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Autonomous form measurement on machining centers for free-form surfaces
International Journal of Machine Tools & Manufacture 44 (2004) 961–969 www.elsevier.com/locate/ijmactool
Autonomous form measurement on machining centers for free-form surfaces Hua Qiu a,, Hironobu Nisitani a, Akio Kubo a, Yong Yue b a
Department of Mechanical Engineering, Kyushu Sangyo University, 2-3-1 Matsukadai, Higashi-ku, Fukuoka City, 813-8503, Japan b Faculty of Creative Arts and Technologies, University of Luton, Park Square, Luton LU1 3JU, UK Received 2 November 2003; accepted 14 January 2004
Abstract This research aims at developing a measurement technique on machining centers for 3D free-form contours. An autonomous measuring principle is proposed and a prototype measuring device applicable to a machining center has been produced. In the measuring device, a laser displacement detector in a narrow range, which directly detects the distance from a point on the measured surface to the reference position of the detector output, is put together with the movable part of a linear encoder on the nut of a ball screw. A stepping motor controls the laser detector position to keep the output at the central value of the detector measuring range by driving the ball screw. Both the motor and the fixed part of the linear encoder are placed on the device base. The linear encoder detects the moving displacement of the screw nut, i.e. the position change of the laser detector. By installing the base on the spindle of a machining center and moving the table along a plane perpendicular to the spindle, the laser detector can automatically follow the contour of a work piece set on the table and measure its form along a scanning line, simultaneously. The displacement of a measured point relative to the reference position of the linear encoder output on the spindle side is just equal to the sum of the outputs of the two sensors, i.e. the laser detector and the linear encoder. Moreover, a simple experimental approach to identifying the sensing direction errors for an assembled measuring device is developed. The results of some experiments are also shown, which sufficiently demonstrate the effectiveness of the proposed inspection method and error identification approach. # 2004 Elsevier Ltd. All rights reserved. Keywords: Autonomous form measurement; Free-form surface; Machining center; Measuring device; Error identification
1. Introduction Coordinate measuring machines (CMM) are widely used for precise measurement of surface profiles in unknown forms such as clay models, and art and craft products in industrial practice. As an expensive machine, however, the CMM cannot be considered the best choice for such jobs. One of the main reasons is that the touch trigger probe, which is accepted as a touch sensor in most CMMs, results in a low efficiency in the measuring a work piece where a large quantity of points must be detected to define and evaluate the
Corresponding author. Tel.: +81-92-673-5619; fax: +81-92-6735699. E-mail address: [email protected] (H. Qiu).
0890-6955/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2004.01.012
profile form. The touch trigger probe also brings an another problem in practice, i.e. an offsetting operation is necessary for obtaining the surface profile from the measurement data. However, it is sometimes very difficult how to perform an appropriate and correct offsetting calculation for a measured profile in unknown forms, especially, in situations where there is a large number of measured points. On the other hand, as an efficient inspection method of complicated profiles, automated laser scanning measurement is a potential technique. However, because several difficult problems such as automatic planning of scanning line and measurement accuracy have not been satisfactorily solved, recent research effort has been attracted to apply the technique to inspecting profiles in a known form, for example, a contour defined by a CAD model, rather than that in unknown forms [1–3].
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With the advances of control and element technologies, in recent years, the motion accuracy of CNC machine tools has been rapidly improved. As a result, an attractive idea naturally grew. That is, if adequate and inexpensive sensors and devices can be developed together with appropriate applications of NC functions, complicated surface profiles can be precisely and efficiently inspected on a CNC type of machine tools. A method for inspecting profiles of machined parts on a machining center (MC) has been reported by Kakino et al. [4]. In the method, a displacement sensor is set up on the spindle and fed by the NC machining program so that the machining errors of the parts are detected as the deviation of the measured points relative to the cutter path in machining. Based on a similar principle, an inspection technique on CNC gear grinder for gear errors has been developed and utilized in practice [5]. Chang and Lin have developed an approach, which contains a controller to machine tool movement and a scanning probe, for inspecting contours in unknown forms on a MC [6]. Because this approach is based on the principle of open CNC apparatus, a high level of knowledge on NC technologies is required for the applications. This research aims at developing a measurement technique on machining centers for 3D free form contours. Section 2 proposes the principle of autonomous measurement and a non-contact type of prototype measuring device, in which two different types of sensors are combined with a servo motor. Section 3 presents a simple experimental approach to identifying the errors of the measuring device in sensing direction. Section 4 illustrates several examples of form measurement applying the prototype device on a MC, with results sufficiently demonstrating the effectiveness of the proposed principle and approach. Finally, Section 5 draws conclusions on the research.
2. The measuring principle and the prototype device 2.1. The principle of autonomous form measurement Some sensors such as electrical comparators and laser displacement detectors (laser styluses) are widely used for direct measurement of a point position on a contour. Such sensors, called the sensing sensor in this paper, have some advantages, for example, compact structure and high accuracy. However, a remarkable disadvantage, i.e. a narrow measuring range, exists in general. On the other hand, some sensors such as a linear encoder, called the position sensor in this paper, have a large measuring range and high accuracy. However, they cannot directly detect the position of a point. Therefore, incorporating both types of sensors with a servo motor to supplement effectively the functions
with each other, a form inspection to automatically follow the measured profile should be realizable. Based on the above consideration, a simple principle of autonomous form measurement is proposed. The principle consists of two parts as follows: 1. A sensing sensor is connected with a position sensor through a servo motor and both sensors are arranged to keep the same sensing direction. Based on the output of sensing sensor, the motor moves the sensing sensor along the sensing direction to keep the distance from the sensor to the measured point at the central value of the sensor measuring range. The position sensor detects the displacement of the sensing sensor. Therefore, the sensing sensor automatically follows the position of a measured point and the displacement from the measured point to the output reference on the position sensor is just equal to the sum of the two sensor outputs. 2. A work piece to be measured is fed at a uniform speed along parallel straight lines on a plane perpendicular to the sensing direction to complete the form inspection. On each straight line, called the scanning line, the position of a sampling point is recorded as three components of the orthogonal coordinates. That is, one in the sensing direction that is directly detected, one in the scanning direction that is decided by the scanning speed, sampling time and order of the point, and another in the perpendicular direction to the scanning line that is decided by the position of the scanning line. By applying such measuring device to a MC, an autonomous form inspection can be easily completed. That is, to install the device on the spindle of the MC making the sensing direction coincident with the spindle axis, set the work piece to be measured on the table and drive the table to perform scanning movement by a simple NC program. As a start position for data record, a reference that can be directly detected by the sensing sensor should be set on the table so that the inspection operation will be entirely independent of the NC apparatus. It is clear that the form defining the measured contour is not necessary prior to the measurement. Therefore, the proposed principle is appropriate for the inspection of a contour in unknown forms. 2.2. Outline of the prototype measuring device A prototype measuring device based on the above principle has been produced. Fig. 1 is a schematic diagram of the prototype device. The device base is fixed on the spindle of a MC by fitting a taper surface into the taper hole of the spindle. A ball screw that is directly driven by a stepping motor is fixed on the base. The screw axis is parallel to the spindle axis, i.e. the z
H. Qiu et al. / International Journal of Machine Tools & Manufacture 44 (2004) 961–969
Fig. 1.
