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Measurement and verification of position-independent geometric errors of a five-axis machine tool using a double ball-bar
International Journal of Machine Tools & Manufacture 70 (2013) 45–52
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Measurement and verification of position-independent geometric errors of a five-axis machine tool using a double ball-bar Kwang-Il Lee, Seung-Han Yang n School of Mechanical Engineering, Kyungpook National University, 80, Daehak-ro, Buk-gu, Daegu, Republic of Korea
art ic l e i nf o
a b s t r a c t
Article history: Received 11 September 2012 Received in revised form 20 March 2013 Accepted 21 March 2013 Available online 30 March 2013
In this study, position-independent geometric errors, including offset errors and squareness errors of rotary axes of a five-axis machine tool are measured using a double ball-bar and are verified through compensation. In addition, standard uncertainties of measurement results are calculated to establish their confidence intervals. This requires two measurement paths for each rotary axis, which are involving control of single rotary axis during measurement. So, the measurement paths simplify the measurement process, and reduce measurement cost including less operator effort and measurement time. Set-up errors, which are inevitable during the installation of the balls, are modeled as constants. Their effects on the measurement results are investigated to improve the accuracy of the measurement result. A novel fixture consisting of flexure hinges and two pairs of bolts is used to minimize set-up error by adjusting the ball's position located at the tool nose. Simulation is performed to check the validation of measurement and to analyze the standard uncertainties of the measurement results. Finally, the position-independent geometric errors of the five-axis machine tool (involving a rotary axis and a trunnion axis) are measured using proposed method. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Position-independent geometric errors Rotary axis Five-axis machine tool Uncertainty analysis Double ball-bar
1. Introduction A five-axis machine tool is essential for manufacturing products with complex shapes requiring high form accuracy. Its main advantage compared to a three-axis machine tool is its capability for flexible control of the relative position and relative orientation between the work-piece and tool. However, the number of errors (i.e., geometric errors, thermally-induced errors, and dynamic errors) also increases inevitably by increasing the number of controlled axes. Positioning accuracy between the work-piece and tool decreases as a result. The geometric errors are the most important because they affect positioning accuracy during all of the running time [1,2]. Geometric errors are categorized as position-dependent geometric errors (PDGEs) and positionindependent geometric errors (PIGEs) [3]. Here, “position” refers to the command for a controlled axis. PDGEs are caused mainly by imperfections in components, and PIGEs are caused mainly during the assembly process. For linear axes, there have been many studies of the measurement of geometric errors [4–11]. However, there are only a few studies available for rotary axes [12]. A laser interferometer with a precision index table is used to measure angular positioning
n
Corresponding author. Tel.: +82 539506569; fax: +82 539506550. E-mail address: [email protected] (S.-H. Yang).
0890-6955/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmachtools.2013.03.010
error [13]. Two LVDT sensors with a precision ball, auto-collimator with a polygon mirror can be used to measure two radial errors and two tilt errors of a rotary axis with investigating set-up errors [14]. For a trunnion axis, a method involving five measurement steps with a single set-up has been developed [15]. A double ballbar has been used to measure PDGEs (except angular positioning error) and the PIGEs of a rotary axis through reverse kinematic approaches [16]. The effects of the PDGEs of a rotary axis have also been investigated [17]. To simplify the measurement process, only PIGEs are considered and these are measured by the simultaneous control of three axes (two linear axes and a rotary axis) using a double ball-bar [18]. In a similar approach, an R-test requiring a three-dimensional probe and a precision ball is available as commercial product [19,20]. However, the results suffer mainly due to the positioning variability, including positioning accuracy and repeatability, of two linear axes that are simultaneously controlled during the measurements. In this study, the PIGEs of a five-axis machine tool with a rotary axis and a trunnion axis are measured using double ball-bar. During measurement, only the single rotary axis is controlled to simplify the measurement and to increase the measurement accuracy within the positioning variability of the rotary axis. Then, the standard uncertainties of the measurement results are calculated for a confidence interval and are analyzed to find the main contributor for the improvement of measurement reliability. Set-up errors, which are inevitable in the installation of a double
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Nomenclature n u ai , ci oia , sia oic , sic ðx; y; zÞ L R U
sample number; standard uncertainty; command angle for rotary axis A, C(i¼1,…, n); offset error and squareness error in the i-direction at axis A(i¼ y, z); offset error and squareness error in the i-direction at axis C(i¼x, y); command for positioning; offset at measurement; nominal length of double ball-bar; measurement uncertainty;
ball-bar, are modeled as constant, and their effects on the measurement results are investigated. A fixture consisting of flexure hinges and two pairs of bolts is used to minimize set-up error by adjusting the ball's position at the tool nose. In Section 2, the PIGEs are briefly described and are summarized by comparison with results from other studies. The measurement paths for the PIGEs are planned and analyzed to investigate the effects of the PIGEs on the measured data of double ball-bar. The standard uncertainties of the measured PIGEs are defined by contributors. In Section 3, it describes a simulation that is performed to check the validity of measurements by analyzing standard uncertainties. Then, the PIGEs of a five-axis machine tool with a rotary axis and a trunnion axis are measured and compensated using measurement results. In Section 4, it summarizes the advantages and contributions of the proposed measurement method.
ΔRij
deviation to the nominal length at i-th measurement and j-th measured value(i¼1,…, 4; j¼1,…, n); {i} coordinate system of axis i(i¼X, Y, Z, A, C); ti ðt xi ; t yi ; t zi Þ design position of the ball at the tool nose at the i-th measurement(i¼ 1,…, 4); wi ðwxi ; wyi ; wzi Þ design position of the ball on the work-table at the i-th measurement(i¼ 1,…, 4); Δti ðΔt xi ; Δt yi ; Δt zi Þ set-up error at the ball at the tool nose at the i-th measurement(i¼ 1,…, 4); Δwi ðΔwxi ; Δwyi ; Δwzi Þ set-up error at the ball on the work-table at the i-th measurement(i ¼1,…, 4); ΔR i n 1 column vector consisting of deviation ΔRij at the i-th measurement(i¼ 1,…, 4).
2.1. Measurement paths of a double ball-Bar Four measurement paths involving control of a single rotary axis during each measurement are shown in Fig. 3. The paths are used to measure all of the PIGEs for rotary axes A and C. It assumes that the positioning accuracies of linear axes X, Y, and Z, which are used to install the balls at set-up, are within tolerances and thus will not deviate significantly from the target positions. 2.1.1. Measurement of two offset errors of rotary axis A One of the two balls is designed to be located at the position of the tool nose, which is the origin of the reference coordinate system {F} on
z y
2. Measurement and standard uncertainty
sza
{A}
As shown in Fig. 1, a five-axis machine tool typically consists of three linear axes and two rotary axes. The position and orientation of the rotary axes are deviated from the designed ones. Two offset errors and two squareness errors for each rotary axis are enough to describe such deviations, as summarized in Fig. 2 and Table 1. The effects of these PIGEs on the end-effector are modeled using an error synthesis model involving homogenous transformation matrices and their sequential multiplication [21–23].
oya
sya
oza
a
x
{F}
c
z
sxc
syc
y oxc
{C} oyc
{A} Fig. 1. Five-axis machine tool involving a rotary axis and a trunnion axis.
x
Fig. 2. PIGEs of rotary axis A and C. (a) Rotary axis A. (b) Rotary axis C.
