Journal of Advanced Mechanical Design, Systems, and Manufacturing

Transcription

1 A Study on Error Compensation on High Precision Machine Tool System Using a 2D Laser Holographic Scale System* (First Report: Scale Development and Two Dimensional Motion Error Compensation Method) Toru FUJIMORI**, ***, Kayoko TANIGUCHI***, Chris ELLIS****, Tojiro AOYAMA** and Kazuo YAMAZAKI**** **Department of System Design Engineering, Keio University, Hiyoshi, Kouhoku-ku, Yokohama-si, Kanagawa-ken , Japan fujimori@mgscale.com ***Magnescale Corporation, 45 Suzukawa, Isehara-si, Kanagawa-ken , Japan ****IMS-Mechatronics Laboratory, Department of Mechanical and Aerospace Engineering, University of California, 2020C Bainer Hall Davis, CA , U.S.A. Abstract Numerical error compensation is a cost-effective way to further improve the accuracy of high precision CNC machine tools. This paper discusses the major techniques for numerical error compensation on high precision machine tools. First, the high accuracy 2D holographic scale system and the dedicated measurement software are introduced. Then, the mathematical modeling of repeatable machine tool errors and the synthesis of volumetric error model are presented, followed by a description on the procedure of NC code modification. Error compensation experiments have been performed on a ballscrew driven machine tool and a linear driven machine tool. The experimental results show that numerical error compensation is generally effective on machine tools with both drive systems, while the linear driver system provides better dynamic performance for further improvement when the control precision is increased. Key words: Holographic Scale, 2D Measurement, Machine Tool Accuracy, Error Compensation 1. Introduction *Received 27 Jan., 2012 (No ) [DOI: /jamdsm.6.999] Copyright 2012 by JSME In order to successfully manufacture high precision parts such as medical devices, MEMS systems, and die/molds for optical devices, etc., high accuracy machine tools are essential [1]. The accuracy of machine tools is degraded by geometric errors of machine components, kinematic errors of the assembly process, thermal errors of the environment, and tracking errors of the control system etc [2]. Various measurement systems have been developed for the evaluation of machine tool accuracy to ensure high precision machining. Double ball bars [3] were developed to evaluate two or three dimensional accuracy of the machine in a rapid way. However, the measurement range is limited due to the small stroke of the magnetic bar. Laser interferometers are the traditional high precision system for the measurement of linear displacements based on the interference of two beams. Even though nanometer resolution can be achieved in laser interferometers, the measurement accuracy is limited by the measurement environment, such as air temperature, pressure, and humidity [4]. On the other hand, measurement accuracy of linear optical scales is not 999

2 heavily influenced by the measurement environment. Sub nanometer resolution is achievable in diffraction optical scales. For example, the holographic scale developed by Magnescale Corp. has a measurement resolution as small as nm [5]. When used to comprehensively evaluate machine tool accuracy, however, laser interferometers and linear scales are not efficient, because they only provide one dimensional measurement capability. The cross grid measurement system (KGM), developed by HEIDENHAIN CORP., provides noncontact two dimensional measurement capability and is suitable for the evaluation of both static and dynamic machine tool errors [6, 7]. Although KGM can be used for the moderate needs in the evaluation of two dimensional machine motion error with the resolution of 0.1 μm, today s evaluation needs of motion error of ultra precision machine tool require much higher resolution such as pico meter under the moving speed of several meters per minute. To validate the machine tool accuracy is only currently the primary application of the various measurement devices. A more attractive application is to capture the repeatable machine tool volumetric errors and then compensate for them. As a pioneer of numerical error compensation, Hocken [8] successfully implemented numerical error compensation on a coordinate measuring machine. Donmez [9] presented a general methodology for the compensation of quasi-static machine tool errors, including geometric errors and thermally induced errors. Sartori and Zhang [10] reviewed the methods for geometric error measurement and compensation that were developed before In their paper, the foundations of numerical compensation were described, including a common language for error measurement and compensation, conditions for carrying out successful error compensation, various error measurement and compensation methods, etc. Recently, Schwenke [2] gave an updated review on the methods for geometric error measurement and compensation from the viewpoint of practical applications, including the appearance of new calibration methods, new concepts in international standards, and growing capabilities of machine tool controllers for error compensation. In this paper, we first briefly introduce a new measurement system, which is composed of a 2D holographic scale, a 2D sensor head, and dedicated measurement software. Then, the mathematical modeling of repeatable machine tool errors and the synthesis of three dimensional volumetric error of the 3-axis machine tool are discussed, as well as a more simplified two-dimensional model that is more suitable for the early stages of such research. Thereafter, by using the two-dimensional error mapping, the procedure for modifying NC code is presented. Finally, experimental results of numerical error compensation on high precision machine tools with different types of drive systems are discussed. 2. Two-dimensional machine tool accuracy measurement system based on holographic scales 2.1 Two-dimensional scale and laser head Holographic gratings produce far less stray light than ruled gratings [11]; as a result, Laserscale based on holographic gratings provide better capability for high accuracy than conventional laser scales. By collaborating with the industry, recently we have developed a 2D accuracy measurement system by using high resolution 2 holographic scales [12]. The most recent 2D holographic scale is developed which is composed of two sets of perpendicular grating lines, one for the measurement of displacement in X direction, and the other for Y direction. The size of the laser scale is 100 mm 100 mm, and the pitch of the gratings is 1um. Signal wavelength supplied out from the 2D head is 250nm. After 4000 times interpolating by the laser scale interpolator, the final measurement resolution is nm. Fig. 1 shows the integrated 2D laser scale. 1000