Schematic diagram of the prototype measuring device.
axis. With the screw rotation, the screw nut moves along the z direction. A linear encoder is separately set, i.e. the scale at the base and the head at the screw nut. A laser displacement detector whose sensing direction is parallel to the z axis is fixed on the screw nut, too. While a measuring process is carried out, the output of the laser detector is transferred to a personal computer at each sampling time through an RS-232C interface and then the computer sends a movement command to the motor driver via a motor control board. The command is decided according to a rule to keep the distance from the measured point to the laser detector at the central value of the detector range, as shown in Fig. 2, where t0 and t are the central value of measuring range and the output value of the laser detector respectively. At the same time, the moving displacement of the screw nut is detected by the linear encoder and transferred into the computer through the encoder amplifier and a counter board.
Fig. 2.
Control block diagram of the servo motor.
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In a measuring operation, a work piece to be measured is set on the table of the MC together with a reference block. The table is moved in turn along each scanning line, which is straight line parallel to the y axis (or the x axis), by a pre-coded simple NC program, as shown in Fig. 3. Because the distance from a measured point to the laser detector is always kept near the central value of the detector measuring range, the detector can automatically follow the work piece profile and detect the point position along a scanning line. The displacement of each measured point in the z direction relative to the reference point on the block is just equal to the sum of the outputs of the two sensors, i.e. the laser detector and the linear encoder. As shown in Fig. 3, the scanning is started at point A along the y direction. Once the position of the reference block edge is detected by the laser detector, the positions of measured points in the z direction are in turn recorded at every sampling time. At the end of the scanning line, point B, the recording is stopped and the table is turned back to point A along the opposite direction. Then the table is fed to the start point of the next scanning line, point C, through a scanning line interval. The above process is repeated for each scanning line until the last one. For each measured point on a scanning line, the position in the x direction is the same as that of the scanning line, and the position in the y direction is equal to the distance from the reference block edge, i.e. scanning speed sampling time the order of the point. An incremental type of linear encoder is used in the prototype device, and its measuring range and resolution are 120 mm and 0.1 lm. A diffusion reflection type of laser displacement detector is used, and the measuring range and resolution are 30 5 mm and 0.1 lm. The mass of the sensing head is only 265 g. The diameter of the laser beam is 30 lm at the central position of the measuring range. As the manufacturer’s
Fig. 3.
Scanning process of form measurement.
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for the assembled measuring device. The approach contains two steps. The first identifies the sensing direction error for the laser detector and the second for the linear encoder. 3.1. Identification of the sensing direction error of the laser detector In the identification, the measuring device is installed on the spindle of a MC and an accurate ball with a proper radius, R, is set at a suitable position on the table, where the laser beam just shines at the highest point of the ball by a visual judgment. The position is taken as a reference and a coordinate frame fixed on the spindle is defined. The origin coincides with the ball center and each axis is respectively parallel to the corresponding one of the machine coordinate frame in the MC. In this coordinate frame, o-xyz, the laser beam can be expressed as: 8 < x ¼ a þ tsinhcos/ y ¼ b þ tsinhsin/ ð1Þ : z ¼ c þ tcosh
Fig. 4.
Photograph of the prototype measuring device.
where, t is the output of laser detector and a, b, c are the coordinates of the reference point for the output. As shown in Fig. 5, the direction error of the laser beam with respect to the z axis is defined by two parameters, h and /.
catalogue claims, the laser detector can measure a surv face with a normal angle of less than 40 relative to the laser beam. The data communication speed of the RS232C interface is 19200 bps. There is a limitation on the inclination of surfaces to be measured; however, the adoption of the laser detector can not only realized non-contact measurement but also eliminate the need for the offsetting operation to obtain the profile form. Fig. 4 is a photograph of the prototype measuring device. 3. Identification of the sensing direction errors in the measuring device From the measuring accuracy point of view, it is very important to set the sensing direction of the measuring device parallel to the theoretic direction, i.e. the z axis of the MC where the device is installed. Therefore, a key point is how to position both the laser detector and linear encoder in device assembly. It is not very difficult to set up the linear encoder because of the characteristic of the device structure. It is usually hard to set up the laser detector because no reference for positioning is provided on the detector case. In order to resolve this problem, a simple experimental approach to identifying the sensing direction errors is developed
Fig. 5. Identification method for the sensing direction errors of sensors.
