Journal Pre-proof Simultaneous geometric error identification of rotary axis and tool setting in an ultraprecision 5-axis machine tool using on-machine measurement Sangjin Maeng, Sangkee Min PII:
To appear in:
Received Date: 28 October 2019 Revised Date:
17 January 2020
Accepted Date: 30 January 2020
Please cite this article as: Maeng S, Min S, Simultaneous geometric error identification of rotary axis and tool setting in an ultra-precision 5-axis machine tool using on-machine measurement, Precision Engineering (2020), doi: https://doi.org/10.1016/j.precisioneng.2020.01.007. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Inc.
Simultaneous geometric error identification of rotary axis and tool setting in an ultra-precision 5-axis machine tool using on-machine measurement
Sangjin Maeng1 and Sangkee Min1,* 1
Department of Mechanical Engineering, University of Wisconsin, Madison, WI, 53706,
Corresponding author: [email protected]
This paper presents a method to identify the position independent geometric errors of rotary axis and tool setting simultaneously using on-machine measurement. Reducing geometric errors of an ultra-precision five-axis machine tool is a key to improve machining accuracy. Five-axis machines are more complicated and less rigid than three axis machine tools, which leads to inevitable geometric errors of the rotary axis. Position deviation in the process of installing a tool on the rotary axis magnifies the machining error. Moreover, an ultra-precision machine tool, which is capable of machining part within sub-micrometer accuracy, is relatively more sensitive to the errors than a conventional machine tool. To improve machining performance, the error components must be identified and compensated. While previous approaches have only measured and identified the geometric errors on the rotary axis without considering errors induced in tool setting, this study identifies the geometric errors of the rotary axis and tool setting. The error components are calculated from a geometric error model. The model presents the error components in a function of tool position and angle of the rotary axis. An approach using on-machine measurement is proposed to measure the tool position in the range of 10s nm. Simulation is conducted to check the sensitivity of the method to noise. The model is validated through experiments. Uncertainty analysis is also presented to validate the confidence of the error identification.
Keywords: Error identification; Position independent geometric errors; Touch probe measurement, Uncertainty analysis
1. Introduction Geometric errors on the axis and induced by tool setting are inevitable on machine tools. The error degrades the form accuracy of machined parts. Especially, the error on an ultra-precision machine tool, which has machining accuracy under one µm, generates “relatively” worse machining quality than conventional machining tools. To improve the accuracy of machining, error sources have to be analyzed and compensated . The geometric errors of a rotary axis in the five-axis machine tools can be categorized as position independent geometric errors (PIGEs) and position dependent geometric errors (PDGEs). Defects in the assembling process cause PIGEs while imperfections of machine parts, which has a different value with respect to position, result in PDGEs . Many researchers have worked to identify PIGEs and PDGEs of a conventional machine tool. An analytical approach for crucial PIGE and PDGE identification was proposed for a multiaxis machine tool based on multibody system theory and global sensitivity analysis. The approach overcame the limitation of local sensitivity analysis and identified significant error components with the systematic approach . Ding et al. suggested two error models; “Error first model” and “Motion first model” which have different modeling sequences for identifying PDGEs of a rotary axis . They observed the excessive difference between geometric errors identified with the two error models. Hong et al. machined a cone frustum to verify the influence of PDGEs on the circularity error . They highlighted that the enlargement of periodic pure radial error motion by the gravity could be a critical error factor on the circularity. PIGEs including two linear errors and two squareness errors were simulated and evaluated by a method using double ball bar (DBB) [6, 7]. The method has
an advantage of reducing the measurement uncertainty by minimizing the number of controlled axes and using an adjusting fixture. However, all the models they proposed have not considered the geometric error components in tool setting. For measuring the geometric errors, several metrologies have been introduced. Schwenke et al. reviewed direct measurement methodologies in the machine tool to measure and compensate the geometric errors . DBB measurement which is one of the direct measurement methods is widely used to measure the geometric errors on a rotary axis in the machine tools [6, 7, 9–12]. However, this approach is not applicable to measure the geometric errors induced from the tool setting because the DBB cannot be installed on the tool holder after the tool setting. On-machine measurement (OMM) based on the touchtrigger probe installed on a tool holder of the machine tool is also used to measure the geometric errors of rotary axes [13, 