3 Fig. 1 2D holographic scale Refer to Fig.2 for the 2D sensor head. When the 2D sensor head moves over the 2D holographic scale, linear displacement along the X direction is measured by the X displacement sensor, and linear displacement along the Y direction is measured by the Y displacement sensor. X axis sensor and Y axis sensor have been mounted in the sensor head such that x-axis sensor and y-axis sensor will target almost same point of the measurement. Thus, the two dimensional motion trajectory of the sensor head relative to the 2D scale can be captured. 1001

4 Fig.2 The sensor head for the 2D accuracy measurement system 2.2 Dedicated measurement software for the 2D accuracy measurement system Dedicated software has been developed using the C++ language and the OpenGL graphical display library. Fig.3 shows a screen shot of the software when performing circular tests on a machine tool. The main functions of the software are as follows: a) User interface: The software user interface supports the control of hardware calibration steps, setting measurement parameters, and setting graphical display parameters. Tools for graphics manipulation, documentation operation, and nominal trajectory definition are also provided. b) NC code generation: This module supports the generation of NC code to be fed into the controller of a CNC machine tool for specific machine tool accuracy measurements, circular tests, general freeform tests, feed rate tests, and interpolation tests [ISO230-4] [ISO ]. c) Data capture: Two data capturing modes are supported. In the data capture according to nominal trajectory mode, only the data points consistent with the nominal trajectory will be captured; the data points corresponding to auxiliary machine tool movement will be discarded. This mode is used when performing machine tool tests. In direct data capture mode, all the data points are captured; this mode is used when calibrating the hardware system. d) Analysis of machine tool tests: This module supports the visualization of measured data, evaluation of machine tool characteristic error parameters, and evaluation of deviation between measured data and nominal trajectory. Visualization of measured data includes visualization of circular test data in a polar coordinate system, visualization of circular test data in an orthogonal coordinate system, and visualization of freeform test data in an orthogonal coordinate system. Evaluation of characteristic parameters from measured data in the circular test is according to the ISO standard. Evaluation of deviation between measured data and nominal trajectory is supported for both circular tests and freeform tests. 1002

5 e) Documentation: This module includes data storage on the hard disk, data retrieval from the hard disk, exporting high resolution graphics for machine tool test results, printing machine tool test result etc. Fig.3 The dedicated software of the 2D accuracy measurement system 3. Machine tool error modeling and error compensation 3.1 Conventional modeling scheme of volumetric errors of 3-axis machine tools The conventional modeling scheme to describe the positioning accuracy of a 3-axis CNC machine tool is based on the six degree of freedom motion of the rigid body in an x-y-z Cartesian coordinate frame, namely, for each of its axes, there is one positioning error, two straightness errors, and three angular errors (roll, pitch, and yaw), refer to Fig.4. Besides those, three perpendicularity errors exist, between each pair of the three axes. Thus, there are in total 21 component errors for a three axis CNC machine tool. These component errors are used in forming the homogeneous transformation matrices. [13, 14], which describe the actual motion of the machine tool axes. The volumetric position errors are acquired by multiplying the homogeneous transformation matrices together, essentially creating a single transformation matrix between the spindle and table, and taking only the displacement component