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Next, the table is continually moved and stopped in a proper distance interval along plural straight lines parallel to the x axis or the y axis, when the power supply to the servo motor in the measuring device is turned off, i.e. the position of laser detector is fixed in o-xyz. At each stopping position, the output of the laser detector, ti, is recorded. After the i-th movement, the ball reached the position sketched in dot lines in Fig. 5 and the coordinates of the ball center are indicated as (xi, yi, 0). The ball surface can be expressed as: ðx xi Þ2 þ ðy yi Þ2 þ z2 ¼ R2
ð2Þ
At this position, because the shining point of the laser beam on the ball surface simultaneously satisfies both Eq. (1) and Eq. (2), the following equation can be derived. 2
ða þ ti sinhcos/ xi Þ þ ðb þ ti sinhsin/ yi Þ þ ðc þ ti cosh zi Þ2 ¼ R2
ð3Þ
Rewriting the practical value of the ball radius as R0 + DR, where R0 is the normal size and DR is the size deviation, the deviation in the radial direction at the i-th recorded point, di, can be obtained as follows: di ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða þ ti sinhcos/ xi Þ2 þ ðb þ ti sinhsin/ yi Þ2 þ ðc þ ti cosh zi Þ2 ðR0 DRÞ ð4Þ
The deviation square sum for all n recorded points, S, is given as follows: S¼
n X
d2i
ð5Þ
i¼1
Using the least square method to solve Eq. (5), the values of parameters, h,/,a,b,c and DR, can be obtained. Therefore, by applying the identified values of h and / to correct the output, it is possible to remove the effect of the sensing direction error of the laser detector from the measurement result. 3.2. Identification of the sensing direction error of the linear encoder After completing the identification of the sensing direction error for the laser detector, a similar process is applied to identifying the sensing direction error of the linear encoder. In the process, only one point is different from those described in the previous subsection, i.e. the servo motor in the measuring device is driven to keep the output of the laser detector at the central value of the measuring range. As shown in Fig. 5, an accurate ball with a radius of R is set on the table and a coordinate frame, o-xyz, is defined in the same way with the previous subsection. In o-xyz, the sensing direction of the linear encoder, as
a straight line, can be expressed as follows: 8 < x ¼ d þ lsinacosb y ¼ e þ lsinasinb : z ¼ f þ lcosa
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ð6Þ
where, l is the output of the linear encoder, and d, e, f are the coordinates of the reference point for the encoder output. The sensing direction error with respect to the z axis is defined by two parameters, a and b. At the i-th stopping position, the shining point of laser beam on the ball surface satisfies the following equation. ðd þ li sinacosb þ ti sinhcos/ xi Þ2 þ ðe þ li sinasinb þ ti sinhsin/ yi Þ2 þ ðf þ li cosa þ ti cosh zi Þ2 ¼ R2
ð7Þ
In Eq. (7), the identified values for h and / should be used. The radial deviation at the i-th recorded point can be given by the following equation. di ¼ fðd þ li sinacosb þ ti sinhcos/ xi Þ2 þ ðe þ li sinasinb þ ti sinhsin/ yi Þ2 þ ðf þ li cosa þ ti cosh zi Þ2 g1=2 ðR0 þ DRÞ
ð8Þ
Therefore, using the least square algorithm to minimize the deviation square sum for all recorded points, the values of parameters, a, b, d, e, f and DR, can be obtained. If the identified values of a and b are applied to correcting the output, it is possible to remove the effect of the sensing direction error of the linear encoder from the measurement result. The above explanation uses an expression of ‘‘moving-stopping’’ for the table movement. Obviously, a scanning movement, as described in subsection 2.2, is also valid for completing the identifying operation. Moreover, instead of a MC, a more precise controllable X-Y stage should be a better choice for the identifying operation in order to improve the identification accuracy. It should be noticed that if the same setting standard is accepted in an assembled measuring device for both operations of form measurement and identification, once the identified error values is coded into the measuring program of the device, the compensation of the error effects can be automatically performed for the form measurement results. 4. Results of measurement experiments and discussions In this section, the prototype measuring device was used in the identification experiment and the measurement of a contour profile in unknown forms on a Washino WMC-4 machining center with a vertical spindle.
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4.1. Experiments for identifying the sensing direction errors of the sensors In identification experiments, a SUJ2 steel bearing ball with a diameter of 50 mm was used. According to the manufacturer’s catalogue, the diameter tolerance is 15 lm, the sphericity is 10 lm and the surface roughness is Ra 0.2 lm (Max). Considering the limitation of laser detector on the inclination of the surface to be measured, the ball was covered by a S45C steel lid so that only part of surface, where the normal angle of a v point is within 45 with respect to the z axis, was exposed as shown in Fig. 6. A step with a height of 2 mm on the lid was taken as the position reference for recording the measured point along a scanning line in the experiments. The scanning direction was the y direction and the experimental conditions included a sampling time of 10 ms for motor control, a scanning speed of 60 mm/min along the y direction, an interval of 1 mm for either the scanning lines or recorded points. An example of scanning result is shown in Fig. 6.
Fig. 6.
Test ball and its measurement result.
In the experiments, all points on the ball surface were detected without any sensing trouble of the laser detector. However, a trend was observed that the radial v deviations at the points with a normal angle over 40 relative to the z axis are remarkably larger than other points. Therefore, only 805 points, where the normal v angle is within 40 , were used in identification calcuv lation. The result was obtained as h¼ 2:146 ; / ¼ v 92:421 and R¼ 24:9910 mm for the laser detector, and v v a¼ 0:134 ; b¼ 209:864 and R¼ 24:9917 mm for the linear encoder. Although the experiments and calculations for the laser detector and the linear encoder were independently performed, the identified values of R agreed very well with each other. After compensating the effects of the sensing direction errors for the measurement results, the radial error at each recorded point with a normal angle less than v 40 was calculated. The maximum difference among these errors is 26.78 lm. Part of the result is illustrated in Fig. 7, where an error is presented as the difference with respect to the average radius, i.e. 24.9917 mm. As a comparison, the same part of the ball surface, where v the normal angle is within 40 with respect to the z axis, was also measured with a Mitsutoyo BHN 506 CMM under the same sampling condition, i.e. the interval of points of 1 1 mm. Part of the result is illustrated in Fig. 8. The least square radius is 24.9934 mm with the maximal radial difference of 12.96 lm. Comparing Fig. 7 with Fig. 8, two opposite trends can be observed in the former. One is the error increase
Fig. 7. device.