K.-I. Lee, S.-H. Yang / International Journal of Machine Tools & Manufacture 70 (2013) 45–52
the rotation axis of the spindle. The other ball is installed at a position far from the origin as nominal length R to direction of y-axis in reference coordinate system {F}, on the work-piece table. During measurement, the ball at the tool nose is stationary and the ball on the work-piece table is rotated about the x-axis in coordinate system {A}, as shown in Fig. 4. Consequently, the measured data of double ball-bar is the distances between two balls and deviated from nominal length R by the disagreement between the ball's position at tool nose and the origin of coordinate system {A}, which is the offset errors. At installation, a fixture called a tool cup is used to fix the ball at the tool nose for measurement, as shown in Fig. 5. However, the tool cup is tilted by the fastening process at the spindle. The position of the ball attached at the tool cup deviates, which is modeled as set-up error from the rotation axis of the spindle. In addition, the position of the ball on the work-piece table deviates from the design position by the set-up error of the ball at the tool nose and by the fastening process using screw. Therefore, the relationship between the deviation of the measured data, the offset errors and the set-up errors is defined as follows: 2
3
2
⋮ ⋮ 6 ΔR 7 6 cosa 4 1i 5 ¼ 4 i ⋮ ⋮
⋮ sinai ⋮
32
−oya þ Δt y1
ΔR1 ¼ A1 x1
PIGEs
Ball on work-piece table
Actual path
Initial set-up
R+ΔR1
⋮ 6 7 17 54 −oza þ Δt z1 5 −Δwy1 ⋮
YOA ZOA BOA COA XOC YOC AOC BOC
a
3
{A}
Δw1 R
(oya, oza)
z
{F}
At ISO oya oza sya sza oxc oyc sxc syc
ð1Þ
Two offset errors combined with set-up error Δt1 are derived by applying the least squares method to Eq. (1). It is difficult to separate only offset errors from Eq. (1) due to singular problem, so it is necessary to minimize the set-up error Δt1 to improve measurement accuracy. However, the set-up error Δw1 only affects to the relative radius of the measured data on the polar plot, and does not affect the offset errors.
Table 1 Symbols to define geometric errors for rotary axes. Presented
47
y
Δt1
Ball at tool nose
Location errors
Nominal path
Fig. 4. The effect of offset errors and set-up errors at first measurement.
Fig. 3. Measurement paths for PIGEs of rotary axis A and C. (a) First measurement for offset errors of rotary axis A. (b) Second measurement for squareness errors of rotary axis A. (c) Third measurement for offset errors of rotary axis C. (d) Fourth measurement for squareness errors of rotary axis C.
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first measurement. Then, offset error oya is combined with offset error oyc because the ball at the tool nose is designed to locate at the origin of reference coordinate system {F}, and not the origin of coordinate system {A}. To calculate the offset error oyc , it is essential to subtract offset error oya , which is derived using Eq. (1), from the solution of the following equation. Thus, 2 3 2 32 3 oxc −Δt x1 ⋮ ⋮ ⋮ ⋮ 6 ΔR 7 6 cosc sinc 1 76 o þ o −Δt 7 yc 4 3i 5 ¼ 4 54 ya y1 5 i i Δwx3 ⋮ ⋮ ⋮ ⋮ ΔR 3 ¼ A3 x3
Fig. 5. Set-up errors at the installation of a double ball-bar.
2.1.2. Measurement of two squareness errors of rotary axis A It measures two squareness errors of rotary axis A. The ball at the tool nose is moved by offset L using linear axis X in the direction of the x-axis in reference coordinate system {F} without reinstallation. The ball on the work-piece table is positioned with offset L to the x-axis by reinstallation, causing set-up error Δw2 . During measurement, the ball at the tool nose is also stationary, and the ball on the work-piece table is rotated about rotary axis A, not the x-axis in reference coordinate system {F}. Two squareness errors affect the measured data by multiplied with offset L. In this case, the relation between the deviation of the measured data, squareness errors and set-up errors is given by Eq. (2). By subtracting Eq. (1) from Eq. (2), two squareness errors (which are not affected by set-up errors) are calculated using Eq. (3). Thus, 2 3 2 32 3 −oya −Lsza þ Δt y1 ⋮ ⋮ ⋮ ⋮ 6 ΔR 7 6 cosa sina 1 76 −oza þ Lsya þ Δt 7 z1 5 4 2i 5 ¼ 4 54 i i −Δwy2 ⋮ ⋮ ⋮ ⋮ ΔR2 ¼ A2 x2 2
3 2 ⋮ ⋮ 6 ΔR −ΔR 7 6 cosa ¼ 4 2i 4 1i 5 i ⋮ ⋮ ΔR2 −ΔR 1 ¼ A2 x21
ð2Þ ⋮ sinai ⋮
3 −Lsza 6 7 7 Lsya 1 54 5 −Δwy2 þ Δwy1 ⋮ ⋮
32
ð3Þ
2.1.3. Measurement of two offset errors of rotary axis C Two offset errors of rotary axis C are measured with control of rotary axis C only during measurement. The position of the ball at the tool nose is positioned, without reinstallation, at the origin of reference coordinate system {F} as same with these at first measurement. The position of the ball on the work-piece table is reinstalled at a position, far from the origin of reference coordinate system {F} at nominal length R in the direction of x-axis. Then, the ball on the work-piece table is rotated about the origin of coordinate system {C} during the measurement. In this case, the relation between the deviation of measured data, offset errors and set-up error is given by Eq. (4). Here, two offset errors of rotary axis C combined with set-up error Δt1 are calculated similar to the
ð4Þ
2.1.4. Measurement of two squareness errors of rotary axis C The ball at the tool nose without reinstallation is moved by offset L in the direction of the z-axis at reference coordinate system {F} using linear axis z. The ball on the work-piece table is reinstalled in a position with offset L, far from the origin of reference coordinate system {F}. Offset L is designed to amplify the effects of the squareness errors, which are regarded as angular errors, with respect to the measured data. The relation between the deviation of measured data, squareness errors, and set-up error is defined by Eq. (5). Similarly, with the results of the third measurement, two squareness errors of rotary axis C are combined with the errors of rotary axis A and the set-up errors because the ball at the tool nose is located on the z-axis of reference coordinate system {F}, and not on the z-axis of coordinate system {A}. Subtraction of Eq. (4) from Eq. (5) removes the offset errors, and set-up errors as described by Eq. (6). In addition, the two derived squareness errors using Eq. (3) must be subtracted from the solution of Eq. (6) to calculate the squareness errors of rotary axis C: 2 3 2 32 3 oxc þ Lðsya þ syc Þ−Δt x1 ⋮ ⋮ ⋮ ⋮ 6 ΔR 7 6 cosc sinc 1 76 o þ o −Ls −Δt 7 yc xc 4 4i 5 ¼ 4 54 ya y1 5 i i Δwx4 ⋮ ⋮ ⋮ ⋮ ΔR 4 ¼ A4 x4 2