6 (x) ε z δ z (x) δ y (x) δ x (x) (x) ε y ε x (x) Fig.4 Six component errors of a linear motion axis As previously mentioned, this research will focus on the 2D case, so the above equations can be reduced. This reduction results in Eq. 1, which contains only positioning and straightness errors in the X and Y directions, as well as the non-perpendicularity error between the X and Y axes, and the yaw error of the X axis. F 2D u = v ( x, y) ( x, y) δ x = ( x) + δ x ( y) + y α xy y ε z ( x) δ ( x) + δ ( y) y y (1) δ, δ (y) : positioning error δ (x), δ (y) :straightness error ε (x) where (x) error x α xy y : non-perpendicularity error. y x z :yaw In the reduced 2D model, there are still 6 component error functions involved in the full error profile. In practice, it is very difficult and tedious to measure each of these, especially as it necessary to update this information as an error behavior changes over time. Instead, it is desirable to find a function that equivalently models the error in a much more concise and simple way. 3.2 Newly proposed compensation method for machine tool motion trajectory error. One of the great advantages of using a two dimensional scale system is to enable the direct observation of the two dimensional motion of the machine tool motion during the machining if the two dimensional scale is attached on the back side of the two dimensional motion table. Since the NC program usually defines the desirable trajectory of the motion, an error of motion can be defined by comparing the actual motion trajectory directly measured through a two dimensional scale with the ideal trajectory defined by the NC program. 1004

7 Based on the concept mentioned above, the new representation method of an error of the motion has been introduced such that an error can consistently be defined for any motion trajectory. The independent variable for the error function is the position along the trajectory. For the general case, this is the distance that has been traversed along the path. The error is defined as being normal to the path, so the nominal position for any measured point is the closest point on the nominally defined trajectory. This is, essentially, the contour error of the machine trajectory. The error, then, is the distance between the nominal position and the measured position, and the sign of the error is negative to the left of the nominal trajectory, and positive to the right, as seen from the nominal position looking forward along the path. For an example, if the trajectory is a strictly circular trajectory, the independent variable for the error function is the angle of the position about the center point and the error is identical to the radial error, but the sign is defined as negative if the measured radius is smaller than the nominal radius, and positive if it is greater. For a counter-clockwise motion, the error is then consistent with the general trajectory error definition, but the sign is reversed for a clockwise motion. In this study, the B-Spline function has been selected to represent the error function. The general for of the B-Spline function can be represented as: f ( x) = n i= 0 [ x ] where s, x e i Ni, p(( x xs) /( xe xs)) Vi are the minimum and maximum value of the i th segment of the B-Spline, V ( ) are ordinates (sampled data points, corresponding to the B-Spline segments), U = { u, u } are basis functions defined on a knot vector,..., 0 1 u n + p+ 1 (5) N i, p u, and p is the degree of basis functions. The knot vector is a sort of parametric parameter that corresponds with the x coordinates. Compared with nominal movement range, the V i magnitude of ordinates are very small, thus uniform B-Spline basis functions are adequate, and the knot vector is chosen as 0 0 i p u i = ( i p) /( n + 1 p) p + 1 i n 1 n + 1 i n + p + 1 (6) Basis functions are defined recursively as N i,0 1 if ui u < ui ( u) = 0 otherwise u u u 1 u i i+ p+ Ni, p( u) = Ni, p 1( u) + Ni+ 1, p 1( u) u u u u + 1 i+ p i i+ p+ 1 i+ 1 (7) 1005