Radial errors of the test ball by the prototype measuring
H. Qiu et al. / International Journal of Machine Tools & Manufacture 44 (2004) 961–969
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Fig. 8. Radial errors of the test ball by a CMM.
from left side to right side, for example, in the scanning lines of x¼ 14 mm; x¼ 3 mm; or x¼ 7 mm, and the another is the error decrease from the left to the right, for example, in the scanning lines of x¼ 0 mm; x¼ 14 mm. The reason causing the phenomenon can be considered as the sensing error to the position reference, i.e. the lid edge. Because of the limitation of the RS-232C interface on data communication speed, the sampling time for sensing the lid edge with the laser detector could not be set less than 10 ms in the prototype measuring device. It resulted in a theoretic maximum sensing error of 10 lm for the reference. The sensing error resulted in the inclination of radial errors along a scanning line being different from those along other scanning lines. On the other hand, it can be also observed in Fig. 7 that the difference between some neighbor points exceeds 10 lm. The value is remarkably larger than that in Fig. 8. The following experiment was performed to confirm the reason. An accurate block gauge with a surface roughness of Rmax 0.03 lm was set on the table of the MC and its surface to be measured was perpendicular to the laser beam. On turning off the power supply of the MC, the distance from the laser detector to the block surface was detected and recorded in a sampling time of 10 ms for 10 s. The result is shown in Fig. 9a. Then a scanning measurement without controlling the servo motor in the measuring device, i.e. the power supply of the servo motor was turned off, was performed along the y direction at a feed speed of 60 mm/min and a sampling time of 10 ms. The result is shown in Fig. 9b. Next, a scanning measurement with the motor control was done under the same conditions and the distance data were recorded as the sum of the outputs from the laser detector and linear encoder of the measuring device. The result is shown in Fig. 9c. From Fig. 9, it can be concluded that the precision of the laser detector itself to detecting a point is about 1 lm; the precision in a scanning measurement is
Fig. 9.
Measurement results of the block gauge surface.
nearly 10 lm; the increase of scanning error is not caused by the servo motor movement of the measuring device since no remarkable difference is observed from both the results with or without the servo motor control. In consideration of the actual condition of the MC, which has been used in technical education courses of metal cutting and NC programming for 13 years with some remarkable motion errors confirmed [7–9], the error increase resulted probably from the effects of the vibration and motion error in the driving system of the MC table. This is also assumed the main cause to the undulation of radial errors in Fig. 7. Based on the above discussions, it can be concluded that the developed identification method is practical and efficient although some factors affecting the identification accuracy existed. Moreover, it is obvious that more accurate identification results can be expected if a more modern MC or a precise controllable X-Y stage is used in the identification experiment. Furthermore, it is
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also concluded that the inspection accuracy of the prototype measuring device is about 20–30 lm under a suitable measuring condition. This is a satisfactory result for most measurements of clay models, and art and craft products. 4.2. Inspection of a craft product surface A plaster Hakata doll for ornamenting wall, whose surface is colorfully drawn as shown in Fig. 10, was inspected by the prototype measuring device with the MC used in the identification experiment. The height in the sensing direction was about 18 mm. A cardboard with a step of 1.5 mm was used in the measurement. The sampling time was 10 ms for motor control. The scanning direction was the y direction with a feed speed of 120 mm/min. Both intervals for scanning lines and recorded points were 2 mm. The measurement result is illustrated in Fig. 10. Some local steep areas with a normal angle larger than v 60 relative to the z axis exist along the circumference edge or inside the spot circled in Fig. 10. The laser detector failed to sense some points belonging to the steep areas and a special value was recorded. In drawing the measurement result, less than 30 of such failed points, were removed from a total of 12,075 measured points, and the front and rear points were directly connected to each failed point. By examining on screen along each scanning line in detail, it has been verified
that the measured results were satisfactory except for a few spots around a removed point. From the above experiment results, it can be concluded that the proposed autonomous form measurement method and the identification method for sensing direction errors in the device are effective and practical. At the same time, two aspects to further improve the performance of the prototype measuring device are also confirmed. That is to introduce an independent sensor for detecting the reference position in a scanning line to improve the precision and efficiency of inspection, and to accept (or develop) a new displacement sensor with less limitation than the present laser detector for the surface measurement. 5. Conclusions The results of this paper can be summarized as follows: 1. A new autonomous measuring principle for free form surfaces on a machining center has been proposed. The fundamental ideas include combining two displacement sensors with a servo motor, keeping the position of the sensor to directly measure surface at the central value of the sensor measuring range by controlling the motor rotation, and performing scanning motion with NC functions of the machining center. Both the control rule and the NC programming are very simple. Based on the principle, a non-contact prototype measuring device has been developed. 2. A practical and efficient experimental approach to identifying the sensing direction errors of the sensors has been developed for an assembled measuring device. 3. Experimental results with the prototype measuring device have sufficiently demonstrated the effectiveness of the proposed principle and identification approach. The results also provide a clear clue for further improvement on the performance of the prototype measuring device. 4. Although the principle of autonomous form measuring has been proposed based on the premise of application to a machining center in this paper, the principle and the corresponding measuring device are also applicable to other types of NC machine tools. References
Fig. 10. A Hakata doll and its measurement result.
[1] A. Bernard, M. Ve´ron, Analysis and validation of 3D laser sensor scanning process, Annals of the CIRP 48(1) (1999) 111–114. [2] F. Xi, C. Shu, CAD-based path planning for 3-D line laser scanning, Computer-Aided Design 31 (1999) 473–479.
H. Qiu et al. / International Journal of Machine Tools & Manufacture 44 (2004) 961–969 [3] S. Son, H. Park, K.H. Lee, Automated laser scanning system for reverse engineering and inspection, International Journal of Machine Tools and Manufacture 42 (2002) 889–897. [4] Y. Kakino, Y. Iwata, Y. Iwasaki, H. Otsubo, Study on a deviation detection type on-the-machine measurement of machined profile, Journal of the Japan Society for Precision Engineering 58(6) (1992) 1059–1064 (in Japanese). [5] H. Qiu, M. Hashitani, H. Nakano, S. Inui, J. Bosaka, A basic analysis of inspecting cylindrical gear accuracy on machine tool with a touch-trigger probe, Transactions of the Japan Society of Mechanical Engineers 65-C 631 (1999) 1148–1155 (in Japanese). [6] M. Chang, P.P. Lin, On-line free form surface measurement via a fuzzy-logic controlled scanning probe, International Journal of Machine Tools and Manufacture 39(4) (1999) 537–552.
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[7] H. Qiu, Y. Li, Y.-B. Li, A new method and device for motion accuracy measurement of NC machine tools. Part 1: Principle and equipment, International Journal of Machine Tools and Manufacture 41(4) (2001) 521–534. [8] H. Qiu, Y.-B. Li, Y. Li, A new method and device for motion accuracy measurement of NC machine tools. Part 2: Device error identification and trajectory measurement of general planar motions, International Journal of Machine Tools and Manufacture 41(4) (2001) 535–554. [9] H. Qiu, H. Nisitani, A. Kubo, J. Yamamoto, I. Hirakawa, Examinations on motion accuracy evaluation based on the ball bar test for a machining center, Bulletin of the Faculty of Engineering Kyushu Sangyo University 40 (2003) (in Japanese, in press).
