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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. 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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