8 The benefit of the proposed B-spline function based error expression is that the error can be defined as a one-dimensional function in reference to the position along the trajectory path. (instead of two dimensional coordinate based expressions.) The component error functions can be acquired from the repetitively measured data by considering the statistical distribution of the residual errors [15]. Fig.5 shows an example of extracting the repeatable error function from the repetitively measured straightness error data on a high precision machine. Fig.6 shows the statistical distribution of the residual errors. Fig.5 Extraction of component error function by least squares fitting Fig.6 Distribution of residual errors for the extraction of component error function Because of the consistent definition of the error, the B-Spline error map can easily be decomposed into X and Y components for every nominal position of the trajectory. For all measurements, the repeatable error is represented by a magnitude and direction from the nominal position along the normal, which allows for that easy decomposition. When all the error values are decomposed, this is equivalent to two functions with X and Y as the inputs, and the X and Y components of the error each as the outputs. Essentially, this is the same effect as what is given in Eq. 5, but with only 2 error functions (one for each axis error component), rather than 6 component error functions. Therefore, the end result is just as accurate but simpler to use, while also being much more straightforward to calculate. 1006

9 Based on the identified error function, it is possible to continuously check if the behavior of the motion trajectory error is repeatable and the established error function is still valid for the compensation. If the result of the check indicates that the error function is not valid for the compensation, a new error function can immediately be generated based on the newest measured motion trajectory. 3.3 Error compensation by NC Code modification In order to achieve numerical error compensation for the machine tool, the nominal trajectory of the tool path can be modified based on the mathematical error map that was constructed from repeated measurements. The error compensation process resulting in compensated NC code is outline in Fig. 7. First, the position is accurately measured and compared to the nominal trajectory to create the error map as described in 3.1. Then inverse error mapping is applied to the nominal trajectory to create the compensated discrete trajectory. Inverse error mapping assumes that the change in error is negligible on the scale of the error itself (if the error is some particular value at some location on the nominal trajectory, it will be nearly the same at some location a few microns over). Based on this, the error is simply subtracted from the nominal trajectory to create the compensated trajectory; for example, if the nominal radius of a circle is 5mm, and the actual movement is measured to have a radius of 5.001mm at some angle, then the adjusted trajectory would have a 4.999mm radius at that angle. Once the compensated trajectory has been created, it is fitted to linear and circular segments that can be understood by the machine tool as NC code. This in itself requires careful consideration, because creating a segment between each discrete point on the trajectory could turn 1 line of NC code into thousands of lines, while having too few segments would not be able to fully make use of the error compensation. Thus, the trajectory is recursively split into smaller and smaller segments until the precision of the machine has been reached, and further splitting would not yield any extra precision in the machine motion. Fig.7 Flowchart of compensation G-code generation 1007

10 4. Experimental validation of error measurement and error compensation on high precision machine tools For high precision machine tools, currently both ballscrew driven and linear driven types are available on the commercial market. Generally, compared with a ballscrew driver, a linear driver has several important advantages, such as lower inertial mass, lower elasticity, zero backlash and hysteresis, and lower friction [16]. Since most of these advantages will influence the performance of the control system, the effectiveness of error compensation must be validated and compared on machine tools with different driver systems. In the experiments, a ballscrew driven machine tool and a linear driven machine tool are used; both machines have a programming precision of 0.1μm. Error measurement and error compensation are performed on the XY plane of the machine tools. Machine tool component errors are measured using the same procedures as in [13]. Fig.8 is a typical setup of the measurement system in the experiment. Fig.8 The complete measurement system installed on a machining center 1008

11 4.1 Experiments on the ballscrew driven machine Fig.9 Kinematic structure of the ballscrew driven machine Fig.9 shows the kinematic structure of the ballscrew driven machine tool used in the experiments. As described in Section 3.2, the volumetric error map on the XY plane can be established by synthesizing the component errors according to the machine structure. However, that merely provides an understanding of the source of the error in each axis. The machine motion must be analyzed and compensated for repeatable machine tool errors, as described in sections 3.1 and 3.3, to generate the modified NC code. In order to check the effectiveness of error compensation, circular tests before and after error compensation have been performed in the XY plane of the machine. The setup of the experimental system is shown in Fig

12 Fig.10 Setup of the experimental system on the ballscrew driven machine tool The testing parameters are chosen as follows: Nominal radius: 20 mm Feedrate: 200 mm/min Testing direction: counter clockwise The testing results are shown in Fig.11. It can be seen that after error compensation, the contouring accuracy has been significantly improved, although the reversal peaks at the quadrant positions of the circle remain the same. The mean values, standard deviations, and root mean square errors of the tests are shown in Table