Autonomous form measurement on machining centers for free-form surfaces Hua Qiu a,, Hironobu Nisitani a, Akio Kubo a, Yong Yue b a
Department of Mechanical Engineering, Kyushu Sangyo University, 2-3-1 Matsukadai, Higashi-ku, Fukuoka City, 813-8503, Japan b Faculty of Creative Arts and Technologies, University of Luton, Park Square, Luton LU1 3JU, UK Received 2 November 2003; accepted 14 January 2004
Abstract This research aims at developing a measurement technique on machining centers for 3D free-form contours. An autonomous measuring principle is proposed and a prototype measuring device applicable to a machining center has been produced. In the measuring device, a laser displacement detector in a narrow range, which directly detects the distance from a point on the measured surface to the reference position of the detector output, is put together with the movable part of a linear encoder on the nut of a ball screw. A stepping motor controls the laser detector position to keep the output at the central value of the detector measuring range by driving the ball screw. Both the motor and the fixed part of the linear encoder are placed on the device base. The linear encoder detects the moving displacement of the screw nut, i.e. the position change of the laser detector. By installing the base on the spindle of a machining center and moving the table along a plane perpendicular to the spindle, the laser detector can automatically follow the contour of a work piece set on the table and measure its form along a scanning line, simultaneously. The displacement of a measured point relative to the reference position of the linear encoder output on the spindle side is just equal to the sum of the outputs of the two sensors, i.e. the laser detector and the linear encoder. Moreover, a simple experimental approach to identifying the sensing direction errors for an assembled measuring device is developed. The results of some experiments are also shown, which sufficiently demonstrate the effectiveness of the proposed inspection method and error identification approach. # 2004 Elsevier Ltd. All rights reserved. Keywords: Autonomous form measurement; Free-form surface; Machining center; Measuring device; Error identification
1. Introduction Coordinate measuring machines (CMM) are widely used for precise measurement of surface profiles in unknown forms such as clay models, and art and craft products in industrial practice. As an expensive machine, however, the CMM cannot be considered the best choice for such jobs. One of the main reasons is that the touch trigger probe, which is accepted as a touch sensor in most CMMs, results in a low efficiency in the measuring a work piece where a large quantity of points must be detected to define and evaluate the
Corresponding author. Tel.: +81-92-673-5619; fax: +81-92-6735699. E-mail address: [email protected] (H. Qiu).
0890-6955/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2004.01.012
profile form. The touch trigger probe also brings an another problem in practice, i.e. an offsetting operation is necessary for obtaining the surface profile from the measurement data. However, it is sometimes very difficult how to perform an appropriate and correct offsetting calculation for a measured profile in unknown forms, especially, in situations where there is a large number of measured points. On the other hand, as an efficient inspection method of complicated profiles, automated laser scanning measurement is a potential technique. However, because several difficult problems such as automatic planning of scanning line and measurement accuracy have not been satisfactorily solved, recent research effort has been attracted to apply the technique to inspecting profiles in a known form, for example, a contour defined by a CAD model, rather than that in unknown forms [1–3].
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With the advances of control and element technologies, in recent years, the motion accuracy of CNC machine tools has been rapidly improved. As a result, an attractive idea naturally grew. That is, if adequate and inexpensive sensors and devices can be developed together with appropriate applications of NC functions, complicated surface profiles can be precisely and efficiently inspected on a CNC type of machine tools. A method for inspecting profiles of machined parts on a machining center (MC) has been reported by Kakino et al. [4]. In the method, a displacement sensor is set up on the spindle and fed by the NC machining program so that the machining errors of the parts are detected as the deviation of the measured points relative to the cutter path in machining. Based on a similar principle, an inspection technique on CNC gear grinder for gear errors has been developed and utilized in practice [5]. Chang and Lin have developed an approach, which contains a controller to machine tool movement and a scanning probe, for inspecting contours in unknown forms on a MC [6]. Because this approach is based on the principle of open CNC apparatus, a high level of knowledge on NC technologies is required for the applications. This research aims at developing a measurement technique on machining centers for 3D free form contours. Section 2 proposes the principle of autonomous measurement and a non-contact type of prototype measuring device, in which two different types of sensors are combined with a servo motor. Section 3 presents a simple experimental approach to identifying the errors of the measuring device in sensing direction. Section 4 illustrates several examples of form measurement applying the prototype device on a MC, with results sufficiently demonstrating the effectiveness of the proposed principle and approach. Finally, Section 5 draws conclusions on the research.
2. The measuring principle and the prototype device 2.1. The principle of autonomous form measurement Some sensors such as electrical comparators and laser displacement detectors (laser styluses) are widely used for direct measurement of a point position on a contour. Such sensors, called the sensing sensor in this paper, have some advantages, for example, compact structure and high accuracy. However, a remarkable disadvantage, i.e. a narrow measuring range, exists in general. On the other hand, some sensors such as a linear encoder, called the position sensor in this paper, have a large measuring range and high accuracy. However, they cannot directly detect the position of a point. Therefore, incorporating both types of sensors with a servo motor to supplement effectively the functions
with each other, a form inspection to automatically follow the measured profile should be realizable. Based on the above consideration, a simple principle of autonomous form measurement is proposed. The principle consists of two parts as follows: 1. A sensing sensor is connected with a position sensor through a servo motor and both sensors are arranged to keep the same sensing direction. Based on the output of sensing sensor, the motor moves the sensing sensor along the sensing direction to keep the distance from the sensor to the measured point at the central value of the sensor measuring range. The position sensor detects the displacement of the sensing sensor. Therefore, the sensing sensor automatically follows the position of a measured point and the displacement from the measured point to the output reference on the position sensor is just equal to the sum of the two sensor outputs. 2. A work piece to be measured is fed at a uniform speed along parallel straight lines on a plane perpendicular to the sensing direction to complete the form inspection. On each straight line, called the scanning line, the position of a sampling point is recorded as three components of the orthogonal coordinates. That is, one in the sensing direction that is directly detected, one in the scanning direction that is decided by the scanning speed, sampling time and order of the point, and another in the perpendicular direction to the scanning line that is decided by the position of the scanning line. By applying such measuring device to a MC, an autonomous form inspection can be easily completed. That is, to install the device on the spindle of the MC making the sensing direction coincident with the spindle axis, set the work piece to be measured on the table and drive the table to perform scanning movement by a simple NC program. As a start position for data record, a reference that can be directly detected by the sensing sensor should be set on the table so that the inspection operation will be entirely independent of the NC apparatus. It is clear that the form defining the measured contour is not necessary prior to the measurement. Therefore, the proposed principle is appropriate for the inspection of a contour in unknown forms. 2.2. Outline of the prototype measuring device A prototype measuring device based on the above principle has been produced. Fig. 1 is a schematic diagram of the prototype device. The device base is fixed on the spindle of a MC by fitting a taper surface into the taper hole of the spindle. A ball screw that is directly driven by a stepping motor is fixed on the base. The screw axis is parallel to the spindle axis, i.e. the z
H. Qiu et al. / International Journal of Machine Tools & Manufacture 44 (2004) 961–969
Fig. 1.
