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Accuracy test of five-axis CNC machine tool with 3D probe-ball. Part II: errors estimation
International Journal of Machine Tools & Manufacture 42 (2002) 1163–1170
Accuracy test of five-axis CNC machine tool with 3D probe-ball. Part II: errors estimation W.T. Lei ∗, Y.Y. Hsu Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu, Taiwan, ROC Received 16 October 2001; received in revised form 12 April 2002; accepted 22 April 2002
Abstract To improve the accuracy of CNC machine tools, error sources and its effects on the overall position and orientation errors must be known. Most motional errors in the error model of five-axis machine tool can be measured with modern laser interferometer devices, but there are still some not measurable geometric errors. These not measurable errors include constant, inaccurate link errors of components such as rotary axes block, main spindle block and tool holder. After setting all measured errors in the error model, a reduced error model is defined, which describes the influence of each unknown and not measurable link error on the overall position errors of the five-axis machine tool. On the other hand, the newly developed probe-ball device can measure the overall position errors of five-axis machine tools directly. Based on the reduced model and the overall position errors, the link errors can be estimated very accurately with the least square estimation method. The error model is then fully known and can be used for advanced purposes such as error prediction and compensation. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Geometric error; Least square estimation; Five-axis machine tool; Probe-ball
1. Introduction The technique of building machine tool’s geometric error model is well developed in the past few years [1– 5]. The error model describes the position and orientation errors of tool relative to workpiece at specific machine position, whereby inaccurate influential factors come from kinematic link parameters and individual error sources. It is well known that the inaccurate motion of a linearly driven axis is associated with six motional errors, including one linear error, two straightness errors, and three rotational errors. With modern measurement devices such as the 6D laser interferometer [6], all six motional errors of the linearly driven axis can be measured rapidly. Based on the error model, the accuracy of three-axis machine tools can be dramatically improved through the error compensation [7,8]. While most accuracy enhancement research focuses on three-axis CNC machine tools, relevant works in the area of five-axis
Corresponding author. Tel.: +886-3-5742921; fax: +886-35722840. E-mail address: [email protected] (W.T. Lei). ∗
machine tools are rather sparse. Kiridena and Ferreira [3] developed a viable method for modeling and visualizing the effects of position errors on the volumetric accuracy of five-axis CNC machine tool. Srivastava et al. [5] used homogenous transformation matrix to develop the geometric and thermal error model in a fiveaxis CNC machine tool. Although the theoretical error model of five-axis machine tool is known, it is still impossible to improve the accuracy of five-axis machine tool with the error compensation technique based on it. The reason is that there are some not directly measurable link errors in the error model, such as the inaccurate link errors of rotary axes block, main spindle block, and tool holder. These errors are constant and exist as deviations between coordinate frames and are difficult to access after the mounting process. It is clear that the key step toward effective accuracy enhancement of five-axis machine tools is the identification of these unknown link errors. The least square estimation (LSE) methods provide mathematical procedures by which a linear model can achieve a best fit to experimental data in the sense of least-squared error. The methods are powerful and welldeveloped mathematical tools that have been proposed
0890-6955/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 0 2 ) 0 0 0 4 8 - 2
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and used in a variety of areas for decades, including adaptive control, signal processing, and statistics [9]. In the field of error estimation of machine tools, some researches also used this method to estimate error components [10–12]. In this article, the estimation of the unknown and not directly measurable link errors in the error model of the five-axis machine tool is addressed. The newly developed probe-ball device presented in Part I of this study offers a new tool to do so. It is capable of measuring the overall position errors of five-axis machine tools directly. The error model of the probe-ball measurement is already derived in Part I in detail. Based on this relationship and the measured overall position errors, the link errors can be estimated very accurately with the LSE method.
2. Basic theory The linear least-squares problem involves the use of a data set of measurements w, which are linearly related to the unknown parameter vector u by the expression w ⫽ Gu ⫹ v
(1)
where w is an m×1 vector, G, an m×n matrix, u, an n×1 vector, v, an m×1 vector of measurement ‘noise’, m, the number of measurements, and n, the number of the unknown parameters to be estimated. The goal is to find the unknown vector, denoted by u˜, which minimizes the sum of squares of the deviations, i.e. minimizes the cost function J J ⫽ (w⫺Gu˜)T(w⫺Gu˜)
(2)
The minimum is obtained when ∂J ⫽0 ∂u˜
(3)
and the Hessian of J is positive semidefinite ∂2J ⱖ0. ∂u˜2
| |
(4)
Differentiating J and setting the result to zero yields GTGu˜ ⫽ GTw
(5)
It can be shown that the second derivative of J, with respect to u˜, is positive semidefinite; and thus Eq. (3) does define a minimum [9]. The least-squares estimate of u˜ is then u˜ ⫽ (GTG)⫺1GTw
(6)
The method mentioned above is now applied to the estimation of not measurable link errors in the five-axis machine tools. The probe-ball device delivers necessary measurement data for the estimation, including three
deviations in X-, Y-, and Z-axis at one sampling. Since each link error has certain contribution to the three axes, it is necessary to rewrite Eq. (1) as follows q ⫽ Ha ⫹ e
(7)
where q is the measurement vector, a, the unknown link error vector, e, a noise vector, H, an error gain matrix which describes the relationship between the link error vector and the measurement output q ⫽ [q1,x % qm,x q1,y % qm,y q1,z % qm,z]T
(8)
a ⫽ [a1 % an]T f1,x(p1) f2,x(p1) % fn,x(p1)
(9)
H⫽
f1,x(p2) f2,x(p2) % fn,x(p2) ⯗
⯗
⯗
⯗
f1,x(pm) f2,x(pm) % fn,x(pm) f1,y(p1) f2,y(p1) % fn,y(p1) f1,y(p2) f2,y(p2) % fn,y(p2) ⯗
⯗
⯗
⯗
f1,y(pm) f2,y(pm) % fn,y(pm) f1,z(p1) f2,z(p2) % fn,z(p1) f1,z(p2) f2,z(p2) % fn,z(p2) M
M
M M
f1,z(pm) f2,z(pm) % fn,z(pm)
(10)
The functions fi,x(pj), fi,y(pj), and fi,z(pj) in matrix H represent the effects of link errors at specified setting position pj and are also called error gain functions in this article. The index n and m represent the number of parameter and sample, respectively. An explicit expression of each error gain function can be obtained from the error model in Part I of this study. The goal is to find a parameter vector, denoted by a˜, which minimizes the sum of squares of the deviations, thus the cost function for LSE
E(a˜) ⫽ eTe ⫽ (q⫺Ha˜)T(q⫺Ha˜)
(11)
An optimal estimation occurs when the cost function is at its minimum, which means ∂E(a˜) ⫽ 2HTHa˜⫺2HTq ⫽ 0 ∂a˜
(12)
The LSE of a˜ is then a˜ ⫽ (HTH)⫺1HTq
(13)
3. Reduced error model for estimation In Part I of study 59 errors are defined in the error model of the probe-ball measurement. To reduce the scope of estimation, a reasonable approach is to measure
W.T. Lei, Y.Y. Hsu / International Journal of Machine Tools & Manufacture 42 (2002) 1163–1170
Table 1 Position-dependent, not measured motional errors Machine axis
X-axis Y-axis Z-axis A-axis C-axis
Translational errors
dx(a),dy(a),dz(a) dx(c),dy(c),dz(c)
Rotational errors
ex(x) ey(y) ez(z) ey(a),ez(a) ex(c),ey(c)
all measurable errors with available devices and set the not measured motional errors to zero. The estimation focuses then only on the unknown, constant, and not measurable link errors. The advantage of this approach is that the estimation can be done even if available measurement devices are limited, for example only linear or 5D laser interferometer is available. Obviously, the accuracy of estimation will be better if more powerful devices are used. The error estimation begins with the sorting and merging of unknown link errors into independent parameters. In the error model of the probe-ball measurement, some errors have the same gain factor. For example, the constant rotational error aya and the position-dependent motional error ey(a) have the same gain factor ⫺ XwoCce ⫹ YwoSce in contributing to the position error ⌬Zp. Besides, the position-dependent motional error dz(a) adds with the constant link errors dza, dzs, and dzh in contributing to the position error ⌬Zp. These errors cannot be estimated separately. To simplify the estimation process and to maintain the physical meanings of error sources, all position-dependent, not measured motional errors are set to zero, as seen in Table 1. The errors with the same gain factor are merged into one error parameter without loss of their geometrical features. For example, the not measured tilt error of A-axis ey(a) is set to zero and the constant rotational error aya Table 2 Error parameters to be estimated Parameter Name a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13
d∗xa d∗ya d∗za aya aza dyc d∗zc axc ayc dxt dyt ext eyt
Sources of error dxa,dxc,dxs,dxh,eys,eyh dya,dyh,dys,exs,exh,ezt dza,dzs,dzh aya aza dyc dzc,dzt axc ayc dxt dyt ext eyt
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is to be estimated. Also, the not measured motional error dz(a) is set to zero and the constant link errors dza, dzs, and dzh are merged together and renamed as error para∗ , since all errors contribute in the Z-direction. meter dza In this way, the total 59 geometric errors are reduced to 13 independent, constant error parameters. These error parameters form the parameter vector a and are shown in Table 2. The position error model (⌬Xp,⌬Yp,⌬Zp) is then divided into two sub-models: the device error model and the reduced error model (Cx,Cy,Cz) (⌬Xp∗,⌬Yp∗,⌬Zp∗). The device error model describes the contribution of all measured motional and not measured motional errors, whereby the not measured motional errors are set to zero. Therefore, the device error model contains no unknown error parameters. In contrast, the reduced error model describes the contribution of unknown error parameters to the overall errors ⌬Xp ⫽ Cx ⫹ ⌬X∗p
(14)
⌬Yp ⫽ Cy ⫹ ⌬Y∗p
(15)
⌬Zp ⫽ Cz ⫹ ⌬Z
(16)
∗ p
The device error model (Cx,Cy,Cz) can be expressed as Cx ⫽ dx(x) ⫹ dx(y)⫺dx(z) ⫹ (ey(x) ⫹ gyx)(Z2 ⫹ YwoSaeCce ⫹ XwoSaeSce ⫹ CaeZwo ⫹ CaeZ3) ⫹ (ez(x) ⫹ gzx)(⫺CaeXwoSce⫺CaeYwoCce
(17)
⫹ SaeZwo ⫹ SaeZ3) ⫹ ez(y)(⫺CaeYwoCce ⫺CaeXwoSce ⫹ SaeZwo ⫹ SaeZ3) ⫹ ey(z)(Lt ⫹ R) ⫹ Xm⫺YwoSce ⫹ XwoCce Cy ⫽ dy(x) ⫹ dy(y)⫺dy(z) ⫹ (ez(x) ⫹ gzx)( ⫺YwoSce ⫹ XwoCce) ⫹ (ex(y) ⫹ gxy)(⫺Z1⫺Z2 ⫺XwoSaeSce⫺YwoSaeCce⫺CaeZwo⫺CaeZ3)
(18)
⫹ ez(y)(Xm⫺YwoSce ⫹ XwoCce) ⫹ ex(z)(⫺Lt⫺R) ⫹ Ym⫺SaeZwo⫺SaeZ3 ⫹ CaeYwoCce ⫹ CaeXwoSce Cz ⫽ dz(x) ⫹ dz(y)⫺dz(z) ⫹ (ey(x) ⫹ gyx)(YwoSce ⫺XwoCce) ⫹ (ex(y) ⫹ gxy)(CaeXwoSce ⫹ CaeYwoCce ⫺SaeZwo⫺SaeZ3) ⫹ ex(z)(⫺Y1) ⫹ Z1⫺Z0 ⫹ Z2
(19)
⫹ XwoSaeSce ⫹ YwoSaeCce ⫹ CaeZwo ⫹ CaeZ3⫺Zm ⫹ Lt ⫹ R The reduced error model (⌬X∗p ,⌬Y∗p ,⌬Z∗p ) is ⌬Xp∗ ⫽ d∗xa ⫹ aya(ZwoCae ⫹ SaeYwoCce ⫹ SaeXwoSce ⫹ CaeZ3) ⫹ aza(⫺CaeXwoSce ⫺CaeYwoCce ⫹ SaeZwo ⫹ SaeZ3) ⫹ ayc(Z3 ⫹ Zwo) ⫹ dxt(Cce) ⫹ dyt(⫺Sce) ⫹ ext(ZwoSce)
(20)
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error parameter a5. Its error gain function f5,x(pj) is ⫺ CaeXwoSce⫺CaeYwoCce ⫹ SaeZwo ⫹ SaeZ3, the third row of Eq. (20). Similarly, the f5,y(pj) is ⫺YwoSce ⫹ XwoCce,, the second row of Eq. (21) and f5,z(pj) is 0 since the link error aza has no contribution to ⌬Z∗p .
⫹ eyt(ZwoCce) ∗ ⌬Y∗p ⫽ dya ⫹ aza(⫺YwoSce ⫹ XwoCce) ⫹ axc(
⫺ZwoCae⫺SaeXwoSce⫺SaeYwoCce⫺CaeZ3) ⫹ ayc( ∗ ⫺SaeYwoSce ⫹ SaeXwoCce) ⫹ dyc(Cae) ⫹ dzc (
(21)
⫺Sae) ⫹ ext(⫺CaeZwoCce⫺SaeYwo) ⫹ eyt(CaeZwoSce ⫹ SaeXwo) ⫹ dxt(CaeSce)
4. Estimation procedure and results
⫹ dyt(CaeCce)
The block diagram of parameter estimation is shown in Fig. 1. Firstly, motional errors are measured with available devices. In this research, the 5D laser interferometer of API Inc. is used, which delivers only five motional errors of each linearly driven axis including three translational and two rotational errors. The roll errors of X-, Y-, and Z-axis are not measurable with this device and are set to zero. Note that the newest 6D laser interferometer can measure all six motional errors of a linearly driven axis. The perpendicularity errors between X-, Y-, and Z-axis are also measurable with the API laser device and a pentaprism. The axial errors of the two rotary axes are measured with a multi-faces optical mirror and an electronic level. Other motional errors of the
∗ ⌬Z∗p ⫽ dza ⫹ aya(⫺XwoCce ⫹ YwoSce) ⫹ axc(
⫺ZwoSae⫺SaeZ3 ⫹ CaeXwoSce ⫹ CaeYwoCce) ⫹ ayc(⫺CaeXwoCce ⫹ CaeYwoSce) ⫹ dyc(Sae)
(22)
⫹ d∗zc(Cae) ⫹ ext(CaeYwo⫺ZwoSaeCce) ⫹ eyt( ⫺CaeXwo ⫹ ZwoSaeSce) ⫹ dxt(SaeSce) ⫹ dyt(SaeCce) The reduced error model builds the mathematical basis for the LSE and can be re-arranged into vector form to obtain the error gain functions fi,x(pj),, fi,y(pj), and fi,z(pj) in the error gain matrix H for each setting position pj. For example, the link error aza is defined as
Fig. 1. The block diagram of error estimation.
W.T. Lei, Y.Y. Hsu / International Journal of Machine Tools & Manufacture 42 (2002) 1163–1170
two rotary axes are not measurable in the laboratory and are set to zero. Note that all motional errors are of random variables in nature with statistical characteristics. The measurements are conducted according to the international standard ISO230-2. The accuracy of used measurement instruments are listed as follows: 앫 5D laser measurement device of API Inc. Linear accuracy: 1 ppm. Straightness accuracy: greater than ± 1µm in the range of measurements. Angular accuracy: greater than ± 1.0arcs in the range of measurements. 앫 Electronic level accuracy: ± 1.0arcs 앫 Probe of Renishaw Inc. Accuracy: ± 2µm over 2 mm range with 1 µm resolution in X, Y, and Z axes. The measurement data are shown in Table 3. The nominal kinematic parameters of the five-axis milling machine tool are calibrated and are shown in Table 4. The five-axis milling machine tool is tested with two different test paths: the helix-like one path F and the circular one path S. To evaluate the results of estimation, the path F is used for estimation and the path S is used for justification. The radius of the
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spherical test surface is 150 mm. The overall position errors and the setting positions pj of each error sampling are registered for the purpose of error estimation. The total number of samples for path F is 118. After the measurements, the values of error gain functions fi,x(pj), fi,y(pj), and fi,z(pj) in the error gain matrix H are computed for each setting position pj. The elements of the measurement vector q are obtained through the subtraction of the results of device error model (Cx,Cy,Cz) from the registered overall position errors also for each setting position pj, (see Eqs. (14)– (16)). With the error gain matrix H and the measurement vector q known, the unknown error parameter vector a˜ is obtained through solving Eq. (13) directly with the help of the mathematical tool software matlab. The solution of error parameter vector a˜ involves great amount of computation with matrices and arrays. No iteration is necessary. The results of estimation based on the test path F are shown in Table 5. Most estimated errors are larger than the measured motional errors. The measured rotational errors of linearly driven axes are within 5 arc s. In contrast, the estimated rotational error aza is ⫺1151 arc s. With the error gain function ⫺ CaeXwoSce⫺CaeYwoCce ⫹ SaeZwo ⫹ SaeZ3, the link error aza plays a dominant role in the volumetric errors of the target five-axis milling machine in the X-direction. For the sake of justification, the estimated link errors are set into the error model to compute the overall pos-
Table 3 Measured motional errors in mean X-axis (mm) dx(x)(µm) dy(x)(µm) dz(x)(µm) ey(x)(arcs) ez(x)(arcs) Y-axis (mm) dx(y)(µm) dy(y)(µm) dz(y)(µm) ex(y)(arcs) ez(y)(arcs) Z-axis (mm) dx(z)(µm) dy(z)(µm) dz(z)(µm) ex(z)(arcs) ey(z)(arcs) Squareness errors (arc s) A-axis(°) ex(a)(arcs) C-axis (°) ez(c)(arcs) Backlash errors (µm/arc s)
0 40 80 120 3 8 12 17 1 1 1 0 ⫺1 0 0 0 2 4 4 5 0 0 0 1 0 20 40 60 1 1 1 1 2 7 8 5 ⫺1 ⫺1 ⫺1 ⫺1 2 3 3 3 ⫺2 ⫺1 ⫺1 ⫺2 0 20 40 60 0 0 1 0 0 ⫺1 0 ⫺1 0 ⫺1 ⫺2 ⫺5 0 1 1 2 0 0 0 ⫺1 gxy ⫽ 8.9;gyx ⫽ ⫺7.5gzx ⫽ 5.5 0 7 14 0 ⫺129 ⫺106 0 30 60 0 1 3 Xb ⫽ 6;Yb ⫽ 10;Zb ⫽ 1Ab ⫽
160 22 0 1 5 1 80 1 9 ⫺1 2 ⫺2 80 ⫺1 0 ⫺9 1 ⫺1
21 28 ⫺167 ⫺178 90 120 6 4 254;Cb ⫽ 96
200 22 ⫺1 0 5 1 100 0 7 0 2 ⫺2 100 0 0 ⫺8 2 0
240 26 ⫺1 1 5 1 120 0 5 ⫺1 2 ⫺3 120 ⫺1 0 ⫺9 2 0
280 30 1 0 5 2 140 0 6 0 3 ⫺3 140 0 0 ⫺11 2 0
320 30 ⫺2 1 5 1 160 0 6 0 3 ⫺3 160 0 1 ⫺13 1 0
360 34 ⫺1 1 5 1 180 ⫺1 2 0 2 ⫺4 180 0 0 ⫺15 1 1
400 39 0 1 4 2 200 ⫺1 5 0 2 ⫺3 200 0 0 ⫺17 0 0
440 38 1 0 6 2 220 ⫺1 7 0 3 ⫺3 220 1 0 ⫺18 1 0
480 42 2 0 6 3 240 ⫺1 5 0 3 ⫺2 240 0 1 ⫺20 2 1
520 46 2 ⫺2 7 4 260 0 8 0 3 ⫺3 260 0 0 ⫺21 1 0
35 ⫺60 150 3
42 ⫺15 180 1
49 ⫺27 210 4
56 ⫺26 240 1
63 32 270 3
70 41 300 0
77 38 330 ⫺12
84 132 360 ⫺3
90 153
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Table 4 Nominal kinematic parameters of the target five-axis machine Parameters
Value (mm)
Parameters
Value (mm)
Parameters
Value (mm)
Z0 Y1 Z1 Z2
887.5 500 200 200
Z3 R Lt Xwo
85.5 150 120.224 49.794
Ywo Zwo
⫺0.279 52.966
Table 5 Results of the parameter estimation Parameters
Name
Value
a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13
d∗xa d∗ya d∗za aya aza dyc d∗zc axc ayc dxt dyt ext eyt
⫺57.4 (µm) ⫺4.8 (µm) 87.0 (µm) 6.8 (arc s) ⫺115.1 (arc s) 36.0 (µm) ⫺115.6 (µm) ⫺12.6 (arc s) 5.6 (arc s) 11.7 (µm) ⫺7.8 (µm) ⫺147.2 (arc s) ⫺6.7 (arc s)
ition errors along the test path F. The results are shown in Fig. 2. The predicted and the really measured overall position errors are very close. The deviations in X-,Y-, and Z-axis are in the range of ± 10 µm. To justify the effectiveness of the estimation further, the predicted and measured position errors along a total different test path S are also compared. The results are shown in Fig. 3. With this test path, the global tendency also matches very well. The deviations are greater and rise to the range of ± 16 µm. The path S causes the two rotary axes to move with velocity reversal (see Fig. 4 in Part I). The C-axis rotates in the opposed direction at the start phase. In test with path S, more error sources are involved, especially the backlash of the two rotary axes. Further measurements show that the backlash of A- and C-axis are stark position dependent and cannot be modeled accurately with a constant value. This explains why the difference between the predicted and the measured errors enlarges. The backlash of rotary axes must be measured carefully and compensated locally. Since all five axes are driven in a certain range to generate the spherical working space, the estimation with path F delivers an optimal solution minimizing the overall position errors in this working space. This is an essential advantage in comparison with traditional calibration methods. As mentioned previously, the unknown link errors in the error model of five-axis machine tool are fully known after the error estimation. Fig. 4 shows the results of
Fig. 2. Comparison between measured and predicted errors of path F.
further analysis of error sources. As expected, the rotary axes block is the major error source, which contributes up to 80% of the overall position errors. In contrast, the three linear axes contribute less than 20%. To predict the machining accuracy, the error model is derived in the workpiece coordinate frame to describe the overall position and orientation errors of tool relative to workpiece. The motional errors and the link errors remain unchanged. Figs. 5 and 6 show the predicted position and orientation errors of the experimental five-axis milling machine while positioning the tool on the spheri-
W.T. Lei, Y.Y. Hsu / International Journal of Machine Tools & Manufacture 42 (2002) 1163–1170
Fig. 3. Comparison between measured and predicted errors of path S.
Fig. 4.
Error sources analysis.
cal surface in the normal direction. It can be seen that the errors at the top point of the spherical test surface have more than one value and direction. The setting tool orientation is in the Z-direction and the machine is at its singular point. The C-axis can take any position. For different C-axis positions, the resultant errors are also different.
5. Conclusions The error modeling technique is very useful in predicting the volumetric errors of CNC machine tools. Until now the implementation of this technique in fiveaxis machine tools faced great problems. Although the majority of motional errors in the error model are measurable with modern measurement devices, there are still some link errors that are not measurable. These not measurable errors include constant link errors of rotary axes block, main spindle block, and tool holder. Much
Fig. 5.
Predicted position errors.
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is to implement the real-time error compensation function in the CNC, which will be published later. References
Fig. 6.
Predicted orientation errors.
worse is the fact that errors coming from the rotary block are dominant. This study presents a new method to estimate these link errors by applying the newly developed probe-ball device described in Part I. A reduced error model is used for LSE to increase the accuracy of estimation and to accelerate the estimation process. Tests with different paths prove that the proposed estimation method delivers very good results. The probe-ball device and the errors estimation method have great impact on the accuracy enhancement of five-axis machine tools. Taking the information flow of five-axis machining into consideration, one immediate way to improve the accuracy of five-axis machining without upgrading the CNC controller’s functionality is to apply the error model in the post-processor of CAD/CAM systems. Another way
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Accuracy test of five-axis CNC machine tool with 3D probe-ball. Part II: errors estimation W.T. Lei ∗, Y.Y. Hsu Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu, Taiwan, ROC Received 16 October 2001; received in revised form 12 April 2002; accepted 22 April 2002
Abstract To improve the accuracy of CNC machine tools, error sources and its effects on the overall position and orientation errors must be known. Most motional errors in the error model of five-axis machine tool can be measured with modern laser interferometer devices, but there are still some not measurable geometric errors. These not measurable errors include constant, inaccurate link errors of components such as rotary axes block, main spindle block and tool holder. After setting all measured errors in the error model, a reduced error model is defined, which describes the influence of each unknown and not measurable link error on the overall position errors of the five-axis machine tool. On the other hand, the newly developed probe-ball device can measure the overall position errors of five-axis machine tools directly. Based on the reduced model and the overall position errors, the link errors can be estimated very accurately with the least square estimation method. The error model is then fully known and can be used for advanced purposes such as error prediction and compensation. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Geometric error; Least square estimation; Five-axis machine tool; Probe-ball
1. Introduction The technique of building machine tool’s geometric error model is well developed in the past few years [1– 5]. The error model describes the position and orientation errors of tool relative to workpiece at specific machine position, whereby inaccurate influential factors come from kinematic link parameters and individual error sources. It is well known that the inaccurate motion of a linearly driven axis is associated with six motional errors, including one linear error, two straightness errors, and three rotational errors. With modern measurement devices such as the 6D laser interferometer [6], all six motional errors of the linearly driven axis can be measured rapidly. Based on the error model, the accuracy of three-axis machine tools can be dramatically improved through the error compensation [7,8]. While most accuracy enhancement research focuses on three-axis CNC machine tools, relevant works in the area of five-axis
Corresponding author. Tel.: +886-3-5742921; fax: +886-35722840. E-mail address: [email protected] (W.T. Lei). ∗
machine tools are rather sparse. Kiridena and Ferreira [3] developed a viable method for modeling and visualizing the effects of position errors on the volumetric accuracy of five-axis CNC machine tool. Srivastava et al. [5] used homogenous transformation matrix to develop the geometric and thermal error model in a fiveaxis CNC machine tool. Although the theoretical error model of five-axis machine tool is known, it is still impossible to improve the accuracy of five-axis machine tool with the error compensation technique based on it. The reason is that there are some not directly measurable link errors in the error model, such as the inaccurate link errors of rotary axes block, main spindle block, and tool holder. These errors are constant and exist as deviations between coordinate frames and are difficult to access after the mounting process. It is clear that the key step toward effective accuracy enhancement of five-axis machine tools is the identification of these unknown link errors. The least square estimation (LSE) methods provide mathematical procedures by which a linear model can achieve a best fit to experimental data in the sense of least-squared error. The methods are powerful and welldeveloped mathematical tools that have been proposed
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and used in a variety of areas for decades, including adaptive control, signal processing, and statistics [9]. In the field of error estimation of machine tools, some researches also used this method to estimate error components [10–12]. In this article, the estimation of the unknown and not directly measurable link errors in the error model of the five-axis machine tool is addressed. The newly developed probe-ball device presented in Part I of this study offers a new tool to do so. It is capable of measuring the overall position errors of five-axis machine tools directly. The error model of the probe-ball measurement is already derived in Part I in detail. Based on this relationship and the measured overall position errors, the link errors can be estimated very accurately with the LSE method.
2. Basic theory The linear least-squares problem involves the use of a data set of measurements w, which are linearly related to the unknown parameter vector u by the expression w ⫽ Gu ⫹ v
(1)
where w is an m×1 vector, G, an m×n matrix, u, an n×1 vector, v, an m×1 vector of measurement ‘noise’, m, the number of measurements, and n, the number of the unknown parameters to be estimated. The goal is to find the unknown vector, denoted by u˜, which minimizes the sum of squares of the deviations, i.e. minimizes the cost function J J ⫽ (w⫺Gu˜)T(w⫺Gu˜)
(2)
The minimum is obtained when ∂J ⫽0 ∂u˜
(3)
and the Hessian of J is positive semidefinite ∂2J ⱖ0. ∂u˜2
| |
(4)
Differentiating J and setting the result to zero yields GTGu˜ ⫽ GTw
(5)
It can be shown that the second derivative of J, with respect to u˜, is positive semidefinite; and thus Eq. (3) does define a minimum [9]. The least-squares estimate of u˜ is then u˜ ⫽ (GTG)⫺1GTw
(6)
The method mentioned above is now applied to the estimation of not measurable link errors in the five-axis machine tools. The probe-ball device delivers necessary measurement data for the estimation, including three
deviations in X-, Y-, and Z-axis at one sampling. Since each link error has certain contribution to the three axes, it is necessary to rewrite Eq. (1) as follows q ⫽ Ha ⫹ e
(7)
where q is the measurement vector, a, the unknown link error vector, e, a noise vector, H, an error gain matrix which describes the relationship between the link error vector and the measurement output q ⫽ [q1,x % qm,x q1,y % qm,y q1,z % qm,z]T
(8)
a ⫽ [a1 % an]T f1,x(p1) f2,x(p1) % fn,x(p1)
(9)
H⫽
f1,x(p2) f2,x(p2) % fn,x(p2) ⯗
⯗
⯗
⯗
f1,x(pm) f2,x(pm) % fn,x(pm) f1,y(p1) f2,y(p1) % fn,y(p1) f1,y(p2) f2,y(p2) % fn,y(p2) ⯗
⯗
⯗
⯗
f1,y(pm) f2,y(pm) % fn,y(pm) f1,z(p1) f2,z(p2) % fn,z(p1) f1,z(p2) f2,z(p2) % fn,z(p2) M
M
M M
f1,z(pm) f2,z(pm) % fn,z(pm)
(10)
The functions fi,x(pj), fi,y(pj), and fi,z(pj) in matrix H represent the effects of link errors at specified setting position pj and are also called error gain functions in this article. The index n and m represent the number of parameter and sample, respectively. An explicit expression of each error gain function can be obtained from the error model in Part I of this study. The goal is to find a parameter vector, denoted by a˜, which minimizes the sum of squares of the deviations, thus the cost function for LSE
E(a˜) ⫽ eTe ⫽ (q⫺Ha˜)T(q⫺Ha˜)
(11)
An optimal estimation occurs when the cost function is at its minimum, which means ∂E(a˜) ⫽ 2HTHa˜⫺2HTq ⫽ 0 ∂a˜
(12)
The LSE of a˜ is then a˜ ⫽ (HTH)⫺1HTq
(13)
3. Reduced error model for estimation In Part I of study 59 errors are defined in the error model of the probe-ball measurement. To reduce the scope of estimation, a reasonable approach is to measure
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Table 1 Position-dependent, not measured motional errors Machine axis
X-axis Y-axis Z-axis A-axis C-axis
Translational errors
dx(a),dy(a),dz(a) dx(c),dy(c),dz(c)
Rotational errors
ex(x) ey(y) ez(z) ey(a),ez(a) ex(c),ey(c)
all measurable errors with available devices and set the not measured motional errors to zero. The estimation focuses then only on the unknown, constant, and not measurable link errors. The advantage of this approach is that the estimation can be done even if available measurement devices are limited, for example only linear or 5D laser interferometer is available. Obviously, the accuracy of estimation will be better if more powerful devices are used. The error estimation begins with the sorting and merging of unknown link errors into independent parameters. In the error model of the probe-ball measurement, some errors have the same gain factor. For example, the constant rotational error aya and the position-dependent motional error ey(a) have the same gain factor ⫺ XwoCce ⫹ YwoSce in contributing to the position error ⌬Zp. Besides, the position-dependent motional error dz(a) adds with the constant link errors dza, dzs, and dzh in contributing to the position error ⌬Zp. These errors cannot be estimated separately. To simplify the estimation process and to maintain the physical meanings of error sources, all position-dependent, not measured motional errors are set to zero, as seen in Table 1. The errors with the same gain factor are merged into one error parameter without loss of their geometrical features. For example, the not measured tilt error of A-axis ey(a) is set to zero and the constant rotational error aya Table 2 Error parameters to be estimated Parameter Name a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13
d∗xa d∗ya d∗za aya aza dyc d∗zc axc ayc dxt dyt ext eyt
Sources of error dxa,dxc,dxs,dxh,eys,eyh dya,dyh,dys,exs,exh,ezt dza,dzs,dzh aya aza dyc dzc,dzt axc ayc dxt dyt ext eyt
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is to be estimated. Also, the not measured motional error dz(a) is set to zero and the constant link errors dza, dzs, and dzh are merged together and renamed as error para∗ , since all errors contribute in the Z-direction. meter dza In this way, the total 59 geometric errors are reduced to 13 independent, constant error parameters. These error parameters form the parameter vector a and are shown in Table 2. The position error model (⌬Xp,⌬Yp,⌬Zp) is then divided into two sub-models: the device error model and the reduced error model (Cx,Cy,Cz) (⌬Xp∗,⌬Yp∗,⌬Zp∗). The device error model describes the contribution of all measured motional and not measured motional errors, whereby the not measured motional errors are set to zero. Therefore, the device error model contains no unknown error parameters. In contrast, the reduced error model describes the contribution of unknown error parameters to the overall errors ⌬Xp ⫽ Cx ⫹ ⌬X∗p
(14)
⌬Yp ⫽ Cy ⫹ ⌬Y∗p
(15)
⌬Zp ⫽ Cz ⫹ ⌬Z
(16)
∗ p
The device error model (Cx,Cy,Cz) can be expressed as Cx ⫽ dx(x) ⫹ dx(y)⫺dx(z) ⫹ (ey(x) ⫹ gyx)(Z2 ⫹ YwoSaeCce ⫹ XwoSaeSce ⫹ CaeZwo ⫹ CaeZ3) ⫹ (ez(x) ⫹ gzx)(⫺CaeXwoSce⫺CaeYwoCce
(17)
⫹ SaeZwo ⫹ SaeZ3) ⫹ ez(y)(⫺CaeYwoCce ⫺CaeXwoSce ⫹ SaeZwo ⫹ SaeZ3) ⫹ ey(z)(Lt ⫹ R) ⫹ Xm⫺YwoSce ⫹ XwoCce Cy ⫽ dy(x) ⫹ dy(y)⫺dy(z) ⫹ (ez(x) ⫹ gzx)( ⫺YwoSce ⫹ XwoCce) ⫹ (ex(y) ⫹ gxy)(⫺Z1⫺Z2 ⫺XwoSaeSce⫺YwoSaeCce⫺CaeZwo⫺CaeZ3)
(18)
⫹ ez(y)(Xm⫺YwoSce ⫹ XwoCce) ⫹ ex(z)(⫺Lt⫺R) ⫹ Ym⫺SaeZwo⫺SaeZ3 ⫹ CaeYwoCce ⫹ CaeXwoSce Cz ⫽ dz(x) ⫹ dz(y)⫺dz(z) ⫹ (ey(x) ⫹ gyx)(YwoSce ⫺XwoCce) ⫹ (ex(y) ⫹ gxy)(CaeXwoSce ⫹ CaeYwoCce ⫺SaeZwo⫺SaeZ3) ⫹ ex(z)(⫺Y1) ⫹ Z1⫺Z0 ⫹ Z2
(19)
⫹ XwoSaeSce ⫹ YwoSaeCce ⫹ CaeZwo ⫹ CaeZ3⫺Zm ⫹ Lt ⫹ R The reduced error model (⌬X∗p ,⌬Y∗p ,⌬Z∗p ) is ⌬Xp∗ ⫽ d∗xa ⫹ aya(ZwoCae ⫹ SaeYwoCce ⫹ SaeXwoSce ⫹ CaeZ3) ⫹ aza(⫺CaeXwoSce ⫺CaeYwoCce ⫹ SaeZwo ⫹ SaeZ3) ⫹ ayc(Z3 ⫹ Zwo) ⫹ dxt(Cce) ⫹ dyt(⫺Sce) ⫹ ext(ZwoSce)
(20)
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error parameter a5. Its error gain function f5,x(pj) is ⫺ CaeXwoSce⫺CaeYwoCce ⫹ SaeZwo ⫹ SaeZ3, the third row of Eq. (20). Similarly, the f5,y(pj) is ⫺YwoSce ⫹ XwoCce,, the second row of Eq. (21) and f5,z(pj) is 0 since the link error aza has no contribution to ⌬Z∗p .
⫹ eyt(ZwoCce) ∗ ⌬Y∗p ⫽ dya ⫹ aza(⫺YwoSce ⫹ XwoCce) ⫹ axc(
⫺ZwoCae⫺SaeXwoSce⫺SaeYwoCce⫺CaeZ3) ⫹ ayc( ∗ ⫺SaeYwoSce ⫹ SaeXwoCce) ⫹ dyc(Cae) ⫹ dzc (
(21)
⫺Sae) ⫹ ext(⫺CaeZwoCce⫺SaeYwo) ⫹ eyt(CaeZwoSce ⫹ SaeXwo) ⫹ dxt(CaeSce)
4. Estimation procedure and results
⫹ dyt(CaeCce)
The block diagram of parameter estimation is shown in Fig. 1. Firstly, motional errors are measured with available devices. In this research, the 5D laser interferometer of API Inc. is used, which delivers only five motional errors of each linearly driven axis including three translational and two rotational errors. The roll errors of X-, Y-, and Z-axis are not measurable with this device and are set to zero. Note that the newest 6D laser interferometer can measure all six motional errors of a linearly driven axis. The perpendicularity errors between X-, Y-, and Z-axis are also measurable with the API laser device and a pentaprism. The axial errors of the two rotary axes are measured with a multi-faces optical mirror and an electronic level. Other motional errors of the
∗ ⌬Z∗p ⫽ dza ⫹ aya(⫺XwoCce ⫹ YwoSce) ⫹ axc(
⫺ZwoSae⫺SaeZ3 ⫹ CaeXwoSce ⫹ CaeYwoCce) ⫹ ayc(⫺CaeXwoCce ⫹ CaeYwoSce) ⫹ dyc(Sae)
(22)
⫹ d∗zc(Cae) ⫹ ext(CaeYwo⫺ZwoSaeCce) ⫹ eyt( ⫺CaeXwo ⫹ ZwoSaeSce) ⫹ dxt(SaeSce) ⫹ dyt(SaeCce) The reduced error model builds the mathematical basis for the LSE and can be re-arranged into vector form to obtain the error gain functions fi,x(pj),, fi,y(pj), and fi,z(pj) in the error gain matrix H for each setting position pj. For example, the link error aza is defined as
Fig. 1. The block diagram of error estimation.
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two rotary axes are not measurable in the laboratory and are set to zero. Note that all motional errors are of random variables in nature with statistical characteristics. The measurements are conducted according to the international standard ISO230-2. The accuracy of used measurement instruments are listed as follows: 앫 5D laser measurement device of API Inc. Linear accuracy: 1 ppm. Straightness accuracy: greater than ± 1µm in the range of measurements. Angular accuracy: greater than ± 1.0arcs in the range of measurements. 앫 Electronic level accuracy: ± 1.0arcs 앫 Probe of Renishaw Inc. Accuracy: ± 2µm over 2 mm range with 1 µm resolution in X, Y, and Z axes. The measurement data are shown in Table 3. The nominal kinematic parameters of the five-axis milling machine tool are calibrated and are shown in Table 4. The five-axis milling machine tool is tested with two different test paths: the helix-like one path F and the circular one path S. To evaluate the results of estimation, the path F is used for estimation and the path S is used for justification. The radius of the
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spherical test surface is 150 mm. The overall position errors and the setting positions pj of each error sampling are registered for the purpose of error estimation. The total number of samples for path F is 118. After the measurements, the values of error gain functions fi,x(pj), fi,y(pj), and fi,z(pj) in the error gain matrix H are computed for each setting position pj. The elements of the measurement vector q are obtained through the subtraction of the results of device error model (Cx,Cy,Cz) from the registered overall position errors also for each setting position pj, (see Eqs. (14)– (16)). With the error gain matrix H and the measurement vector q known, the unknown error parameter vector a˜ is obtained through solving Eq. (13) directly with the help of the mathematical tool software matlab. The solution of error parameter vector a˜ involves great amount of computation with matrices and arrays. No iteration is necessary. The results of estimation based on the test path F are shown in Table 5. Most estimated errors are larger than the measured motional errors. The measured rotational errors of linearly driven axes are within 5 arc s. In contrast, the estimated rotational error aza is ⫺1151 arc s. With the error gain function ⫺ CaeXwoSce⫺CaeYwoCce ⫹ SaeZwo ⫹ SaeZ3, the link error aza plays a dominant role in the volumetric errors of the target five-axis milling machine in the X-direction. For the sake of justification, the estimated link errors are set into the error model to compute the overall pos-
Table 3 Measured motional errors in mean X-axis (mm) dx(x)(µm) dy(x)(µm) dz(x)(µm) ey(x)(arcs) ez(x)(arcs) Y-axis (mm) dx(y)(µm) dy(y)(µm) dz(y)(µm) ex(y)(arcs) ez(y)(arcs) Z-axis (mm) dx(z)(µm) dy(z)(µm) dz(z)(µm) ex(z)(arcs) ey(z)(arcs) Squareness errors (arc s) A-axis(°) ex(a)(arcs) C-axis (°) ez(c)(arcs) Backlash errors (µm/arc s)
0 40 80 120 3 8 12 17 1 1 1 0 ⫺1 0 0 0 2 4 4 5 0 0 0 1 0 20 40 60 1 1 1 1 2 7 8 5 ⫺1 ⫺1 ⫺1 ⫺1 2 3 3 3 ⫺2 ⫺1 ⫺1 ⫺2 0 20 40 60 0 0 1 0 0 ⫺1 0 ⫺1 0 ⫺1 ⫺2 ⫺5 0 1 1 2 0 0 0 ⫺1 gxy ⫽ 8.9;gyx ⫽ ⫺7.5gzx ⫽ 5.5 0 7 14 0 ⫺129 ⫺106 0 30 60 0 1 3 Xb ⫽ 6;Yb ⫽ 10;Zb ⫽ 1Ab ⫽
160 22 0 1 5 1 80 1 9 ⫺1 2 ⫺2 80 ⫺1 0 ⫺9 1 ⫺1
21 28 ⫺167 ⫺178 90 120 6 4 254;Cb ⫽ 96
200 22 ⫺1 0 5 1 100 0 7 0 2 ⫺2 100 0 0 ⫺8 2 0
240 26 ⫺1 1 5 1 120 0 5 ⫺1 2 ⫺3 120 ⫺1 0 ⫺9 2 0
280 30 1 0 5 2 140 0 6 0 3 ⫺3 140 0 0 ⫺11 2 0
320 30 ⫺2 1 5 1 160 0 6 0 3 ⫺3 160 0 1 ⫺13 1 0
360 34 ⫺1 1 5 1 180 ⫺1 2 0 2 ⫺4 180 0 0 ⫺15 1 1
400 39 0 1 4 2 200 ⫺1 5 0 2 ⫺3 200 0 0 ⫺17 0 0
440 38 1 0 6 2 220 ⫺1 7 0 3 ⫺3 220 1 0 ⫺18 1 0
480 42 2 0 6 3 240 ⫺1 5 0 3 ⫺2 240 0 1 ⫺20 2 1
520 46 2 ⫺2 7 4 260 0 8 0 3 ⫺3 260 0 0 ⫺21 1 0
35 ⫺60 150 3
42 ⫺15 180 1
49 ⫺27 210 4
56 ⫺26 240 1
63 32 270 3
70 41 300 0
77 38 330 ⫺12
84 132 360 ⫺3
90 153
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Table 4 Nominal kinematic parameters of the target five-axis machine Parameters
Value (mm)
Parameters
Value (mm)
Parameters
Value (mm)
Z0 Y1 Z1 Z2
887.5 500 200 200
Z3 R Lt Xwo
85.5 150 120.224 49.794
Ywo Zwo
⫺0.279 52.966
Table 5 Results of the parameter estimation Parameters
Name
Value
a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13
d∗xa d∗ya d∗za aya aza dyc d∗zc axc ayc dxt dyt ext eyt
⫺57.4 (µm) ⫺4.8 (µm) 87.0 (µm) 6.8 (arc s) ⫺115.1 (arc s) 36.0 (µm) ⫺115.6 (µm) ⫺12.6 (arc s) 5.6 (arc s) 11.7 (µm) ⫺7.8 (µm) ⫺147.2 (arc s) ⫺6.7 (arc s)
ition errors along the test path F. The results are shown in Fig. 2. The predicted and the really measured overall position errors are very close. The deviations in X-,Y-, and Z-axis are in the range of ± 10 µm. To justify the effectiveness of the estimation further, the predicted and measured position errors along a total different test path S are also compared. The results are shown in Fig. 3. With this test path, the global tendency also matches very well. The deviations are greater and rise to the range of ± 16 µm. The path S causes the two rotary axes to move with velocity reversal (see Fig. 4 in Part I). The C-axis rotates in the opposed direction at the start phase. In test with path S, more error sources are involved, especially the backlash of the two rotary axes. Further measurements show that the backlash of A- and C-axis are stark position dependent and cannot be modeled accurately with a constant value. This explains why the difference between the predicted and the measured errors enlarges. The backlash of rotary axes must be measured carefully and compensated locally. Since all five axes are driven in a certain range to generate the spherical working space, the estimation with path F delivers an optimal solution minimizing the overall position errors in this working space. This is an essential advantage in comparison with traditional calibration methods. As mentioned previously, the unknown link errors in the error model of five-axis machine tool are fully known after the error estimation. Fig. 4 shows the results of
Fig. 2. Comparison between measured and predicted errors of path F.
further analysis of error sources. As expected, the rotary axes block is the major error source, which contributes up to 80% of the overall position errors. In contrast, the three linear axes contribute less than 20%. To predict the machining accuracy, the error model is derived in the workpiece coordinate frame to describe the overall position and orientation errors of tool relative to workpiece. The motional errors and the link errors remain unchanged. Figs. 5 and 6 show the predicted position and orientation errors of the experimental five-axis milling machine while positioning the tool on the spheri-
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Fig. 3. Comparison between measured and predicted errors of path S.
Fig. 4.
Error sources analysis.
cal surface in the normal direction. It can be seen that the errors at the top point of the spherical test surface have more than one value and direction. The setting tool orientation is in the Z-direction and the machine is at its singular point. The C-axis can take any position. For different C-axis positions, the resultant errors are also different.
5. Conclusions The error modeling technique is very useful in predicting the volumetric errors of CNC machine tools. Until now the implementation of this technique in fiveaxis machine tools faced great problems. Although the majority of motional errors in the error model are measurable with modern measurement devices, there are still some link errors that are not measurable. These not measurable errors include constant link errors of rotary axes block, main spindle block, and tool holder. Much
Fig. 5.
Predicted position errors.
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is to implement the real-time error compensation function in the CNC, which will be published later. References
Fig. 6.
Predicted orientation errors.
worse is the fact that errors coming from the rotary block are dominant. This study presents a new method to estimate these link errors by applying the newly developed probe-ball device described in Part I. A reduced error model is used for LSE to increase the accuracy of estimation and to accelerate the estimation process. Tests with different paths prove that the proposed estimation method delivers very good results. The probe-ball device and the errors estimation method have great impact on the accuracy enhancement of five-axis machine tools. Taking the information flow of five-axis machining into consideration, one immediate way to improve the accuracy of five-axis machining without upgrading the CNC controller’s functionality is to apply the error model in the post-processor of CAD/CAM systems. Another way
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