3
ð5Þ 2
⋮ ⋮ 6 ΔR −ΔR 7 6 cosc 4 4i 3i 5 ¼ 4 i ⋮ ⋮
⋮ sinci ⋮
⋮
32
6 17 54 ⋮
ΔR 4 −ΔR3 ¼ A4 x43
Lðsya þ syc Þ −Lsxc
3 7 5
Δwx4 −Δwx3 ð6Þ
The eight PIGEs of rotary axes A and C are measured by the proposed paths with Eqs. (1)–(6) according to the procedure as show in Fig. 6: however, all of the offset errors are affected by setup error Δt1 which is mainly caused by operator. So the measured offset errors are dependent on the skill of operator degrading the measurement accuracy. To improve the measurement accuracy, a fixture to adjust the ball's position at the tool nose is used to minimize set-up error Δt1 . 2.2. Novel fixture for set-up error Set-up error Δt1 is the difference between the ball's position at the tool nose and the rotation axis of the spindle, as shown in Fig. 5. By using a tool setting system, set-up error Δt z1 is easily measured and is compensated in terms of tool length. However, it requires an additional method to minimize set-up error Δt x1 , Δt y1 . To minimize these set-up errors, a fixture is designed to adjust the ball's position to the rotation axis of the spindle, as shown in Fig. 7. The fixture consists of two flexure hinges and two pairs of bolts to adjust the ball's position. Two flexure hinges are designed to be perpendicular to each other in a plane. Thus, the equal sensitivity of operator's handling to the two adjusting direction is achievable for convenience, and the mass of the moving part at the fixture can be decreased to increase the resonance frequency of the designed
K.-I. Lee, S.-H. Yang / International Journal of Machine Tools & Manufacture 70 (2013) 45–52
49
respectively:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u n n u 2 uðoya Þ ¼ t m2111 ∑ cos2 ai þ m2112 ∑ sin ai þ m2113 n u2DBB þ u2Δty1 i¼1
i¼1
i¼1
i¼1
i¼1
i¼1
i¼1
i¼1
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u n n u 2 uðoza Þ ¼ t m2121 ∑ cos2 ai þ m2122 ∑ sin ai þ m2123 n u2DBB þ u2Δtz1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi pffiffiffi u n n 2u t m2 ∑ cos2 a þ m2 ∑ sin2 a þ m2 n u uðsya Þ ¼ DBB i i 221 222 223 L i¼1 i¼1 ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffi u n n 2u t m2 ∑ cos2 a þ m2 ∑ sin2 a þ m2 n u uðsza Þ ¼ DBB i i 211 212 213 L i¼1 i¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u n n u 2 uðoxc Þ ¼ t m2311 ∑ cos2 ci þ m2312 ∑ sin ci þ m2313 n u2DBB þ u2Δtx1 ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u n n u 2 t 2 2 2 2 2 2 m321 ∑ cos ci þ m322 ∑ sin ci þ m323 n uDBB þ uΔty1 uðoyc Þ ¼
Fig. 6. Measurement procedure.
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffi u n n 2u t m2 ∑ cos2 c þ m2 ∑ sin2 c þ m2 n u uðsxc Þ ¼ DBB i i 421 422 423 L i¼1 i¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u n n 1u uðsyc Þ ¼ t2 m2411 ∑ cos2 ci þ m2412 ∑ sin2 ci þ m2413 n u2DBB þ L2 u2 ðsya Þ L i¼1 i¼1
ð7Þ where uðΔt x1 Þ ¼ uðΔt y1 Þ ¼ uðΔt z1 Þ ¼ uTOOL
Fig. 7. Fixture to adjust the ball's position.
fixture. In addition, the fixture robustly keeps the ball's position at the tool nose during installation with high axial stiffness due to its monolithic structure. The designed fixture, which is made of stainless steel, has its first resonance frequency at 914 Hz; this is sufficient for quasi-static measurements. Using this fixture, set-up error Δt x1 and Δt y1 are minimized by an adjustment of the ball's position using a rotating the spindle and by measuring the radial deviation with an indicator.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2ADJUST þ u2INDICATOR
SETTING
The standard uncertainties uDBB , uΔt x1 , uΔt y1 , and uΔtz1 affect the standard uncertainties of the measured PIGEs. Set-up error Δt1 always affects all of the offset errors, whereas it does not affect the squareness errors. However, the standard uncertainty uDBB affects the standard pffiffiffi uncertainty of squareness errors with multiplication of factor 2 by the effect of subtraction between two data. In addition, the rotation range affects the standard uncertainties of the PIGEs by combining with standard uncertainty uDBB . The coefficients, which are multiplied with the standard uncertainty uDBB of the offset errors for rotary axis A in Eq. (7), are calculated according to the rotation range as shown in Fig. 8. As the rotation range increases, the coefficients decrease dramatically at [701,1801] and converge to 0.002 approximately at rotation range as 3601. This means the effect of standard uncertainty uDBB to the standard uncertainty of measured PIGEs is decreased as increase of rotation range. Only 0.2% of standard uncertainty uDBB affects the
2.3. Standard uncertainties for measurement It is necessary to calculate the standard uncertainty of measured PIGEs to quantify the confidence interval and to increase measurement accuracy by minimizing the effect of the most important contributor. Therefore, the standard uncertainties of PIGEs in Eqs. (1)–(6) are defined by Eq. (7) according to the ISO 230-9 standard [24,25]. Here, mijk (i¼ 1,…,4; j, k ¼1, 2, 3) refers to the factor at the j-th row and k-th column of matrix ðATi Ai Þ−1 at i-th measurement. In addition, uDBB , uADJUST , uINDICATOR , and uTOOL SETTING refer to standard uncertainties of the double ball-bar, adjusting result, indicator, and tool setting system,
Fig. 8. Coefficients at standard uncertainties versus the rotation range.
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measurement results. So, set-up error Δt1 is the main contributor for offset errors and must be minimized to improve the standard uncertainty of the measured offset errors. To improve the standard uncertainty of measured squareness errors, the standard uncertainty uDBB must be improved, and large offset L must be designed in work-space.