14]. However, the precision of the probe sensor is not suitable to an ultra-precision machine tool. A laser tracker is used to track three points and identifies geometric errors on a rotary axis in the range of 50 µm in conventional machine tools . However, the laser tracker with the maximum deviation of 0.80 µm and the repeatability of 0.33 µm cannot be easily installed and applied to identify geometric errors in ultra-precision machine tools. A capacitive sensor was used to measure squareness and straightness errors of the three prismatic axes simultaneously at the precision range. However, the approach is limited to prismatic axes and cannot be applied to identify geometric errors of the rotary axis . The advantage of the study is to simultaneously identify geometric errors of a rotary axis and tool setting, while previous studies have established error models of the rotary axis excluding the tool setting. In addition, the geometric errors are accurately identified using a 4
high precision OMM. The proposed methodology will identify the geometric errors through a geometric error model. The model will present the position and orientation errors of the rotary axis and the tool setting in a function of tool position and angle of the rotary axis. An approach to precisely measure the tool position in 10s nm in precision with an OMM will be introduced. Its experimental validation on a commercial ultra-precision five-axis machine tool will be presented. 2. PIGEs and model 2.1. Machine configuration of 5-axis machine tool An ultra-precision five-axis machine tool (ROBONANO α-0iB, FANUC Corporation) used in this research has the configuration of RPPPR as seen in Figure 1. The configuration of RPPPR or RPPR is widely used in ultra-precision machine tools as listed in Table 1. “R” and “P” stand for rotary and prismatic axis, respectively. B-axis is mounted on X-axis and C-axis is attached on Y-axis that is laid on Z-axis. Many ultra-precision machine tools offer the flexibility of processing by allowing replacement of process modules, but it introduces geometric errors. The machine tool used in the study can perform scribing, milling and turning process by replacing modules. Conversely, the amount of errors is adjustable and the effect of the error compensation on machining can be clearly observed. Hence, systematic approach to identify, assess, and compensate errors is developed. Milling module is selected for validating the error identification. The module is mounted on C-axis and a workpiece is fixed on B-axis.
Table 1 Industrial ultra-precision machine tools
Command resolution [nm]
Nanotech 650 FG
Based on the NACHI machine catalog
Figure 1. Schematic diagram of the five-axis machine tool (RPPPR Configuration)
2.2. PIGE components
Prismatic and rotary joints in five-axis machine tools have six error parameters individually . According to Ref [2, 6], the position and orientation errors of axis average line and error motions of axis defined in ISO230-1 and -7 [23, 24] are categorized into PIGEs and PDGEs, respectively.. Causes of PDGEs are bearing error motion and structural error motion while a defect in the assembly process results in PIGEs . The axis average line of a rotary axis represents the mean position and orientation of a rotary axis over its full rotation. Its position and orientation errors from its nominal location are considered as PIGEs. The PIGEs are, by definition, constant regardless of angular position of a rotary axis. As it is assumed that the error motions from C-axis average line is sufficiently small, the PDGEs of C-axis are ignored in this study. The geometric errors of X-, Y-, Z-, and B-axis would be small enough to be ignored because the range of movement in these axes was marginal. The errors in tool setting process and tool holder installation on a rotary axis can also be PIGEs because both are the assembly process and have constant error components with respect to the angular position of the rotary axis. As the tool holder and the tool setting are coupled without rotation matrix, error components of the tool holder and the tool setting can be merged into a tool frame. There are ten error components to represent geometric errors from a tool frame and a rotary axis frame. The PIGEs are described in Figure 2 and listed in Table 2. Once a rotary axis ideally rotates, an arbitrary position, P1 on the rotary axis will rotate along origin O1,2 and reach to P2 as shown in Figure 2 (a). However, as PIGEs of the axis are inevitable, the
position, P1 and O1 would move to P’2 and O’2, respectively. If a tool are perfectly assembled on the rotary axis, the relative tool tip position, O1 on the rotary axis should correspond to the tool length. Considering the PIGEs introduced in tool setting, the tool tip position, O1 would be oriented and deviated to O’1. Four errors of the ten error components are introduced by C-axis. Two errors in orientation in A- and B-axis direction and two errors in position in X- and Y-axis direction with respect to the nominal position of C-axis exist whenever a machine artifact is installed on C-axis. Zero position error of tool, represents how the tool frame rotates from the zero angular position of C-axis. Error in the position of tool in Z-axis direction,
indicates how the tool frame deviates from the
nominal position of C-axis in Z-axis direction.
