数据分别如下15： （1）.建构一个错误查找表6参数错误（线性位移， 2直线，唱名，音高和偏航）为每个轴。初始化所有参赛作品在查找表到零。 （2）.在读测量数据。 （3）.补偿的热膨胀为定位误差。如果计量数据已赔偿，与时俱进，以第4步。 （4）.转移的坐标从测量坐标系统，以机器坐标系统。 （5）.推断的测量数据，以支付整系列的轴在机器中的工作区。 （6）.阿贝抵销补偿翻译错误。阿贝抵销是瞬时值垂直之间的距离，位移测量系统的一台机器（级）和测量线位移正在精确的协调14，因为该阿贝抵销翻译错误往往是复杂的影响角错误。 （7）.直线的数据，计算出最适合路线，通过补偿的数据和存储残值。 （8）.计算矩形的错误使用最适合线获得的第7步。试验研究卧式加工中心和讨论我们使用的测量所得的数据，按5三维激光系统17从横向主轴加工中心，以核实我们的通用模型。显示在图6 ，卧式机床主轴，可列为XFZY机。因为第一轴是一个移动表，运用法治的移动表方程（27） -（29） ，我们有 (30) (31) (32)最后，我们以一般的轴线与界定的轴线。在XFZY机，I= X,= Z，= y ，1231（ X,Y,Z形成一个右手坐标系统）。因此，具体的错误模型卧式加工中心主轴将图6 运动链的横向主轴加工中心 (33) (34) (35)我们还尝试推导出具体的误差模型，由齐次坐标变换。 (36)适用于方程（2） ,（18）和（19）方程（36） ，我们可以确认该具体模型所得的，我们的通用模型是完全一样的获得由齐次坐标变换。它也可以看到，通用模型是更直接的和需要较少的计算和模拟的努力。人们没有渊博的知识，在运动学和齐次坐标变换仍然能从通用模型获得机器误差模型。为了进一步测试通用模型和虚拟数控程序的有效率，从加工中心使用对角线的测量14数据。对角线测量是一个简单的线性测量发生沿对角线的机器工作容积，这表明的综合效果，错误的议案3轴。图7显示，对角线测试为卧式加工中心，其中测量直线位移误差在11均匀分布的角点，反反复复。预测从虚拟数控程序也显示。可以看出，虚拟数控程序预测误差在对角线试验井（数微米） 。对角线距离（mm）图7 对角线测试和模型预测结论通用运动学误差模型可以描绘各类3轴机床的几何误差。它可以处理不同的机器配置和轴的定义。相比与齐次坐标的转换方法，通用运动学模型是更有效率，更容易实施，大大减少建模和执行工作。虚拟数控程序可以执行的通用模型和模拟机床的几何误差。有严格的程序解码从机床的计量数据的21个参数错误，并使用他们在通用模型预测机器的错误议案和工具的方向。对角线试验结果表明，虚拟数控程序可以预测机床误差，并有助于减少机床误差至数微米。通用模型将测试与更多的数据。对通用模型与更多的轴线进行进一步的研究工作。鸣谢一部分支持工作是由美国国家科学基金会资助，第DMII - 9624966 。支持是极大的感谢。作者想感谢国家标准和技术研究所的Johannes Soons博士，自动化精确度的理查德杨先生，乔治华盛顿大学的Inc 和Sungho Moon先生的有用的讨论。参考1. Mou, J., A Systematic Approach to Enhance Machine Tool Accuracy for Precision Manufacturing, International Journal of Machine Tools & Manufacture, Vol. 37, No.5, 669-685
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