3. Simulation and experiment Simulation consists of three steps: the generation of all PIGEs of rotary axes A and C, the calculation of double ball-bar data involving set-up errors, and the estimation of all PIGEs using the proposed method. The simulation conditions are given in Table 2. The rotation ranges are the nominal ranges of a five-axis machine
tool used in the experiment. The accuracies of the double ball-bar and indicator are given by the manufacturer. Set-up error Δt1 , is determined experimentally after adjustment, and is a few onehundreds of a μm before adjustment. Due to the effects of the PIGEs and set-up errors, the calculated data of double ball-bar deviates from nominal length R as shown in Fig. 9. The simulation results are summarized in Table 3. The discrepancies between generated PIGEs and estimated PIGEs are within 1 μm and 1 μrad, respectively, thereby showing the validity of the suggested method. The measurement uncertainties with a coverage factor of 2 are only a few μm and few μrad, thus showing the reliability of the measurement results. The measurement uncertainty of offset error oyc is relatively larger than the other offset errors because it
Table 2 Simulation condition. Parameter
Unit
Value
Nominal length, R Offset, L Rotation range
mm mm degree degree μm μm μm μm μm μm
100 150 [−30, 120] [0, 360] 71 (1, 1, 1) [−30, 30] 71 71 73
Rotary axis A Rotary axis C
Measurement noise Set-up error, Δt1 Set-up error, Δwi Accuracy of double ball-bar Deviation after adjustment Accuracy of indicator
Fig. 9. Calculated deviations for simulation.
Fig. 10. Discrepancies between generated errors and estimated errors for given value. (a) Discrepancy at offset errors. (b) Discrepancy at squareness errors.
Table 3 Simulation result. Parameter
Unit
Given value
Estimated value
Discrepancy
Standard uncertainty
Measurement uncertainty
oya oza sya sza oxc oyc sxc syc
μm μm μrad μrad μm μm μrad μrad
−21.0 13.7 −37.0 −24.0 12.8 −24.5 25.0 18.0
−22.0 14.7 −37.0 −24.1 11.8 −24.5 25.0 18.0
1.0 −1.0 0.0 0.1 1.0 0.0 0.0 0.0
1.8 1.8 2.0 2.0 1.8 2.5 0.1 2.0
3.5 3.5 4.0 4.0 3.5 5.0 0.3 4.0
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is combined with the measurement uncertainty of offset error oya as shown in Eq. (4). In addition, the discrepancies are calculated at the given value of PIGEs, as shown in Fig. 10, to investigate the size effect of the PIGEs with respect to the measurement results. Here, equal value is given for offset errors and squareness errors, respectively. As a result, the discrepancies increase as the values of the PIGEs increases. This is caused by the elimination of higher-order terms of the PIGEs and set-up errors, which are regarded as negligible, to derive the linearized equations of Eqs. (1)–(6). Thus,
51
re-measurement must be performed after compensation of the measured PIGEs if they exhibit large values in order to measure the PIGEs more accurately. The proposed method is applied to a five-axis machine tool (VARIAXIS 500-5X III, Mazak, Japan) with a double ball-bar (QC20W, Renishaw, United Kingdom) as shown in Fig. 11. The nominal range of rotary axis A is [−301, 1201] as shown in Table 2. But the rotation range of rotary axis A is given as [−201, 751] for experiment to avoid collision between the spindle and work-piece table. The range of rotary axis C, is given as [01, 3601]. Measured data show deviations from nominal length R, as shown in Fig. 12. It takes approximately 20 min for all measurements, including installation and data acquisition. By applying the proposed method, the eight PIGEs of rotary axis A and C are measured and are given in Table 4, with measurement uncertainties. To test the
Table 4 Experiment result. Parameter Unit Without comp.
Fig. 11. Experiment at five-axis machine tool.
oya oza sya sza oxc oyc sxc syc
μm μm μrad μrad μm μm μrad μrad
25.4 −9.1 30.7 −4.5 22.3 3.3 20.2 −4.6
With comp.
Standard uncertainty
Measurement uncertainty
−1.0 −0.2 −4.0 −0.4 4.0 −2.0 0.0 18.5
3.6 2.4 15.0 29.2 1.8 4.0 0.2 15.0
7.1 4.7 30.1 58.4 3.5 7.9 0.4 30.1
Fig. 12. Measured data with and without compensation of PIGEs. (a) Data at first measurement. (b) Data at second measurement. (c) Data at third measurement. (d) Data at fourth measurement.
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validity of the measured PIGEs, the ball's position at the tool nose is compensated and re-measurement is performed. All PIGEs are decreased into measurement uncertainties after compensation, but squareness error syc is increased after compensation although it is into measurement uncertainty. This phenomenon occurs in most cases of additional measurement of all PIGEs and not for a specific error. The PIGEs after compensation shows few μm and within 20 μrad for offset errors and squareness errors, respectively at additional measurement. This might be caused by the positioning variability of linear axes that are used for compensation: The offset errors are directly affected by the positioning variability of linear axes. On the contrary to this, the squareness errors are affected by a combination of positioning variability of linear axes and offset L. During experiment with offset L as 150 mm, positioning variability with 2.78 μm in x-direction might cause squareness error syc to increase to 18.5 μrad as shown in Table 4. For rotary axis A, the coefficient of standard uncertainty uDBB with the rotation range [−201, 751] during experiment is 0.4, and 0.07 at nominal range [−301, 1201]. Therefore, the effect of standard uncertainty uDBB on the measurement results increased six fold, thereby showing relatively lager values compared to the simulation result in Table 3. In addition, the standard uncertainty of PIGEs of rotary axis A affects to the rotary axis C because it is combined as shown in Eq. (7). Thus the standard uncertainty of PIGEs of rotary axis C is also increased compared to the simulation result in Table 3. 4. Conclusion In this study, the offset errors and the squareness errors for rotary axes of a five-axis machine tool are measured using a double ball-bar. Measurement paths involving single axis control during measurement are planned to decrease measurement cost and improve measurement accuracy by minimizing the number of controlled axes, although they are assumed to be in-tolerance. In addition, the effects of set-up error on the measurement results are investigated and minimized using an adjusting fixture. The measurement uncertainty of the measured offset errors and squareness errors are calculated by using contributors. Finally, the proposed method is tested through simulation and applied to a five-axis machine tool to measure and compensate measured errors. The main advantage of using the proposed method alone is the use of a single controlled axis during all measurements. By minimizing the number of controlled axes, the measurement process is simplified. This results in a decrease in measurement cost, including less effort by operators and decreased measurement time. Especially, a decrease in measurement time ensures the robustness of measurement results against the changing measurement condition. Efficient measurement can be performed using the proposed method for offset errors and squareness errors of a five-axis machine tool. Using proposed method, the positioning accuracy of a five-axis machine tool can be significantly improved by practical use at assemble process, and for maintenance. Acknowledgments This work was supported by the Priority Research Centers Programs through the National Research Foundation of Korea
(NRF) funded by the Ministry of Education, Science and Technology (2012–0005856).