Figure 2. PIGEs of (a) the rotary axis and (b) the tool setting Table 2 Symbols for PIGEs of rotary axis C and tool Frame C axis
Description Error in the position of C-axis average line in X-
axis direction Error in the position of C-axis average line in Yaxis direction ( (
Error in the orientation of C-axis average line in the A-axis direction Error in the orientation of C-axis average line in the B-axis direction
Error in the position of tool in X-axis direction
Error in the position of tool in Y-axis direction
Error in the position of tool in Z-axis direction
Error in the orientation of tool in A-axis direction
Error in the orientation of tool in B-axis direction
Zero position error of tool
2.3. Kinematic modeling of the rotary axis and tool setting
The kinematic model of the rotary axis frame and tool frame is defined on C-axis coordinate system, which is fixed to Y-axis frame. Many researchers have established the kinematic model of a rotary axis with the well-defined error matrix [2, 25, 26] as the homogeneous matrix form. Homogeneous transformation matrix of C-axis frame and tool frame with the PIGEs is represented in Eq. (1). ( )=
(1) ( ) are position error matrix, rotation error matrix, and rotation
matrix from C-axis to the tool frame, and
are position error matrix and
rotation error matrix from the tool frame to the tool position, respectively. The error matrices and rotation matrix are represented in Eqs. (2) - (6). 1 0 = 0 0
1 0 = 0 0
0 1 0 0
0 0 1 0
!" 0 0
0 1 0 0
0 0 1 0
0 0 1 0
+ %+ &+ 1
0 0 0 1
0 0 0 1
0 0 0 1
3. Measurement and Error identification 3.1. Measuring procedure
Precision OMM based on touch probe is utilized to measure a tool position. The OMM device (NanoChecker, FANUC Corporation) is capable of measuring the position of an object with the precision of 30 nm and in range of 5 mm. A body connected with a
stylus in the OMM floats on air to decrease friction. Compressed air pushes the stylus to a tool and is controlled to minimize elastic deformation by contact force. Once a ruby ball attached on edge of the stylus contacts the tool, a laser scale embedded on the body starts to measure the displacement. The surface profile of the tool can be measured with the parallel movement to the direction in which the stylus pushes. Profile of the tool nose can be probed by moving the stylus in parallel direction to tool nose as shown in Figure 3 (b). Scanning the tool in flank angle direction can allow measuring the profile of a tool in flank angle direction. Probing, therefore, in tool nose and flank angle directions can generate a tool shape in three dimensions.
Figure 3. (a) Scanning direction of ruby stylus on surface of the tool and (b) measurement procedure with OMM As four error components of C-axis frame require four or more simultaneous equations to identify the defined PIGEs, three positions measured at three different angles of C-axis are required at least. Comparison of the position in X-, Y-, and Z-axis measured at two different angles can generate three equations. If the positions were measured at three
different angles, the number of equations would be nine. The geometric errors of the tool frame is a function of the tool position and the error components of C-axis. In this paper, the tool positions are measured at the angle of C-axis,
from -40 to 40 degrees in
increments of 10 degree. The milling module and the OMM are mounted on C-axis plate and the B-axis plate, respectively as shown in Figure 4. Measurement procedure is listed as follows: 1. Set up the reference of the measurement device and calibrate parallelism of the device to C-axis plate: A hemisphere ball on a reference bar mounted on C-axis is scanned with the OMM. Center of rotation of C-axis and parallelism can be calculated from the measured profile. The highest point of the profile data is defined as the center of C-axis which becomes the origin of X-, Y-, and Z-axis. Parallelism between the measurement device and C-axis plate can be measured by comparing radius of the reference ball with respect to the scanning position. 2. Surface profile of a tool is measured at
= 0, where
is arbitrary position because
C-axis is symmetric. All angles measured in the paper are relative position to the initial point,
Shape of the tool determines scanning direction to prevent the ruby stylus
from deviating from the surface of the tool or being overloaded by a sharp edge of the tool. In case of a ball end milling tool, the probe moves in direction of tool nose and flank angle sequentially as shown on Figure 3 (b). Surface profile of the tool in three dimensions can be more accurately generated by combining the profiles scanned in vertical and horizontal direction.