Schematic diagram of the prototype measuring device.
axis. With the screw rotation, the screw nut moves along the z direction. A linear encoder is separately set, i.e. the scale at the base and the head at the screw nut. A laser displacement detector whose sensing direction is parallel to the z axis is fixed on the screw nut, too. While a measuring process is carried out, the output of the laser detector is transferred to a personal computer at each sampling time through an RS-232C interface and then the computer sends a movement command to the motor driver via a motor control board. The command is decided according to a rule to keep the distance from the measured point to the laser detector at the central value of the detector range, as shown in Fig. 2, where t0 and t are the central value of measuring range and the output value of the laser detector respectively. At the same time, the moving displacement of the screw nut is detected by the linear encoder and transferred into the computer through the encoder amplifier and a counter board.
Fig. 2.
Control block diagram of the servo motor.
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In a measuring operation, a work piece to be measured is set on the table of the MC together with a reference block. The table is moved in turn along each scanning line, which is straight line parallel to the y axis (or the x axis), by a pre-coded simple NC program, as shown in Fig. 3. Because the distance from a measured point to the laser detector is always kept near the central value of the detector measuring range, the detector can automatically follow the work piece profile and detect the point position along a scanning line. The displacement of each measured point in the z direction relative to the reference point on the block is just equal to the sum of the outputs of the two sensors, i.e. the laser detector and the linear encoder. As shown in Fig. 3, the scanning is started at point A along the y direction. Once the position of the reference block edge is detected by the laser detector, the positions of measured points in the z direction are in turn recorded at every sampling time. At the end of the scanning line, point B, the recording is stopped and the table is turned back to point A along the opposite direction. Then the table is fed to the start point of the next scanning line, point C, through a scanning line interval. The above process is repeated for each scanning line until the last one. For each measured point on a scanning line, the position in the x direction is the same as that of the scanning line, and the position in the y direction is equal to the distance from the reference block edge, i.e. scanning speed sampling time the order of the point. An incremental type of linear encoder is used in the prototype device, and its measuring range and resolution are 120 mm and 0.1 lm. A diffusion reflection type of laser displacement detector is used, and the measuring range and resolution are 30 5 mm and 0.1 lm. The mass of the sensing head is only 265 g. The diameter of the laser beam is 30 lm at the central position of the measuring range. As the manufacturer’s
Fig. 3.
Scanning process of form measurement.
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for the assembled measuring device. The approach contains two steps. The first identifies the sensing direction error for the laser detector and the second for the linear encoder. 3.1. Identification of the sensing direction error of the laser detector In the identification, the measuring device is installed on the spindle of a MC and an accurate ball with a proper radius, R, is set at a suitable position on the table, where the laser beam just shines at the highest point of the ball by a visual judgment. The position is taken as a reference and a coordinate frame fixed on the spindle is defined. The origin coincides with the ball center and each axis is respectively parallel to the corresponding one of the machine coordinate frame in the MC. In this coordinate frame, o-xyz, the laser beam can be expressed as: 8 < x ¼ a þ tsinhcos/ y ¼ b þ tsinhsin/ ð1Þ : z ¼ c þ tcosh
Fig. 4.
Photograph of the prototype measuring device.
where, t is the output of laser detector and a, b, c are the coordinates of the reference point for the output. As shown in Fig. 5, the direction error of the laser beam with respect to the z axis is defined by two parameters, h and /.
catalogue claims, the laser detector can measure a surv face with a normal angle of less than 40 relative to the laser beam. The data communication speed of the RS232C interface is 19200 bps. There is a limitation on the inclination of surfaces to be measured; however, the adoption of the laser detector can not only realized non-contact measurement but also eliminate the need for the offsetting operation to obtain the profile form. Fig. 4 is a photograph of the prototype measuring device. 3. Identification of the sensing direction errors in the measuring device From the measuring accuracy point of view, it is very important to set the sensing direction of the measuring device parallel to the theoretic direction, i.e. the z axis of the MC where the device is installed. Therefore, a key point is how to position both the laser detector and linear encoder in device assembly. It is not very difficult to set up the linear encoder because of the characteristic of the device structure. It is usually hard to set up the laser detector because no reference for positioning is provided on the detector case. In order to resolve this problem, a simple experimental approach to identifying the sensing direction errors is developed
Fig. 5. Identification method for the sensing direction errors of sensors.
