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徐 州 工 程 學(xué) 院 畢 業(yè) 設(shè) 計(jì) 論 文 1 附錄 英文原文 A GENERIC KINEMATIC ERROR MODEL FOR MACHINE TOOLS Yizhen Lin Yin Lin Shen Department of Mechanical and Aerospace Engineering The George Washington University Washington DC 20052 ABSTRACT 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 INTRODUCTION Global 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 8 Currently extensive research has been conducted to model the geometric and thermal errors of machine tools 3 11 These research works have proposed effective approaches in modeling the 徐 州 工 程 學(xué) 院 畢 業(yè) 設(shè) 計(jì) 論 文 2 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 ERRORS Among 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 15 The 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 15 The 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 徐 州 工 程 學(xué) 院 畢 業(yè) 設(shè) 計(jì) 論 文 3 the kinematic error model by using the homogeneous coordinate transformation which represent the error motion as9 13 1 101zyxzyxzRTTrams KINEMATIC MODELS By 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 1 11000 xRzSyxzXxTxzSyRyH 3 1 100yRzyyTxzxSzYyxS 4 1100zRzTxyHzyxZ 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 徐 州 工 程 學(xué) 院 畢 業(yè) 設(shè) 計(jì) 論 文 4 Figure 1 Bridge type moving gantry machine Also we have the tool link offset vector 5 1TTXpYZ According to the machine kinematic chain 6 HsytemxyzA Apply Equations 2 5 to Equation 6 we have the kinematic equations for the FXYZ machine trueXpTxRzSxyYpxRySzZp AA 7 RZzYpZA trYyyzX 8 xSX truepzzxpxySpyxSzYpAA 9 XR AA 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 1HsytemxyHzT A 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 trueXpxyzxRSyYpxRySzZp AA 11 yRZYpZz AA 徐 州 工 程 學(xué) 院 畢 業(yè) 設(shè) 計(jì) 論 文 5 Figure 2 Machine with a moving table X axis Figure 3 Machine with moving tables X Y trueYpxTyzyxRZpxzSyXp AA 12 RSZpXRAtreZY 13 zYz For the machines with two moving tables X Y axis they can be classified into the XYFZ group as shown in Figure 3 For XYFZ machine type HyxstemzT AA Therefore we have 14 1HsytemxyzT A Apply Equations 2 5 to Equation 14 we have trueXpxRzSxyYpxRySzZp AA 15 yRZyzYpZz treYT X 16 xSXA truepzzxxySzp A 17 yyRpzYR A 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 1100yRzyTxYHyx 徐 州 工 程 學(xué) 院 畢 業(yè) 設(shè) 計(jì) 論 文 6 Figrue 4 Machine with horizontal spindle 19 1 100zRzyzTxzxSyHyxSZ Also by the kinematic chain 20 sytemxyHzTA Applying Equations 2 18 and 19 to Equation 20 we have trueXpxRzSxyYpxRySzZp AA 21 zRZzYpZ trYTyyzX 22 xSXA truepzzxxySpzxSyYpAAA 23 XRp GENERIC KINEMATIC ERROR MODEL Although 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 1231312123 trueIIpTRSIpRSIp AA 24 2 233 RIpI AA 徐 州 工 程 學(xué) 院 畢 業(yè) 設(shè) 計(jì) 論 文 7 231132 trueIIpTRIpRSIp AA 25 213231 RSI A1 13 tre I 26 213I 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 trueIp 1132123123 TRSIpRSIp AAzone 27 223 I 3 2 trueIp 121132123 TRIRSIp AAzone 28 2I 33 trueIIp 131123123 TRSIp AAzone 徐 州 工 程 學(xué) 院 畢 