3. Measure surface profile of the tool for different angles with
With rotating C-axis
plate from -40° to 40° in increment of 10°, the second step is iterated as shown in Figure 3 (a).
Figure 4. Setup to measure the tool profiles; tool is installed on C-axis and the OMM is mounted on B-axis
3.2. Offset calibration of ruby ball stylus on OMM
Once the OMM measures surface of a tool, the gathered data is distorted. Contact points of the ruby ball and the surface of the tool are apart from measured data points by the radius of the ruby ball in tangential direction as shown in Figure 5 (a). The distortion varies depending on the contact angle of the tool, thereby the confidence of error identification is not reliable. To compensate the distortion, the measured surface profile of the tool can be calibrated by 0 1 !" ,* * * 1 ) = +- / = ) + + 0 / * 0 !" 1 .
where )* , P, r, θ, and φ are the measured surface profile, actual surface profile, the radius of the ruby ball, azimuthal angle and polar angles, respectively. The normal angles of the measured point can be calculated from adjacent points to it.
Ruby ball on end of stylus
Figure 5. (a) Offset between the measured object and data gathered with ruby ball stylus and (b) geometric parameter of ruby ball for calibrating the offset
3.3. Algorithm to identify PIGEs
The proposed algorithm identifies ten PIGEs listed in Table 2 with the actual surface profiles of the tool. The simultaneous equations to identify the errors in terms of angle of C-axis and the tool position are solved. The tool positions measured at different angles of C-axis are associated with the relative positions on the tool and is represented by ,2, 45 -2, 45 .2, 45 = 1
,2 -2 ( ' ) . 2 1
means rotation angle of C-axis frame and (,2, 45 , -2, 45 , .2, 45 ) is the position vector
of )2 at the rotation angle,
Position vector, (,2 , -2 , .2 ) is the relative position from the
reference point on the tool surface, )( . Note that the highest point of the tool is defined as the reference point, )( and value of the points, ,( , -( , and .( is zero. When C-axis rotates value
( ( = 0°,
= 20°...), the error in position of
the tool can be calculated by Eqs. (10) and (11) where points of the relative position on the
tool, (,( , -( , .( ) is (0, 0, 0). The relationship between the position error and the tool position can be drawn from Eq. (9) as follows:
,(, 48 − ,(, 49 -(, 48 − -(, 49 : .(, 48 − .(, 49 = 1 = =
( 6) −
0 0 ( ( ); 0 1
<=9, >8 ?=9, >9 @ 89 A8 ?B9, >9 @C89 C89 8 A 89 8
?<=9, >8 ?=9, >9 @C89 A8 ?B9, >9 @ 89 C89 8 A 89 8
where DEF = !"
F , GEF
Orientation errors of C-axis frame are associated with the measured profile data for
three different rotation angles and position errors of the tool frame.
derived by using Eqs. (9) and (12) and presented in Eqs. (13) and (14): ,(, 4H − ,(, 49 -(, 4H − -(, 49 : .(, 4H − .(, 49 = 1 = J(K
( 7) −
0 0 ( ( ); 0 1
H ?I9, >9 @J(KLM AN)C89 ?KOM 89 P?8 ?I9, >9 @J(KLM AN)CH9 ?KOM H9 P
LM AN) H9 AKOM CH9 PJ(KLM AN)C89 ?KOM 89 P?J(KLM AN) 89 AKOM C89 PJ(KLM AN)CH9 ?KOM H9 P
H ?I9, >9 @J(KLM AN) 89 AKOM C89 P?8 ?I9, >9 @J(KLM AN) H9 AKOM CH9 P LM AN)CH9 ?KOM H9 PJ(KLM AN) 89 AKOM C89 P?J(KLM AN)C89 ?KOM 89 PJ(KLM AN) H9 AKOM CH9 P
Errors in position of C-axis average line and error in position of the tool in z-axis direction mean how the frame deviates from the reference of C-axis plate. The reference was defined by procedure 1 in section 3.2 or the absolute origin of C-axis frame that the
manufacturer set up. Thus, the measured profile vector, (,(, 49 , -(, 49 , .(, 49 ) has to contain relative position data from the reference. The errors can be identified as follows:
= ,(, 49 + Q !"