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Next, the table is continually moved and stopped in a proper distance interval along plural straight lines parallel to the x axis or the y axis, when the power supply to the servo motor in the measuring device is turned off, i.e. the position of laser detector is fixed in o-xyz. At each stopping position, the output of the laser detector, ti, is recorded. After the i-th movement, the ball reached the position sketched in dot lines in Fig. 5 and the coordinates of the ball center are indicated as (xi, yi, 0). The ball surface can be expressed as: ðx xi Þ2 þ ðy yi Þ2 þ z2 ¼ R2
ð2Þ
At this position, because the shining point of the laser beam on the ball surface simultaneously satisfies both Eq. (1) and Eq. (2), the following equation can be derived. 2
ða þ ti sinhcos/ xi Þ þ ðb þ ti sinhsin/ yi Þ þ ðc þ ti cosh zi Þ2 ¼ R2
ð3Þ
Rewriting the practical value of the ball radius as R0 + DR, where R0 is the normal size and DR is the size deviation, the deviation in the radial direction at the i-th recorded point, di, can be obtained as follows: di ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða þ ti sinhcos/ xi Þ2 þ ðb þ ti sinhsin/ yi Þ2 þ ðc þ ti cosh zi Þ2 ðR0 DRÞ ð4Þ
The deviation square sum for all n recorded points, S, is given as follows: S¼
n X
d2i
ð5Þ
i¼1
Using the least square method to solve Eq. (5), the values of parameters, h,/,a,b,c and DR, can be obtained. Therefore, by applying the identified values of h and / to correct the output, it is possible to remove the effect of the sensing direction error of the laser detector from the measurement result. 3.2. Identification of the sensing direction error of the linear encoder After completing the identification of the sensing direction error for the laser detector, a similar process is applied to identifying the sensing direction error of the linear encoder. In the process, only one point is different from those described in the previous subsection, i.e. the servo motor in the measuring device is driven to keep the output of the laser detector at the central value of the measuring range. As shown in Fig. 5, an accurate ball with a radius of R is set on the table and a coordinate frame, o-xyz, is defined in the same way with the previous subsection. In o-xyz, the sensing direction of the linear encoder, as
a straight line, can be expressed as follows: 8 < x ¼ d þ lsinacosb y ¼ e þ lsinasinb : z ¼ f þ lcosa
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ð6Þ
where, l is the output of the linear encoder, and d, e, f are the coordinates of the reference point for the encoder output. The sensing direction error with respect to the z axis is defined by two parameters, a and b. At the i-th stopping position, the shining point of laser beam on the ball surface satisfies the following equation. ðd þ li sinacosb þ ti sinhcos/ xi Þ2 þ ðe þ li sinasinb þ ti sinhsin/ yi Þ2 þ ðf þ li cosa þ ti cosh zi Þ2 ¼ R2
ð7Þ
In Eq. (7), the identified values for h and / should be used. The radial deviation at the i-th recorded point can be given by the following equation. di ¼ fðd þ li sinacosb þ ti sinhcos/ xi Þ2 þ ðe þ li sinasinb þ ti sinhsin/ yi Þ2 þ ðf þ li cosa þ ti cosh zi Þ2 g1=2 ðR0 þ DRÞ
ð8Þ
Therefore, using the least square algorithm to minimize the deviation square sum for all recorded points, the values of parameters, a, b, d, e, f and DR, can be obtained. If the identified values of a and b are applied to correcting the output, it is possible to remove the effect of the sensing direction error of the linear encoder from the measurement result. The above explanation uses an expression of ‘‘moving-stopping’’ for the table movement. Obviously, a scanning movement, as described in subsection 2.2, is also valid for completing the identifying operation. Moreover, instead of a MC, a more precise controllable X-Y stage should be a better choice for the identifying operation in order to improve the identification accuracy. It should be noticed that if the same setting standard is accepted in an assembled measuring device for both operations of form measurement and identification, once the identified error values is coded into the measuring program of the device, the compensation of the error effects can be automatically performed for the form measurement results. 4. Results of measurement experiments and discussions In this section, the prototype measuring device was used in the identification experiment and the measurement of a contour profile in unknown forms on a Washino WMC-4 machining center with a vertical spindle.
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4.1. Experiments for identifying the sensing direction errors of the sensors In identification experiments, a SUJ2 steel bearing ball with a diameter of 50 mm was used. According to the manufacturer’s catalogue, the diameter tolerance is 15 lm, the sphericity is 10 lm and the surface roughness is Ra 0.2 lm (Max). Considering the limitation of laser detector on the inclination of the surface to be measured, the ball was covered by a S45C steel lid so that only part of surface, where the normal angle of a v point is within 45 with respect to the z axis, was exposed as shown in Fig. 6. A step with a height of 2 mm on the lid was taken as the position reference for recording the measured point along a scanning line in the experiments. The scanning direction was the y direction and the experimental conditions included a sampling time of 10 ms for motor control, a scanning speed of 60 mm/min along the y direction, an interval of 1 mm for either the scanning lines or recorded points. An example of scanning result is shown in Fig. 6.
Fig. 6.
Test ball and its measurement result.
In the experiments, all points on the ball surface were detected without any sensing trouble of the laser detector. However, a trend was observed that the radial v deviations at the points with a normal angle over 40 relative to the z axis are remarkably larger than other points. Therefore, only 805 points, where the normal v angle is within 40 , were used in identification calcuv lation. The result was obtained as h¼ 2:146 ; / ¼ v 92:421 and R¼ 24:9910 mm for the laser detector, and v v a¼ 0:134 ; b¼ 209:864 and R¼ 24:9917 mm for the linear encoder. Although the experiments and calculations for the laser detector and the linear encoder were independently performed, the identified values of R agreed very well with each other. After compensating the effects of the sensing direction errors for the measurement results, the radial error at each recorded point with a normal angle less than v 40 was calculated. The maximum difference among these errors is 26.78 lm. Part of the result is illustrated in Fig. 7, where an error is presented as the difference with respect to the average radius, i.e. 24.9917 mm. As a comparison, the same part of the ball surface, where v the normal angle is within 40 with respect to the z axis, was also measured with a Mitsutoyo BHN 506 CMM under the same sampling condition, i.e. the interval of points of 1 1 mm. Part of the result is illustrated in Fig. 8. The least square radius is 24.9934 mm with the maximal radial difference of 12.96 lm. Comparing Fig. 7 with Fig. 8, two opposite trends can be observed in the former. One is the error increase
Fig. 7. device.