業(yè) 設(shè) 計(jì) 論 文 8 29 231232123 TRSIpI AAzone I 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 MODEL A 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 15 徐 州 工 程 學(xué) 院 畢 業(yè) 設(shè) 計(jì) 論 文 9 Figrue 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 DISCUSSION We 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 徐 州 工 程 學(xué) 院 畢 業(yè) 設(shè) 計(jì) 論 文 10 1231312123 trueIpITRSIpRSIp AA 30 2 33 RIp AA2 tre 31 133112SIII 13 3 trueIpTpSIp AA 32 223213 RIpR A Figure 6 Kinematic china of a horizontal spindle machining center Finally 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 trueXpxTRySxzZpxRzSyYpzTx AA 33 zYTyZA treyXSyZA 34 z trueZpxzpxSpzA 35 RSRyyxYRXp A We also try to derive the specific error model by the homogeneous coordinate transformation 36 1HsytmyHzT 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 徐 州 工 程 學(xué) 院 畢 業(yè) 設(shè) 計(jì) 論 文 11 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 prediction CONCLUSION The 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 徐 州 工 程 學(xué) 院 畢 業(yè) 設(shè) 計(jì) 論 文 12 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 ACKNOWLEDGEMENT This 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 REFERENCES 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 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 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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 徐 州 工 程 學(xué) 院 畢 業(yè) 設(shè) 計(jì) 論 文 14 中文譯文 一個(gè)通用的運(yùn)動(dòng)誤差模型機(jī)床 Yizhen Lin Yin Lin Shen 機(jī)械和航空航天工程部 喬治華盛頓大學(xué) 華盛頓哥倫比亞特區(qū) 20052 摘要 一個(gè)通用的運(yùn)動(dòng)誤差模型 提出了表征幾何誤差的機(jī)床 首先 建模方面取得了 移動(dòng)橋門(mén)式機(jī) 移動(dòng)表機(jī)和臥式機(jī)床主軸分別為手段的傳統(tǒng)的齊次坐標(biāo)變換 然后這些 模型被推斷獲得普通錯(cuò)誤模型 在各種 3 軸機(jī)床上能容納不同的配置和軸定義 最后 通用運(yùn)動(dòng)學(xué)模型在一臺(tái)虛擬數(shù)控電腦上被實(shí)施 已嚴(yán)格的程序來(lái)解釋機(jī)械工具計(jì)量數(shù)據(jù) 轉(zhuǎn)化為 21 參數(shù)錯(cuò)誤 通過(guò)使用測(cè)量數(shù)據(jù)普通錯(cuò)誤模型的有效率來(lái)測(cè)試一個(gè)臥式加工中心 對(duì)角位移測(cè)試的結(jié)果提出并且與式樣預(yù)言比較 顯示普通運(yùn)動(dòng)學(xué)模型是高效率和容易實(shí) 施 可能極大減少建模和執(zhí)行工作 導(dǎo)言 在制造業(yè)中 水平精度和生產(chǎn)率在全球競(jìng)爭(zhēng)中有了越來(lái)越嚴(yán)格的要求 1 由于精確的 制造工件是密切相關(guān)的準(zhǔn)確性機(jī)床 2 大量的研究工作被進(jìn)行以提高機(jī)床的準(zhǔn)確性和減少 運(yùn)作成本 為建立一個(gè)高度精確的制造系統(tǒng) 3 8 機(jī)床的性能評(píng)估和實(shí)時(shí)誤差補(bǔ)償提供了 一個(gè)有效的方法 目前 廣泛的研究開(kāi)展模型的幾何和熱誤差的機(jī)床 這些研究工作在建模容積誤差 機(jī)床 3 11 上提出了有效的方法 然而 這些模式大多是用于專(zhuān)用機(jī)床而不是通用機(jī)床 他們無(wú)法提供一個(gè)普遍適用 并準(zhǔn)備到實(shí)施模型的各種不同的機(jī)床 在這里 主要的挑 戰(zhàn)是在車(chē)間如何發(fā)展一個(gè)可容納不同的機(jī)床配置和軸的定義的通用的機(jī)床誤差模型 12 舉 例來(lái)說(shuō) 齊次坐標(biāo)變換 13 最廣泛使用的技術(shù)在建模的幾何誤差機(jī)床 不僅提供了一般方 法而且被證明是效率較低 為每一個(gè)新機(jī)的配置和軸的定義的 人們必須通過(guò)相同的建 模程序 為此目的 我們開(kāi)發(fā)了一個(gè)通用的誤差模型 機(jī)床可以用來(lái)表征各種 3 軸機(jī)床的 迅速和高效率 通用誤差模型已在一個(gè)虛擬數(shù)控電腦程式上實(shí)施 測(cè)試結(jié)果表明 該通 用模型可以預(yù)測(cè)的機(jī)床的幾何誤差 機(jī)床誤差 徐 州 工 程 學(xué) 院 畢 業(yè) 設(shè) 計(jì) 論 文 15 其中的錯(cuò)誤歸因于機(jī)床在制造業(yè)中的系統(tǒng) 準(zhǔn)靜態(tài)誤差 包括幾何和熱誤差 是機(jī) 床安置不確定性的主要貢獻(xiàn)者 這些錯(cuò)誤 估計(jì)占機(jī)床誤差的 70 已觀察到高達(dá) 70 至 120 m 的生產(chǎn)一流的機(jī)床 11 對(duì)于這些錯(cuò)誤