= .(, 49 − (Q
= -(, 49 − Q ) !"
!" − (Q
Three orientation errors of the tool frame can be drawn by combining Eqs. (18) and
(19) where position vector, (,( , -( , .( ) is the highest point of the tool and position vectors, (,6 , -6 , .6 ) and (,7 , -7 , .7 ) are both ends of the tool. Defined with the measured position data, the different measured angles, and orientation errors on C-axis, simultaneous
equations of the orientation on the tool frame errors are presented on Eq. (20). Note that relationship of X- and Y-axis and orientation errors of the tool frame is only linked to orientation errors of C-axis frame, while one of Z-axis and the errors are linked to orientation errors of C-axis frame and position errors of the tool frame. Thus, three measured positions in X- and Y-axis direction are selected to identify the orientation errors of the tool frame by isolating the uncertainty from C-axis movement. ,6, 49 − ,(, 49 + 6, 49 − -(, 49 / = .6, 49 − .(, 49 ,7, 49 − ,(, 49 +-7, 49 − -(, 49 / = .7, 49 − .(, 49
,6 − ,( ( ( ) +-6 − -( / .6 − .(
,7 − ,( ( ( ) +-7 − -( / .7 − .(
/ = R?( S
-6 R = + -7 −-6
−,6 −,7 ,6
−,6 !" −,7 !" ,6
,6, 49 − ,(, 49 − ,6 S = T,7, 49 − ,(, 49 − ,7 -6, 49 − -(, 49 − -6 !"
− -6 ( − -7 ( − -6 !" (
+ ,( !" ( + ,( !" ( − -( (
4. Simulation and experimental verification 4.1. Simulation
The simulation was conducted to verify how sensitive to noise the proposed algorithm is. The error identification is simulated with the tool position generated from arbitrary errors and noise. The arbitrary error components are PIGEs of C-axis and tool setting, and values are listed in Table 3. Range of noise for each error components was determined based on the uncertainty contributors of the machine tool as listed in Table 7. The procedure of simulation analogous to the one presented in  is presented in Figure 6 and listed as follows: 1. Create the arbitrary PIGEs of C-axis and the tool frame. 2. Generate the position data of tool from noise in a range of uncertainty contributors of the machine tools used in the experiment and the generated PIGEs with the kinematic model of the machine tool.
3. Calculate contributors of the geometric error from the position data generated in the previous step by using the proposed model. The tool position can be generated from the kinematic model derived in Eqs. (1) – (6) with simulation parameter. Radius, length, and tolerance condition of a tool used in the simulation is 500 µm, 5 mm, ±1 µm, respectively. Deviation of the calculated tool positions from ideal position are shown in Figure 7 with respect to angle in 15° increment from 0° to 360°. The data indicates how geometric errors affect the deviation from the ideal line in X-, Y-, and Z-axis direction. The initial and the estimated PIGEs are compared in Table 3. The proposed method can be considered reliable to identify PIGEs since the difference values of position and orientation errors are within 0.5 µm and 0.7 µrad, respectively. Discrepancy between initial and estimated value of
is relatively larger than others.