Radial errors of the test ball by the prototype measuring
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Fig. 8. Radial errors of the test ball by a CMM.
from left side to right side, for example, in the scanning lines of x¼ 14 mm; x¼ 3 mm; or x¼ 7 mm, and the another is the error decrease from the left to the right, for example, in the scanning lines of x¼ 0 mm; x¼ 14 mm. The reason causing the phenomenon can be considered as the sensing error to the position reference, i.e. the lid edge. Because of the limitation of the RS-232C interface on data communication speed, the sampling time for sensing the lid edge with the laser detector could not be set less than 10 ms in the prototype measuring device. It resulted in a theoretic maximum sensing error of 10 lm for the reference. The sensing error resulted in the inclination of radial errors along a scanning line being different from those along other scanning lines. On the other hand, it can be also observed in Fig. 7 that the difference between some neighbor points exceeds 10 lm. The value is remarkably larger than that in Fig. 8. The following experiment was performed to confirm the reason. An accurate block gauge with a surface roughness of Rmax 0.03 lm was set on the table of the MC and its surface to be measured was perpendicular to the laser beam. On turning off the power supply of the MC, the distance from the laser detector to the block surface was detected and recorded in a sampling time of 10 ms for 10 s. The result is shown in Fig. 9a. Then a scanning measurement without controlling the servo motor in the measuring device, i.e. the power supply of the servo motor was turned off, was performed along the y direction at a feed speed of 60 mm/min and a sampling time of 10 ms. The result is shown in Fig. 9b. Next, a scanning measurement with the motor control was done under the same conditions and the distance data were recorded as the sum of the outputs from the laser detector and linear encoder of the measuring device. The result is shown in Fig. 9c. From Fig. 9, it can be concluded that the precision of the laser detector itself to detecting a point is about 1 lm; the precision in a scanning measurement is
Fig. 9.
Measurement results of the block gauge surface.
nearly 10 lm; the increase of scanning error is not caused by the servo motor movement of the measuring device since no remarkable difference is observed from both the results with or without the servo motor control. In consideration of the actual condition of the MC, which has been used in technical education courses of metal cutting and NC programming for 13 years with some remarkable motion errors confirmed [7–9], the error increase resulted probably from the effects of the vibration and motion error in the driving system of the MC table. This is also assumed the main cause to the undulation of radial errors in Fig. 7. Based on the above discussions, it can be concluded that the developed identification method is practical and efficient although some factors affecting the identification accuracy existed. Moreover, it is obvious that more accurate identification results can be expected if a more modern MC or a precise controllable X-Y stage is used in the identification experiment. Furthermore, it is
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also concluded that the inspection accuracy of the prototype measuring device is about 20–30 lm under a suitable measuring condition. This is a satisfactory result for most measurements of clay models, and art and craft products. 4.2. Inspection of a craft product surface A plaster Hakata doll for ornamenting wall, whose surface is colorfully drawn as shown in Fig. 10, was inspected by the prototype measuring device with the MC used in the identification experiment. The height in the sensing direction was about 18 mm. A cardboard with a step of 1.5 mm was used in the measurement. The sampling time was 10 ms for motor control. The scanning direction was the y direction with a feed speed of 120 mm/min. Both intervals for scanning lines and recorded points were 2 mm. The measurement result is illustrated in Fig. 10. Some local steep areas with a normal angle larger than v 60 relative to the z axis exist along the circumference edge or inside the spot circled in Fig. 10. The laser detector failed to sense some points belonging to the steep areas and a special value was recorded. In drawing the measurement result, less than 30 of such failed points, were removed from a total of 12,075 measured points, and the front and rear points were directly connected to each failed point. By examining on screen along each scanning line in detail, it has been verified
that the measured results were satisfactory except for a few spots around a removed point. From the above experiment results, it can be concluded that the proposed autonomous form measurement method and the identification method for sensing direction errors in the device are effective and practical. At the same time, two aspects to further improve the performance of the prototype measuring device are also confirmed. That is to introduce an independent sensor for detecting the reference position in a scanning line to improve the precision and efficiency of inspection, and to accept (or develop) a new displacement sensor with less limitation than the present laser detector for the surface measurement. 5. Conclusions The results of this paper can be summarized as follows: 1. A new autonomous measuring principle for free form surfaces on a machining center has been proposed. The fundamental ideas include combining two displacement sensors with a servo motor, keeping the position of the sensor to directly measure surface at the central value of the sensor measuring range by controlling the motor rotation, and performing scanning motion with NC functions of the machining center. Both the control rule and the NC programming are very simple. Based on the principle, a non-contact prototype measuring device has been developed. 2. A practical and efficient experimental approach to identifying the sensing direction errors of the sensors has been developed for an assembled measuring device. 3. Experimental results with the prototype measuring device have sufficiently demonstrated the effectiveness of the proposed principle and identification approach. The results also provide a clear clue for further improvement on the performance of the prototype measuring device. 4. Although the principle of autonomous form measuring has been proposed based on the premise of application to a machining center in this paper, the principle and the corresponding measuring device are also applicable to other types of NC machine tools. References
Fig. 10. A Hakata doll and its measurement result.
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H. Qiu et al. / International Journal of Machine Tools & Manufacture 44 (2004) 961–969 [3] S. Son, H. Park, K.H. Lee, Automated laser scanning system for reverse engineering and inspection, International Journal of Machine Tools and Manufacture 42 (2002) 889–897. [4] Y. Kakino, Y. Iwata, Y. Iwasaki, H. Otsubo, Study on a deviation detection type on-the-machine measurement of machined profile, Journal of the Japan Society for Precision Engineering 58(6) (1992) 1059–1064 (in Japanese). [5] H. Qiu, M. Hashitani, H. Nakano, S. Inui, J. Bosaka, A basic analysis of inspecting cylindrical gear accuracy on machine tool with a touch-trigger probe, Transactions of the Japan Society of Mechanical Engineers 65-C 631 (1999) 1148–1155 (in Japanese). [6] M. Chang, P.P. Lin, On-line free form surface measurement via a fuzzy-logic controlled scanning probe, International Journal of Machine Tools and Manufacture 39(4) (1999) 537–552.
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[7] H. Qiu, Y. Li, Y.-B. Li, A new method and device for motion accuracy measurement of NC machine tools. Part 1: Principle and equipment, International Journal of Machine Tools and Manufacture 41(4) (2001) 521–534. [8] H. Qiu, Y.-B. Li, Y. Li, A new method and device for motion accuracy measurement of NC machine tools. Part 2: Device error identification and trajectory measurement of general planar motions, International Journal of Machine Tools and Manufacture 41(4) (2001) 535–554. [9] H. Qiu, H. Nisitani, A. Kubo, J. Yamamoto, I. Hirakawa, Examinations on motion accuracy evaluation based on the ball bar test for a machining center, Bulletin of the Faculty of Engineering Kyushu Sangyo University 40 (2003) (in Japanese, in press).