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Contents lists available at SciVerse ScienceDirect
International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool
Measurement and verification of position-independent geometric errors of a five-axis machine tool using a double ball-bar Kwang-Il Lee, Seung-Han Yang n School of Mechanical Engineering, Kyungpook National University, 80, Daehak-ro, Buk-gu, Daegu, Republic of Korea
art ic l e i nf o
a b s t r a c t
Article history: Received 11 September 2012 Received in revised form 20 March 2013 Accepted 21 March 2013 Available online 30 March 2013
In this study, position-independent geometric errors, including offset errors and squareness errors of rotary axes of a five-axis machine tool are measured using a double ball-bar and are verified through compensation. In addition, standard uncertainties of measurement results are calculated to establish their confidence intervals. This requires two measurement paths for each rotary axis, which are involving control of single rotary axis during measurement. So, the measurement paths simplify the measurement process, and reduce measurement cost including less operator effort and measurement time. Set-up errors, which are inevitable during the installation of the balls, are modeled as constants. Their effects on the measurement results are investigated to improve the accuracy of the measurement result. A novel fixture consisting of flexure hinges and two pairs of bolts is used to minimize set-up error by adjusting the ball's position located at the tool nose. Simulation is performed to check the validation of measurement and to analyze the standard uncertainties of the measurement results. Finally, the position-independent geometric errors of the five-axis machine tool (involving a rotary axis and a trunnion axis) are measured using proposed method. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Position-independent geometric errors Rotary axis Five-axis machine tool Uncertainty analysis Double ball-bar
1. Introduction A five-axis machine tool is essential for manufacturing products with complex shapes requiring high form accuracy. Its main advantage compared to a three-axis machine tool is its capability for flexible control of the relative position and relative orientation between the work-piece and tool. However, the number of errors (i.e., geometric errors, thermally-induced errors, and dynamic errors) also increases inevitably by increasing the number of controlled axes. Positioning accuracy between the work-piece and tool decreases as a result. The geometric errors are the most important because they affect positioning accuracy during all of the running time [1,2]. Geometric errors are categorized as position-dependent geometric errors (PDGEs) and positionindependent geometric errors (PIGEs) [3]. Here, “position” refers to the command for a controlled axis. PDGEs are caused mainly by imperfections in components, and PIGEs are caused mainly during the assembly process. For linear axes, there have been many studies of the measurement of geometric errors [4–11]. However, there are only a few studies available for rotary axes [12]. A laser interferometer with a precision index table is used to measure angular positioning
n
Corresponding author. Tel.: +82 539506569; fax: +82 539506550. E-mail address: [email protected] (S.-H. Yang).
0890-6955/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmachtools.2013.03.010
error [13]. Two LVDT sensors with a precision ball, auto-collimator with a polygon mirror can be used to measure two radial errors and two tilt errors of a rotary axis with investigating set-up errors [14]. For a trunnion axis, a method involving five measurement steps with a single set-up has been developed [15]. A double ballbar has been used to measure PDGEs (except angular positioning error) and the PIGEs of a rotary axis through reverse kinematic approaches [16]. The effects of the PDGEs of a rotary axis have also been investigated [17]. To simplify the measurement process, only PIGEs are considered and these are measured by the simultaneous control of three axes (two linear axes and a rotary axis) using a double ball-bar [18]. In a similar approach, an R-test requiring a three-dimensional probe and a precision ball is available as commercial product [19,20]. However, the results suffer mainly due to the positioning variability, including positioning accuracy and repeatability, of two linear axes that are simultaneously controlled during the measurements. In this study, the PIGEs of a five-axis machine tool with a rotary axis and a trunnion axis are measured using double ball-bar. During measurement, only the single rotary axis is controlled to simplify the measurement and to increase the measurement accuracy within the positioning variability of the rotary axis. Then, the standard uncertainties of the measurement results are calculated for a confidence interval and are analyzed to find the main contributor for the improvement of measurement reliability. Set-up errors, which are inevitable in the installation of a double
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Nomenclature n u ai , ci oia , sia oic , sic ðx; y; zÞ L R U
sample number; standard uncertainty; command angle for rotary axis A, C(i¼1,…, n); offset error and squareness error in the i-direction at axis A(i¼ y, z); offset error and squareness error in the i-direction at axis C(i¼x, y); command for positioning; offset at measurement; nominal length of double ball-bar; measurement uncertainty;
ball-bar, are modeled as constant, and their effects on the measurement results are investigated. A fixture consisting of flexure hinges and two pairs of bolts is used to minimize set-up error by adjusting the ball's position at the tool nose. In Section 2, the PIGEs are briefly described and are summarized by comparison with results from other studies. The measurement paths for the PIGEs are planned and analyzed to investigate the effects of the PIGEs on the measured data of double ball-bar. The standard uncertainties of the measured PIGEs are defined by contributors. In Section 3, it describes a simulation that is performed to check the validity of measurements by analyzing standard uncertainties. Then, the PIGEs of a five-axis machine tool with a rotary axis and a trunnion axis are measured and compensated using measurement results. In Section 4, it summarizes the advantages and contributions of the proposed measurement method.
ΔRij
deviation to the nominal length at i-th measurement and j-th measured value(i¼1,…, 4; j¼1,…, n); {i} coordinate system of axis i(i¼X, Y, Z, A, C); ti ðt xi ; t yi ; t zi Þ design position of the ball at the tool nose at the i-th measurement(i¼ 1,…, 4); wi ðwxi ; wyi ; wzi Þ design position of the ball on the work-table at the i-th measurement(i¼ 1,…, 4); Δti ðΔt xi ; Δt yi ; Δt zi Þ set-up error at the ball at the tool nose at the i-th measurement(i¼ 1,…, 4); Δwi ðΔwxi ; Δwyi ; Δwzi Þ set-up error at the ball on the work-table at the i-th measurement(i ¼1,…, 4); ΔR i n 1 column vector consisting of deviation ΔRij at the i-th measurement(i¼ 1,…, 4).
2.1. Measurement paths of a double ball-Bar Four measurement paths involving control of a single rotary axis during each measurement are shown in Fig. 3. The paths are used to measure all of the PIGEs for rotary axes A and C. It assumes that the positioning accuracies of linear axes X, Y, and Z, which are used to install the balls at set-up, are within tolerances and thus will not deviate significantly from the target positions. 2.1.1. Measurement of two offset errors of rotary axis A One of the two balls is designed to be located at the position of the tool nose, which is the origin of the reference coordinate system {F} on
z y
2. Measurement and standard uncertainty
sza
{A}
As shown in Fig. 1, a five-axis machine tool typically consists of three linear axes and two rotary axes. The position and orientation of the rotary axes are deviated from the designed ones. Two offset errors and two squareness errors for each rotary axis are enough to describe such deviations, as summarized in Fig. 2 and Table 1. The effects of these PIGEs on the end-effector are modeled using an error synthesis model involving homogenous transformation matrices and their sequential multiplication [21–23].
oya
sya
oza
a
x
{F}
c
z
sxc
syc
y oxc
{C} oyc
{A} Fig. 1. Five-axis machine tool involving a rotary axis and a trunnion axis.
x
Fig. 2. PIGEs of rotary axis A and C. (a) Rotary axis A. (b) Rotary axis C.
K.-I. Lee, S.-H. Yang / International Journal of Machine Tools & Manufacture 70 (2013) 45–52
the rotation axis of the spindle. The other ball is installed at a position far from the origin as nominal length R to direction of y-axis in reference coordinate system {F}, on the work-piece table. During measurement, the ball at the tool nose is stationary and the ball on the work-piece table is rotated about the x-axis in coordinate system {A}, as shown in Fig. 4. Consequently, the measured data of double ball-bar is the distances between two balls and deviated from nominal length R by the disagreement between the ball's position at tool nose and the origin of coordinate system {A}, which is the offset errors. At installation, a fixture called a tool cup is used to fix the ball at the tool nose for measurement, as shown in Fig. 5. However, the tool cup is tilted by the fastening process at the spindle. The position of the ball attached at the tool cup deviates, which is modeled as set-up error from the rotation axis of the spindle. In addition, the position of the ball on the work-piece table deviates from the design position by the set-up error of the ball at the tool nose and by the fastening process using screw. Therefore, the relationship between the deviation of the measured data, the offset errors and the set-up errors is defined as follows: 2
3
2
⋮ ⋮ 6 ΔR 7 6 cosa 4 1i 5 ¼ 4 i ⋮ ⋮
⋮ sinai ⋮
32
−oya þ Δt y1
ΔR1 ¼ A1 x1
PIGEs
Ball on work-piece table
Actual path
Initial set-up
R+ΔR1
⋮ 6 7 17 54 −oza þ Δt z1 5 −Δwy1 ⋮
YOA ZOA BOA COA XOC YOC AOC BOC
a
3
{A}
Δw1 R
(oya, oza)
z
{F}
At ISO oya oza sya sza oxc oyc sxc syc
ð1Þ
Two offset errors combined with set-up error Δt1 are derived by applying the least squares method to Eq. (1). It is difficult to separate only offset errors from Eq. (1) due to singular problem, so it is necessary to minimize the set-up error Δt1 to improve measurement accuracy. However, the set-up error Δw1 only affects to the relative radius of the measured data on the polar plot, and does not affect the offset errors.
Table 1 Symbols to define geometric errors for rotary axes. Presented
47
y
Δt1
Ball at tool nose
Location errors
Nominal path
Fig. 4. The effect of offset errors and set-up errors at first measurement.
Fig. 3. Measurement paths for PIGEs of rotary axis A and C. (a) First measurement for offset errors of rotary axis A. (b) Second measurement for squareness errors of rotary axis A. (c) Third measurement for offset errors of rotary axis C. (d) Fourth measurement for squareness errors of rotary axis C.
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first measurement. Then, offset error oya is combined with offset error oyc because the ball at the tool nose is designed to locate at the origin of reference coordinate system {F}, and not the origin of coordinate system {A}. To calculate the offset error oyc , it is essential to subtract offset error oya , which is derived using Eq. (1), from the solution of the following equation. Thus, 2 3 2 32 3 oxc −Δt x1 ⋮ ⋮ ⋮ ⋮ 6 ΔR 7 6 cosc sinc 1 76 o þ o −Δt 7 yc 4 3i 5 ¼ 4 54 ya y1 5 i i Δwx3 ⋮ ⋮ ⋮ ⋮ ΔR 3 ¼ A3 x3
Fig. 5. Set-up errors at the installation of a double ball-bar.
2.1.2. Measurement of two squareness errors of rotary axis A It measures two squareness errors of rotary axis A. The ball at the tool nose is moved by offset L using linear axis X in the direction of the x-axis in reference coordinate system {F} without reinstallation. The ball on the work-piece table is positioned with offset L to the x-axis by reinstallation, causing set-up error Δw2 . During measurement, the ball at the tool nose is also stationary, and the ball on the work-piece table is rotated about rotary axis A, not the x-axis in reference coordinate system {F}. Two squareness errors affect the measured data by multiplied with offset L. In this case, the relation between the deviation of the measured data, squareness errors and set-up errors is given by Eq. (2). By subtracting Eq. (1) from Eq. (2), two squareness errors (which are not affected by set-up errors) are calculated using Eq. (3). Thus, 2 3 2 32 3 −oya −Lsza þ Δt y1 ⋮ ⋮ ⋮ ⋮ 6 ΔR 7 6 cosa sina 1 76 −oza þ Lsya þ Δt 7 z1 5 4 2i 5 ¼ 4 54 i i −Δwy2 ⋮ ⋮ ⋮ ⋮ ΔR2 ¼ A2 x2 2
3 2 ⋮ ⋮ 6 ΔR −ΔR 7 6 cosa ¼ 4 2i 4 1i 5 i ⋮ ⋮ ΔR2 −ΔR 1 ¼ A2 x21
ð2Þ ⋮ sinai ⋮
3 −Lsza 6 7 7 Lsya 1 54 5 −Δwy2 þ Δwy1 ⋮ ⋮
32
ð3Þ
2.1.3. Measurement of two offset errors of rotary axis C Two offset errors of rotary axis C are measured with control of rotary axis C only during measurement. The position of the ball at the tool nose is positioned, without reinstallation, at the origin of reference coordinate system {F} as same with these at first measurement. The position of the ball on the work-piece table is reinstalled at a position, far from the origin of reference coordinate system {F} at nominal length R in the direction of x-axis. Then, the ball on the work-piece table is rotated about the origin of coordinate system {C} during the measurement. In this case, the relation between the deviation of measured data, offset errors and set-up error is given by Eq. (4). Here, two offset errors of rotary axis C combined with set-up error Δt1 are calculated similar to the
ð4Þ
2.1.4. Measurement of two squareness errors of rotary axis C The ball at the tool nose without reinstallation is moved by offset L in the direction of the z-axis at reference coordinate system {F} using linear axis z. The ball on the work-piece table is reinstalled in a position with offset L, far from the origin of reference coordinate system {F}. Offset L is designed to amplify the effects of the squareness errors, which are regarded as angular errors, with respect to the measured data. The relation between the deviation of measured data, squareness errors, and set-up error is defined by Eq. (5). Similarly, with the results of the third measurement, two squareness errors of rotary axis C are combined with the errors of rotary axis A and the set-up errors because the ball at the tool nose is located on the z-axis of reference coordinate system {F}, and not on the z-axis of coordinate system {A}. Subtraction of Eq. (4) from Eq. (5) removes the offset errors, and set-up errors as described by Eq. (6). In addition, the two derived squareness errors using Eq. (3) must be subtracted from the solution of Eq. (6) to calculate the squareness errors of rotary axis C: 2 3 2 32 3 oxc þ Lðsya þ syc Þ−Δt x1 ⋮ ⋮ ⋮ ⋮ 6 ΔR 7 6 cosc sinc 1 76 o þ o −Ls −Δt 7 yc xc 4 4i 5 ¼ 4 54 ya y1 5 i i Δwx4 ⋮ ⋮ ⋮ ⋮ ΔR 4 ¼ A4 x4 2
3
ð5Þ 2
⋮ ⋮ 6 ΔR −ΔR 7 6 cosc 4 4i 3i 5 ¼ 4 i ⋮ ⋮
⋮ sinci ⋮
⋮
32
6 17 54 ⋮
ΔR 4 −ΔR3 ¼ A4 x43
Lðsya þ syc Þ −Lsxc
3 7 5
Δwx4 −Δwx3 ð6Þ
The eight PIGEs of rotary axes A and C are measured by the proposed paths with Eqs. (1)–(6) according to the procedure as show in Fig. 6: however, all of the offset errors are affected by setup error Δt1 which is mainly caused by operator. So the measured offset errors are dependent on the skill of operator degrading the measurement accuracy. To improve the measurement accuracy, a fixture to adjust the ball's position at the tool nose is used to minimize set-up error Δt1 . 2.2. Novel fixture for set-up error Set-up error Δt1 is the difference between the ball's position at the tool nose and the rotation axis of the spindle, as shown in Fig. 5. By using a tool setting system, set-up error Δt z1 is easily measured and is compensated in terms of tool length. However, it requires an additional method to minimize set-up error Δt x1 , Δt y1 . To minimize these set-up errors, a fixture is designed to adjust the ball's position to the rotation axis of the spindle, as shown in Fig. 7. The fixture consists of two flexure hinges and two pairs of bolts to adjust the ball's position. Two flexure hinges are designed to be perpendicular to each other in a plane. Thus, the equal sensitivity of operator's handling to the two adjusting direction is achievable for convenience, and the mass of the moving part at the fixture can be decreased to increase the resonance frequency of the designed
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49
respectively:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u n n u 2 uðoya Þ ¼ t m2111 ∑ cos2 ai þ m2112 ∑ sin ai þ m2113 n u2DBB þ u2Δty1 i¼1
i¼1
i¼1
i¼1
i¼1
i¼1
i¼1
i¼1
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u n n u 2 uðoza Þ ¼ t m2121 ∑ cos2 ai þ m2122 ∑ sin ai þ m2123 n u2DBB þ u2Δtz1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi pffiffiffi u n n 2u t m2 ∑ cos2 a þ m2 ∑ sin2 a þ m2 n u uðsya Þ ¼ DBB i i 221 222 223 L i¼1 i¼1 ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffi u n n 2u t m2 ∑ cos2 a þ m2 ∑ sin2 a þ m2 n u uðsza Þ ¼ DBB i i 211 212 213 L i¼1 i¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u n n u 2 uðoxc Þ ¼ t m2311 ∑ cos2 ci þ m2312 ∑ sin ci þ m2313 n u2DBB þ u2Δtx1 ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u n n u 2 t 2 2 2 2 2 2 m321 ∑ cos ci þ m322 ∑ sin ci þ m323 n uDBB þ uΔty1 uðoyc Þ ¼
Fig. 6. Measurement procedure.
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffi u n n 2u t m2 ∑ cos2 c þ m2 ∑ sin2 c þ m2 n u uðsxc Þ ¼ DBB i i 421 422 423 L i¼1 i¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u n n 1u uðsyc Þ ¼ t2 m2411 ∑ cos2 ci þ m2412 ∑ sin2 ci þ m2413 n u2DBB þ L2 u2 ðsya Þ L i¼1 i¼1
ð7Þ where uðΔt x1 Þ ¼ uðΔt y1 Þ ¼ uðΔt z1 Þ ¼ uTOOL
Fig. 7. Fixture to adjust the ball's position.
fixture. In addition, the fixture robustly keeps the ball's position at the tool nose during installation with high axial stiffness due to its monolithic structure. The designed fixture, which is made of stainless steel, has its first resonance frequency at 914 Hz; this is sufficient for quasi-static measurements. Using this fixture, set-up error Δt x1 and Δt y1 are minimized by an adjustment of the ball's position using a rotating the spindle and by measuring the radial deviation with an indicator.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2ADJUST þ u2INDICATOR
SETTING
The standard uncertainties uDBB , uΔt x1 , uΔt y1 , and uΔtz1 affect the standard uncertainties of the measured PIGEs. Set-up error Δt1 always affects all of the offset errors, whereas it does not affect the squareness errors. However, the standard uncertainty uDBB affects the standard pffiffiffi uncertainty of squareness errors with multiplication of factor 2 by the effect of subtraction between two data. In addition, the rotation range affects the standard uncertainties of the PIGEs by combining with standard uncertainty uDBB . The coefficients, which are multiplied with the standard uncertainty uDBB of the offset errors for rotary axis A in Eq. (7), are calculated according to the rotation range as shown in Fig. 8. As the rotation range increases, the coefficients decrease dramatically at [701,1801] and converge to 0.002 approximately at rotation range as 3601. This means the effect of standard uncertainty uDBB to the standard uncertainty of measured PIGEs is decreased as increase of rotation range. Only 0.2% of standard uncertainty uDBB affects the
2.3. Standard uncertainties for measurement It is necessary to calculate the standard uncertainty of measured PIGEs to quantify the confidence interval and to increase measurement accuracy by minimizing the effect of the most important contributor. Therefore, the standard uncertainties of PIGEs in Eqs. (1)–(6) are defined by Eq. (7) according to the ISO 230-9 standard [24,25]. Here, mijk (i¼ 1,…,4; j, k ¼1, 2, 3) refers to the factor at the j-th row and k-th column of matrix ðATi Ai Þ−1 at i-th measurement. In addition, uDBB , uADJUST , uINDICATOR , and uTOOL SETTING refer to standard uncertainties of the double ball-bar, adjusting result, indicator, and tool setting system,
Fig. 8. Coefficients at standard uncertainties versus the rotation range.
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measurement results. So, set-up error Δt1 is the main contributor for offset errors and must be minimized to improve the standard uncertainty of the measured offset errors. To improve the standard uncertainty of measured squareness errors, the standard uncertainty uDBB must be improved, and large offset L must be designed in work-space.
3. Simulation and experiment Simulation consists of three steps: the generation of all PIGEs of rotary axes A and C, the calculation of double ball-bar data involving set-up errors, and the estimation of all PIGEs using the proposed method. The simulation conditions are given in Table 2. The rotation ranges are the nominal ranges of a five-axis machine
tool used in the experiment. The accuracies of the double ball-bar and indicator are given by the manufacturer. Set-up error Δt1 , is determined experimentally after adjustment, and is a few onehundreds of a μm before adjustment. Due to the effects of the PIGEs and set-up errors, the calculated data of double ball-bar deviates from nominal length R as shown in Fig. 9. The simulation results are summarized in Table 3. The discrepancies between generated PIGEs and estimated PIGEs are within 1 μm and 1 μrad, respectively, thereby showing the validity of the suggested method. The measurement uncertainties with a coverage factor of 2 are only a few μm and few μrad, thus showing the reliability of the measurement results. The measurement uncertainty of offset error oyc is relatively larger than the other offset errors because it
Table 2 Simulation condition. Parameter
Unit
Value
Nominal length, R Offset, L Rotation range
mm mm degree degree μm μm μm μm μm μm
100 150 [−30, 120] [0, 360] 71 (1, 1, 1) [−30, 30] 71 71 73
Rotary axis A Rotary axis C
Measurement noise Set-up error, Δt1 Set-up error, Δwi Accuracy of double ball-bar Deviation after adjustment Accuracy of indicator
Fig. 9. Calculated deviations for simulation.
Fig. 10. Discrepancies between generated errors and estimated errors for given value. (a) Discrepancy at offset errors. (b) Discrepancy at squareness errors.
Table 3 Simulation result. Parameter
Unit
Given value
Estimated value
Discrepancy
Standard uncertainty
Measurement uncertainty
oya oza sya sza oxc oyc sxc syc
μm μm μrad μrad μm μm μrad μrad
−21.0 13.7 −37.0 −24.0 12.8 −24.5 25.0 18.0
−22.0 14.7 −37.0 −24.1 11.8 −24.5 25.0 18.0
1.0 −1.0 0.0 0.1 1.0 0.0 0.0 0.0
1.8 1.8 2.0 2.0 1.8 2.5 0.1 2.0
3.5 3.5 4.0 4.0 3.5 5.0 0.3 4.0
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is combined with the measurement uncertainty of offset error oya as shown in Eq. (4). In addition, the discrepancies are calculated at the given value of PIGEs, as shown in Fig. 10, to investigate the size effect of the PIGEs with respect to the measurement results. Here, equal value is given for offset errors and squareness errors, respectively. As a result, the discrepancies increase as the values of the PIGEs increases. This is caused by the elimination of higher-order terms of the PIGEs and set-up errors, which are regarded as negligible, to derive the linearized equations of Eqs. (1)–(6). Thus,
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re-measurement must be performed after compensation of the measured PIGEs if they exhibit large values in order to measure the PIGEs more accurately. The proposed method is applied to a five-axis machine tool (VARIAXIS 500-5X III, Mazak, Japan) with a double ball-bar (QC20W, Renishaw, United Kingdom) as shown in Fig. 11. The nominal range of rotary axis A is [−301, 1201] as shown in Table 2. But the rotation range of rotary axis A is given as [−201, 751] for experiment to avoid collision between the spindle and work-piece table. The range of rotary axis C, is given as [01, 3601]. Measured data show deviations from nominal length R, as shown in Fig. 12. It takes approximately 20 min for all measurements, including installation and data acquisition. By applying the proposed method, the eight PIGEs of rotary axis A and C are measured and are given in Table 4, with measurement uncertainties. To test the
Table 4 Experiment result. Parameter Unit Without comp.
Fig. 11. Experiment at five-axis machine tool.
oya oza sya sza oxc oyc sxc syc
μm μm μrad μrad μm μm μrad μrad
25.4 −9.1 30.7 −4.5 22.3 3.3 20.2 −4.6
With comp.
Standard uncertainty
Measurement uncertainty
−1.0 −0.2 −4.0 −0.4 4.0 −2.0 0.0 18.5
3.6 2.4 15.0 29.2 1.8 4.0 0.2 15.0
7.1 4.7 30.1 58.4 3.5 7.9 0.4 30.1
Fig. 12. Measured data with and without compensation of PIGEs. (a) Data at first measurement. (b) Data at second measurement. (c) Data at third measurement. (d) Data at fourth measurement.
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validity of the measured PIGEs, the ball's position at the tool nose is compensated and re-measurement is performed. All PIGEs are decreased into measurement uncertainties after compensation, but squareness error syc is increased after compensation although it is into measurement uncertainty. This phenomenon occurs in most cases of additional measurement of all PIGEs and not for a specific error. The PIGEs after compensation shows few μm and within 20 μrad for offset errors and squareness errors, respectively at additional measurement. This might be caused by the positioning variability of linear axes that are used for compensation: The offset errors are directly affected by the positioning variability of linear axes. On the contrary to this, the squareness errors are affected by a combination of positioning variability of linear axes and offset L. During experiment with offset L as 150 mm, positioning variability with 2.78 μm in x-direction might cause squareness error syc to increase to 18.5 μrad as shown in Table 4. For rotary axis A, the coefficient of standard uncertainty uDBB with the rotation range [−201, 751] during experiment is 0.4, and 0.07 at nominal range [−301, 1201]. Therefore, the effect of standard uncertainty uDBB on the measurement results increased six fold, thereby showing relatively lager values compared to the simulation result in Table 3. In addition, the standard uncertainty of PIGEs of rotary axis A affects to the rotary axis C because it is combined as shown in Eq. (7). Thus the standard uncertainty of PIGEs of rotary axis C is also increased compared to the simulation result in Table 3. 4. Conclusion In this study, the offset errors and the squareness errors for rotary axes of a five-axis machine tool are measured using a double ball-bar. Measurement paths involving single axis control during measurement are planned to decrease measurement cost and improve measurement accuracy by minimizing the number of controlled axes, although they are assumed to be in-tolerance. In addition, the effects of set-up error on the measurement results are investigated and minimized using an adjusting fixture. The measurement uncertainty of the measured offset errors and squareness errors are calculated by using contributors. Finally, the proposed method is tested through simulation and applied to a five-axis machine tool to measure and compensate measured errors. The main advantage of using the proposed method alone is the use of a single controlled axis during all measurements. By minimizing the number of controlled axes, the measurement process is simplified. This results in a decrease in measurement cost, including less effort by operators and decreased measurement time. Especially, a decrease in measurement time ensures the robustness of measurement results against the changing measurement condition. Efficient measurement can be performed using the proposed method for offset errors and squareness errors of a five-axis machine tool. Using proposed method, the positioning accuracy of a five-axis machine tool can be significantly improved by practical use at assemble process, and for maintenance. Acknowledgments This work was supported by the Priority Research Centers Programs through the National Research Foundation of Korea
(NRF) funded by the Ministry of Education, Science and Technology (2012–0005856).
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