This is caused by the errors correlated to other error components. For instance, component is calculated by Eq. (17) where orientation errors of C-axis frame and position errors of the tool frame are multiplied and summed. The multiplication and summation magnify the discrepancy of
Modelling Arbitrary error components
Generate tool position
Configuration of a machine tool
Establish kinematic model
Establish inversekinematic model
Compute the error components
Calculate required correction Generate compensated NC code
Figure 6. The flow chart of the simulation and experiment
Measure tool position
Figure 7. Calculated deviation of a tool with respect to angle for simulation: (a) total deviation, (b) deviation in X-axis direction, (c) deviation in Y-axis direction, and (d) deviation in Z-axis direction. Table 3. Initial value and result of the simulation Frame C-axis
Estimated value Discrepancy 23.29 -48.14
6.28 42.38 -66.47 12.08
6.25 42.44 -66.41 11.46
-0.03 0.06 0.06 -0.62
A series of experiments were conducted to validate the proposed method on the ultraprecision five-axis machine tool with the precision OMM device. A ball end mill was used with nose radius of 1.5133 µm. Material and radius of the stylus on the measurement system is ruby and 500.433 µm with sphericity of 0.13 µm. The range of rotation angle of C-axis is given from -40° to 40° in increment of 10° to prevent the stylus of the measurement from being damaged. Three points on the tool (the top and both ends of the tool) were measured at nine angles (
= -40°, -30°, -20° …, 40°), sequentially and the
measurement data is listed in Table 4. Deviation of the measured positions from the ideal position is shown in Figure 8 with respect to angle of C-axis. Position errors of C-axis and tool frame are dominant to affect the deviation in X- and Y-axis, whereas orientation error influences deviation in Z-axis direction. Position errors not only shift the center of deviation in X- and Y-axis direction but also enlarge radius of the deviation. Thus, the deviation in three directions can estimate the effect of the PIGEs on measured profiles. In the experiment, the deviations in X- and Y-
axis direction are relatively larger than the deviation in Z-axis direction, which means position errors dominantly influences total deviation rather than orientation error. Comparison of the deviation of the tool with and without compensation on Figure 8 supports the validation of the approach. The largest value of total deviation and deviation in X-, Y-, and Z-axis direction with compensation is 13.18, 1.19, 1.16, and -13.10 µm, respectively. Deviation in three axis directions is reduced from about 200 µm to under 14 µm after compensating the geometric errors. The deviation of Z-axis mainly influences the
total deviation, which means the error estimation associated to Z-axis is not accurately identified. Uncertainty of the error estimation in Z-axis is relatively larger than X-axis and Y-axis as the error components in Z-axis is computed from the other components. For instance, uncertainties of
includes uncertainty of
The contributors of the PIGEs are listed in Table 5 with measurement uncertainty. Note that -10°, 0°, and 10° are chosen as three different angles required to calculate the PIGE because combination of the position data at the three angles among nine angles has the smallest uncertainty. Figure 8 shows that the errors in position of the tool mainly result in the deviation of tool positioning. Since the machine tool is designed to switch milling, turning, and scribing module on B- or C-axis plate, the tool frame may induce relatively larger geometric errors than ones of rotary axis.
Table 4. Tool position measurement result; P1 is the top of the tool and P2 and P3 are both ends of the tool. Angle
0.172287 10.090681 -0.112978 -0.191130
9.966671 -0.102549 -1.874879
9.535798 -0.196686 -1.152143
9.539975 -0.089476 -3.501553
9.064808 -0.183606 -2.813458
8.823555 -0.074157 -5.021728
8.318505 -0.168453 -4.389182
7.839182 -0.057059 -6.389214
7.319564 -0.151686 -5.831435
Figure 8. Measured deviation of a tool with OMM: (a) total deviation, (b) deviation in Xaxis direction, (c) deviation in Y-axis direction, and (d) deviation in Z-axis direction.