13 Fig.11 Circular tests before and after error compensation on the ballscrew driven machine Table 1 Error characteristics before and after error compensation (Unit: μm) 1011

14 4.2 Experiments on the linear driven machine Fig.12 Kinematic structure the linear driven machine To fully evaluate the proposed error compensation method by using the developed 2D accuracy measurement system, experiments were also performed on a linear driven high precision machine tool. Fig.12 shows the kinematic structure of the linear driven machine, from which the 3D volumetric error map can be generated according to Section 3.2. The experimental setup on the linear driven high precision machine is shown in Fig.13. Fig.13 Setup of the experimental system on the linear driven machine tool 1012

15 The test parameters used are the same as those used for the circular tests on the ballscrew driven machine. The testing results are shown in Fig.14. Fig.14 Circular tests before and after error compensation on the linear driven machine The statistical information of the error analysis of the circular tests on the linear driven machine is shown in Table 2. Table 2 Error characteristics before and after error compensation (Unit: μm) After error compensation, contouring accuracy achieved an even better improvement than in the case of the ballscrew driven machine tool. In the circular tests on the ballscrew machine, high frequency dynamic error and reversal spikes are obvious. On the linear driven machine, such errors are of minor importance and not obvious. Thus, when a control system with finer precision is used, further improvement on the compensated trajectory of the linear driven machine can be expected. On the ballscrew driven machine, however, further improvement on the compensated trajectory is limited by the high frequency dynamic errors and reversal spikes. 1013

16 5. Conclusions With the developed machine tool accuracy measurement system based on a 2D holographic scale and a 2D sensor, component machine tool errors can be accurately measured for the purpose of numerical error compensation. Effective compensation G-code can be generated by using the 3D volumetric error map synthesized from various component error functions. Error measurement and error compensation have been performed on the XY plane of a ballscrew driven machine tool and a linear driven machine tool. Experimental results show that error compensation is generally effective on machine tools with both drive systems. It is also worthy to note that better results of error compensation can be expected on linear driven machines when the precision of the control system is improved. References [1] McKeown P. The Role of Precision Engineering in of the Future, Annals of the CIRP 36(2) (1987) [2] Schwenke H, Knapp W,Haitjema H, et al. Geometric Error Measurement and Compensation of Machines An Update, Annals of the CIRP 2008; 57(2): [3] Bryan J. A simple method for testing measuring machines and machine tools Part 1: Principles and applications, Precision Engineering, 4(2) (1982): [4] Schelleke P, Rosielle N, Vermeulen H, et al., Design for Precision: Current Status and Trends, Annals of the CIRP 1998; 47(2): [5] [6] [7] Ibaraki S, Hata T, Matsubara A. A new formulation of laser step-diagonal measurement two-dimensional case, Precision Engineering 33 (1) (2009) [8] Hocken R, Simpson J, Borchardt B, et al., Three dimensional metrology, Annals of the CIRP 26 (2) (1977) [9] Donmez A, Blomquist DS, Hocken R, et al., A general methodology for machine tool accuracy enhancement by error compensation, Precision Engineering 8 (4) (1986) [10] Sartori S, Zhang GX. Geometric Error Measurement and Compensation of Machines, Annals of the CIRP 1995; 44 (2): [11] Eugene Hecht, Optics, Fourth Edition, Addison Wesley, 2002; [12] Zhu W, Wang Z, Yamazaki K. Development of a Nanometer Resolution Measurement System by Using 2D Holographic Grid Scales, Proceedings of MTTRF 2009 Annual Meeting Shanghai, China, July (2009). [13] Zhang, G.X., Veale, R., Chartton, T., Borchardt, B., Hocken, R., 1985, Error compensation of co-ordinate measuring machines, Annals of the CIRP, 341: [14] Zhu W, Wang Z, Yamazaki K. Machine tool component error extraction and error compensation by incorporating statistical analysis, International Journal of Machine Tools and Manufacture, 50(9), 2010, [15] Huang PS, Ni J. On-line error compensation of coordinate measuring machines. Int J Mach Tools Manufact, 1995, 35: [16] Pritschow G. A Comparison of Linear and Conventional Electromechanical Dives, Annals of the CIRP, 47(2), 1998,