Table 5. Experiment result and uncertainty Frame
4.3 Sine wave machining
Freeform surface machining is effective to verify the proposed error identification and compensation since form accuracy of the workpiece directly reflects the performance of the five-axis machine tools. As the proposed method focuses on PIGEs of C-axis and tool setting, the form accuracy evaluation of a structure machined with movements in C-axis and linear axes validates how the method corrects the error components well. Machining sine wave shape requires a tool path with movement in X-axis, Y-axis, and C-axis. Sine wave shape with 6 mm in pitch, 300 µm in amplitude, and 2 mm in length was machined as a free-form machining example. Machining parameters are presented in Table 6. NC code is generated with consideration of C-axis to maintain the same contact angle between the diamond tool and the workpiece with respect to the slope of the sine wave. The tool path
allows validating the effect of the geometric errors on X-, Y-, and C-axis simultaneously. As shown in Figure 6, the required correction is calculated based on the identified PIGEs and volumetric error compensation was implemented to NC-code for the sine wave. Figure 9 (a) presents the profile of the part machined with and without compensation. The machined part without the error compensation shows asymmetric shape and uneven slope with machining accuracy of 29.6 µm. The errors in position of C-axis and tool, , tool,
, and ,
induce the asymmetric shape and the errors in orientation of C-axis and ,
make the uneven slope in machining of the freeform. As the
tool holder (milling, turning, or scribing module) is switched for purpose, the errors in position of tool would become relatively large. Therefore, the errors in position and orientation of C-axis average line and tool should be compensated in ultra-precision machining. After the error compensation, form accuracy of the freeform surface is improved to 0.8 µm.
Table 6. Machining parameters for the sine wave form Parameters
Materials of the workpiece
Material of the tool
Type of the tool
Ball end mill
Nose radius of the tool [mm]
Spindle speed [rpm]
Figure 9. (a) Surface profile of the sine wave machined with and without compensation and (b) machined surface with the compensation in cross section 5. Uncertainty Analysis Finding standard uncertainty is important task to verify the confidence of measurement procedure in the experiment. Tool position measurement procedure requires linear movement in X-, Y-, and Z- direction. The movement generates inevitable uncertainty that degrades the reliability of the measurement. Many studies have been 27
verifying confidence of error identification results with uncertainty analysis [7, 27–29]. Standard uncertainties of measured data in X-, Y-, and Z- axis direction are defined by ISO 230-9  and presented in Eqs. (21) – (23).
6 6 V= = WVXYZ['E\ ]^^_,= + VX\Y`ab\F\E]
6 6 VB = WVXYZ['E\ ]^^_,B + VX\Y`ab\F\E]
6 6 VI = WVXYZ['E\ ]^^_,I + VX\Y`ab\F\E]
where VXYZ['E\ ]^^_ and VX\Y`ab\F\E] are the uncertainties of linear axes and the
measurement device separately. Uncertainty of thermal effect is ignored in this analysis because the temperature of the room in which the machine tool is located is maintained 23±0.1° C. Errors of OMM are the common contributor for the uncertainty of the linear axes and
must be minimized to improve the reliability of the measured data. The contributor values were taken from the measurement based on ISO-230-2  and error inspection report of the machine tool provided by the manufacturer. Related to the OMM, unidirectional repeatability and sphericity of ruby ball is taken into account. As the scanned distance of the OMM is less than 0.2 mm, geometric errors would insignificantly influence the measurement. In addition, the drift of the OMM was small enough to be ignored under the maintained temperature within 23±0.1° C. The OMM was calibrated and compensated every time before scanning the tool position in order to minimize the measurement errors.
Three linear axes have five principal contributors that are repeatability, two orientation errors and two straightness errors in perpendicular directions. All principal contributors in three linear axes are considered to analyze the uncertainties from linear movement. Table 5 shows the standard uncertainty of PIGEs calculated through Eqs. (21) – (23). The uncertainty of linear axes is the summation of the correlated contributor to each axis listed in Table 7. For example, the uncertainty of X-axis is calculated from errors of OMM, repeatability of X- axis and squareness and straightness errors to X- axis of Y- and Z- axis. Standard uncertainty of the PIGEs is calculated from the measured profiles with the uncertainty of linear axes by the same process introduced in section 3.4 and listed in Table 5. Uncertainties of
have relatively larger value, since the error
components is computed from the other error components. For instance, uncertainties of includes uncertainty of
. In addition, indistinguishable tool
position measurement can cause significant uncertainty in identification of the PIGEs. As the difference between the tool positions used in identification of the errors,
are within 1 mm, it would induce uncertainty. Errors in position and orientation of Caxis average line are accurately identified from the position measurement over the full rotation of C-axis. As angular range of C-axis (from -40° to 40°) is limited in the proposed method, the uncertainty could become large. Uncertainty contributors to the OMM dominantly influence the standard uncertainty of the PIGEs. On the other hand, the contributors of linear axes would primarily increase the standard uncertainty parallel to the axes and secondarily affect the standard uncertainty perpendicular to the axes. The effect of the contributors of linear axes on the standard
uncertainty under small scanning distance would be marginal. Therefore, major sources of the standard uncertainty in the measurement are uncertainty contributors of the OMM.
Table 7. Contributors of uncertainty Part
Sphericity of stylus
Straightness error to Y-axis
Straightness error to Z-axis
Squareness error to Y-axis
Squareness error to Z-axis
Straightness error to X-axis
Straightness error to Z-axis
Squareness error to X-axis
Squareness error to Z-axis
Straightness error to X-axis
Straightness error to Y-axis
Squareness error to X-axis
Squareness error to Y-axis
6. Conclusion The new method is proposed to identify PIGEs of a rotary axis and tool setting using the precision OMM in this paper. The most significant advantage of this method is to calculate position errors and orientation errors of the tool setting and the rotary axis directly and simultaneously. While other methodologies only identify deviation of the rotary axis or ignore the error from tool setting, the proposed method can estimate geometric errors from rotary axis to tool frame, thereby it can enhance machining accuracy by compensating the error components. The method also suggests the measurement procedure with the OMM to scan the surface profile of a tool with the precision of 30 nm. Tool surface profiles were measured at different angles and distortion from geometric shape of the ruby ball of the OMM device was calibrated to improve the reliability of measured data. The mathematical model is established to identify the error components. Simulation and experiment result shows the proposed method identifies the PIGEs of rotary axis and tool frame. To reduce the effect of uncertainties on identifying the error components linked to Z-axis, improved approach with different scanning direction and geometric error models have to be considered in the future. The proposed method is limited to an application that movement in axes except for C-axis in the error measurement is small enough to be negligible. An error model of the entire machine tool system with PDGEs and PIGEs should be studied to extend the application range of the method in future work.
Acknowledgement Authors thanks Dr. Soichi Ibaraki from Department of Mechanical Systems Engineering, Hiroshima University, Japan who provided thorough check of error matrices and expertise that greatly assisted the paper and gratefully acknowledge the financial support and the donation of the ROBONANO α-0iB to MIN LAB at UW-Madison from the FANUC Corporation, Japan. Appendix Solutions for the PIGEs,
were gathered from simultaneous equation in Eq.
(9). Detail procedure is derived below: Simultaneous equations (24) and (25) can be gathered from Eq. (9). ,(, 48 − ,(, 49 =
-(, 48 − -(, 49 =
where DEF = !"
G6( − D6( +
Solving the equations,
F , GEF
is sequentially derived as Eqs. (26) and (27).
D6( <-(, 48 − -(, 49 @ + G6( <,(, 48 − ,(, 49 @ = D6( ( D6( )
D6( <-(, 48 − -(, 49 @ + G6( <,(, 48 − ,(, 49 @ = =
<=9, >8 ?=9, >9 @ 89 A8 ?B9, >9 @C89 C89 8 A 89 8
G6( ) + G6( (
(26) (27) (10)
is also sequentially derived as Eqs. (28) and (29) same as Eq. (10).
−D6( <,(, 48 − ,(, 49 @ + G6( <-(, 48 − -(, 49 @ = −D6( ( G6( (
−D6( <,(, 48 − ,(, 49 @ + G6( <-(, 48 − -(, 49 @ =
D6( ) +
?<=9, >8 ?=9, >9 @C89 A8 ?B9, >9 @ 89 C89 8 A 89 8
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The novel method is proposed to identify PIGEs (position independent geometric errors) of a rotary axis and tool setting with on-machine measurement system.
The most significant advantage of this method is to calculate offset errors and squareness errors of the tool setting and the rotary axis directly and simultaneously.
The method also suggests the measurement procedure with the on-machine measurement system using a touch probe in order to scan the surface profile of a tool installed on a machine tool.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: