Highly efficient and accurate calibration method for the position-dependent geometric errors of the rotary axes of a five-axis machine tool

Definition of the machine tool structure and geometric error

The five-axis machine tool in this study consists of three linear axes (X-, Y-, and Z-axes), a tilting head (B-axis) and a rotary table (C-axis), as shown in Figure 2. According to the nature of rigid body motion, each motion axis has six degrees of freedom in the Cartesian coordinate system, and six motion errors exist in the corresponding direction of the motion axis.

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Figure 2.

Schematic of the five-axis machine tool.

According to the definition of ISO 230-1, ²⁷ PDGEs can be expressed with δ_ij and ε_ij, which represent the position error and angle error, respectively, wherein j represents the position coordinate and i represents the direction of error.

For the rotary axes in Figure 2, the six motion errors of the B-axis of the five-axis machine tool are position errors in the X, Y and Z directions, such as δxb, δyb, and δzb, and angle errors around the X, Y and Z-axes, such as εyb, εzb and εxb. For the C-axis, which contains position errors δxc, δyc, and δzc and angle errors are εxc, εyc and εzc.

Establishment of a measurement model for the PDGEs of the rotary axes

PDGEs can be described in the local coordinate system of the corresponding motion axis when the error model is established based on multibody system theory. The geometric error terms of the motion axis are defined in the local coordinate system of the motion axis itself rather than relative to the reference coordinate system of the computer numerical control (CNC) machine tool. This means that the measurement and identification of PDGEs should be performed in the local coordinate system of the motion axis itself. The local coordinate systems of the B- and C-axes are shown in Figure 2.

Modelling the motion error of the rotary axis

DBB is widely used to evaluate the performance of multiaxis machine tools. The PDGEs of the machine tool motion axis can be identified by circular trajectory tests with well-developed identification algorithms. The structure of the DBB is relatively simple, consisting of two precision balls and a magnetic telescoping bar.

Precision balls O₁ and O₂ are installed on the tool cup and centre mount, respectively. When there is relative motion between the measured parts in the machine tool, the displacement of the precision balls can be measured with the displacement sensor in the magnetic telescoping bar. The error of the motion radius and the error of the circular trajectory will be formed, and the PDGEs and PIGEs can be identified with an established geometric error model.

According to the multibody system theory and homogeneous transformation method, the position relationship between the feature body k and the adjacent low-order body unit l of five-axis machine tools can be expressed by the homogeneous transformation matrix as follows:

E l k = E p l k E pe l k E m l k E me l k

(1)

where E p l k is the position transformation matrix, E pe l k is the position error transformation matrix, E m l k is the motion transformation matrix and E me k j is the motion error transformation matrix.

The positions of O₁ and O₂ in the measuring coordinate system are P₁= [x₁, y₁, z₁, 0]^T and P₂= [x₂, y₂, z₂, 0]^T, respectively. The accuracy of the rotary axes would not be affected by the PDGEs under ideal conditions, and the ideal transformation matrices of O₁ and O₂ are shown in equation (2).

When asynchronous motion exists in the C- and B-axes, one precision ball moves with the tool cup or centre mount while the other remains stationary. Hence, the matrix responding to the stationary axis is the unit matrix, and the C- and B-axes are set to exhibit asynchronous motion to enable efficient and accurate measurements.

E m d = I 4 × 4 I 4 × 4 [ cos c − sin c 0 0 sin c cos c 0 0 0 0 1 0 0 0 0 1 ] [ cos b 0 sin b 0 0 1 0 0 − sin b 0 cos b 0 0 0 0 1 ] I 4 × 4 P m

(2)

where P_m is the position of O₁ and O₂ in the measuring coordinate system.

The accuracy of the rotary axes would be affected by PDGEs under actual conditions, and the transformation matrices of O₁ and O₂ are expressed as follows:

E m a = I 4 × 4 [ cos ε z θ − sin ε z θ 0 0 sin ε z θ cos ε z θ 0 0 0 0 1 0 0 0 0 1 ] [ 1 0 0 0 0 cos ε y θ − sin ε y θ 0 0 sin ε y θ cos ε y θ 0 0 0 0 1 ] [ cos ε x θ 0 sin ε x θ 0 0 1 0 0 − sin ε x θ 0 cos ε x θ 0 0 0 0 1 ] [ 1 0 0 δ x θ 0 1 0 δ y θ 0 0 0 δ z θ 0 0 0 1 ] [ cos c − sin c 0 0 sin c cos c 0 0 0 0 1 0 0 0 0 1 ] [ cos b 0 sin b 0 0 1 0 0 − sin b 0 cos b 0 0 0 0 1 ] I 4 × 4 P m

(3)

where θ represents the rotation angle of the C- and B-axes.

The distance between the precision balls of the DBB in the ideal state and the actual state can be determined by the positions of O₁ and O₂. The expression of the distance between the precision balls of the DBB is as follows:

e a − e d = E 2 a − E 1 a − E 2 i − E 1 i

(4)

The length variation between the precision balls of the DBB is a space error vector in three directions, which can be expressed as follows:

[ Δ x Δ y Δ z 0 ] T = e a − e d

(5)

The expression of the error vector can be determined by substituting the homogeneous transformation matrix. Conducting asynchronous motion of the C- and B-axes can effectively avoid the influence of servo error and is conducive to identifying the PDGEs of the rotary axes accurately and effectively. Therefore, when measuring the geometric error of the C-axis, it is necessary to ensure that the precision ball at the tool cup is not affected by the motion of the B-axis, and the motion of the C-axis and linear axes should be minimized when performing the circular trajectory test to identify the geometric error of the B-axis. The length variations along the orthogonal directions of the spatial error vector of the C- and B-axes can be obtained by circular trajectory measurements. These length variations can be expressed as follows:

{ Δ x cn = ε zc x m sin c − y m ε zc cos c + z m ε yc + δ xc Δ y cn = ε zc x m cos c − y m ε zc sin c − z m ε xc + δ yc Δ z cn = x m ( ε xc sin c − ε yc cos c ) + y m ( ε yc sin c + ε xc cos c ) + δ zc Δ x bn = − x m ε yb sin b − y m ε zb + z m ε yb cos b + δ xb Δ y bn = x m ( ε zb cos b + ε xb sin b ) + z m ( ε zb sin b − ε xb cos b ) + δ yb Δ z bn = − x m ε yb cos b + y m ε xb − z m ε yb sin b + δ zb

(6)

where n represents the measurement mode under different installation positions.

The PDGEs of the rotary axes can be identified by the least squares method, and different measuring positions can be predefined. The analytical solutions of six geometric errors are determined from the measured data of the circular trajectory.

Setup and measurement mode for the C-axis

While the position of the precision ball on the tool cup moves along the measuring coordinate system, which is defined by the C-axis of the machine tool, and the other motion axes remain stationary, the length variation of the DBB will be affected by the precision ball that is clamped in the centre mount when the C-axis rotates. From the expression of the measurement model of PDGEs (equation (6)), it can be found that the establishment of a non-singular matrix of PDGEs is an essential condition for identifying six geometric errors of the rotary axis. Thus, six linear independent positive definite equations are constructed. Note that x_i, y_i and z_i (i = 1, 2… 6) in equation (6) correspond to the initial installation position of the precision ball on one side of the worktable in the measuring coordinate system. Three installations and six measurement patterns were planned for identifying the PDGEs of the rotary table on the five-axis machine tool with a tilting head. The installation and measurement pattern are shown in Figure 3.

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Figure 3.

DBB setup and measurement mode for the C-axis: (a) First setup, (b) Second setup, (c) Third setup.

In the first installation of the DBB, the telescoping bar of the DBB is installed along the direction of the X-, Y- and Z-axes. In the second installation of the DBB, the telescoping bar of the DBB is installed along the direction of the X- and Y-axes after resetting the centre mount along the C-axis. In the third installation of the DBB, O₂ of the DBB is installed along the direction of the angular bisector between the X-axis and the negative direction of the Y-axis. The C-axis rotates in the XY plane, while the B-axis and other axes are kept stationary. The initial locations of O₁ and O₂ in the measuring coordinate system are shown in Figure 3.

Setup and measurement mode for the B-axis

The location of the precision ball that is attached to the tool cup does not coincide with the B-axis because there is a larger distance between the cutting tool end and the B-axis, which prevents the tool cup of the DBB from being accurately installed in line with the B-axis. To perform circular or arc trajectory tests, it is necessary to conduct multiaxis motion of the B-axis and linear axes.

The local coordinate system of the B-axis is used as the reference coordinate system for measuring the PDGEs of the B-axis. The installation and measurement pattern are shown in Figure 4.

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Figure 4.

DBB setup and measurement mode for the B-axis: (a) First DBB setup, (b) Second DBB setup, (c) Third DBB setup, (d) Fourth DBB setup.

In the first installation of the DBB, the telescoping bar of the DBB is installed along the direction of the X- and Y-axes, O₁ is installed in the direction of the spindle, O₂ is installed in the direction of the Y- or X-axes, and the distance between the precision balls is L+L₁. In the second installation of the DBB, the telescoping bar of the DBB is installed along the direction of the X- and Z- axes, O₁ is installed in the direction of the spindle, O₂ is installed in the direction of X-axis, and the distance between the precision balls is L. In the third installation of the DBB, the telescoping bar of the DBB is installed and parallel to the direction of the X-axis, O₁ is installed along the direction of the angular bisector between the X- and Y-axes, and the distance between the precision balls is L. In the fourth installation of the DBB, the telescoping bar of the DBB is installed along the direction of the X-axis, O₁ is installed in the direction of the spindle, the distance between the precision balls is L, and O₂ is on the centre mount that is clamped on the rotary table with a distance of H₁. The B-axis rotates, the X- and Z-axes exhibit synchronous motion, and the other axes are kept stationary when performing circular or arc trajectory tests.

Continuous measurement mode based on the DBB setup

Multiple installations and measurements are bound to reduce the accuracy of the measurement results; however, circular trajectory testing with a DBB in different planes can be carried out continuously at the same initial installation position, and the DBB needs to be installed only once. Based on the above characteristics of the test setups of the DBB, to perform a volumetric test, five DBB setups were proposed in this section, wherein the DBB needed to move through the test trajectory shown in Figure 5.

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Figure 5.

Measurement process based on five DBB setups.

As shown in Figure 5, the initial installation position of the DBB is consistent for the measurement mode of the C- and B-axes in the 4th DBB setup. Upon DBB installation, the influence of the installation error can be substantially avoided, and the measurement efficiency can be effectively improved.

PDGEs identification based on the adaptive LASSO method

The geometric errors of the rotary axis are related to the command position of the rotary axis, that is, the motion errors change with respect to the command position. Therefore, the motion errors of the C- and B-axes (the rotary axes) can be expressed as functions of the rotation angles. The PDGEs can be expressed as polynomial functions of the motion variables.^9,20,27 The polynomial model of translational and angular errors of the rotary axes can be defined as follows:

{ δ ij = ∑ k = 0 n α ij , k i k ( i , j = x , y , z ) ε ij = ∑ k = 0 n β ij , k i k ( i , j = x , y , z )

(7)

where α_ij and β_ij are the coefficients of the translational and angular error polynomials, respectively, and k is the order of the geometric error polynomials.

By combining equations (6) and (7), a linear mapping model between the geometric error source and the variation in DBB can be expressed as follows:

Δ l = DE

(8)

where Δl is the variation in the DBB in the five different setup positions, D is composed of the initial coordinates of O₁ and O₂ and the command value of the motion axes, and E is a vector consisting of translational and angular errors, which are functions of the coefficients and order of the geometric error polynomials.

The values of geometric error can be identified by solving equation (6), and the ill-conditioned problem can be effectively avoided when the initial position of the magnetic ball is installed properly. For the PDGEs of the rotary axes of the machine tool, the function forms are nonlinear, the coefficients of the geometric error polynomials are biased estimations, and the accuracy of the identified geometric error is always susceptible to measurement uncertainty. Hence, the adaptive LASSO regression method is adopted to overcome this problem in this paper. ²⁸ The adaptive LASSO regression method is a regularization approach that simultaneously chooses variables and estimates parameters, and different weights are assigned to coefficients of the translational and angular error polynomials. The expression of the adaptive LASSO method model is defined as follows:

E ^ = arg min E { ‖ Δ l − DE ‖ 2 + λ ∑ m = 1 n 1 | E m | r | E m | } ( γ > 0 ) , m = 1 , 2 , … , n

(9)

where λ and r are adjustment parameters.

In the adaptive LASSO method, different parameters of polynomial coefficients can be adjusted with a penalty function r; the more important the parameters of the polynomial coefficients, the smaller the adjustment parameters, and vice versa.

The adaptive LASSO estimates in equation (9) can be determined with the least angle regression (LARS) algorithm. ²⁹ The basic steps are as follows:

1) Define x j * * = x j / w j ∧ j = 1 , 2 , ⋯ , p

2) All adjustment parameters λ and n can be determined by solving the LASSO problem

β ^ * * = arg min β ‖ y − ∑ j = 1 p x j * * β j ‖ 2 + λ n ∑ j = 1 p | β j |

(10)

β ^ * ( n ) = β ∧ * * / w j ∧ j = 1 , 2 , ⋯ , p

(11)

According to the properties of the geometric error terms, the corresponding values are generated in the form of cubic polynomials.^20,30 The measurement results show that the identification precision will be affected by the number of measurement points Mp and the order n of the polynomial.

DBB setup and measurement

The travel range of the C-axis is from 0° to 360°, whereas the travel range of the B-axis is from −5° to 5°. The measured range of the B-axis is from −20° to 90° due to the limit of travel of the linear axis. The angular overshoot of the arc trajectories for the C- and B-axes is 5°. The total measurement processes include twelve measurement tests under five DBB setups. The measurement processes of the 1st DBB setup are depicted in Figure 3(a), and the details are as follows:

The measurement processes of the 2nd DBB setup are depicted in Figures 3(b) and 4(d), and the details are as follows:

The measurement processes of the 3rd DBB setup are depicted in Figure 4(c), and the details are as follows:

The measurement process of the 4th DBB setup are depicted in Figure 3(c) and Figure 4(b), and the details are as follows:

The measurement processes of 5th DBB setup are depicted in Figure 4(a), and the details are as follows:

During the C-axis measurement tests only the C-axis rotation is needed, and the six measurement tests required just three DBB setups, thereby greatly enhancing the measurement efficiency. The measurement process does not include synchronous motions of other axes, which increases the measurement accuracy. The B-axis measurement tests also minimize the number of DBB setups, and during the B-axis measurement tests, the other two axes are kept stationary, thereby offering better measurement consistency.

Identification result of the PDGEs of the rotary axes

In the twelve measurement modes of the five DBB setups, circular trajectory tests were carried out three times under the same experimental conditions to avoid disturbance according to ISO 230-1. Note that Δl can be determined with the measurement data of the circular trajectory test with the DBB. In this paper, the number of measuring points Mp is 180 and 45 for the C- and B-axes, respectively, and n = 3 is applied to establish the identification model. The LARS algorithm was used to obtain the adaptive LASSO parameter, and the adjustment parameters λ are 8.31 × 10⁻⁴ and 7.92 × 10⁻⁴ for the C- and B-axes, respectively. The identified values of PDGEs can be determined by substituting the values of the adjustment parameters into equations (4)–(6). The identified PDGEs of the C-axis are shown in Figure 6(a)–(d).

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Figure 6.

Identified PDGE results without compensation: (a) Linear errors of the C-axis, (b) Angle errors of the C-axis, (c) Linear errors of the B-axis, (d) Angle errors of the B-axis.

Figure 6 shows the mean of three measurements, and the error bar represents the expanded uncertainty. The maximum linear errors and angle errors of the C-axis were 1.4 μm and 3.2″ for the measured stroke, respectively. The maximum linear errors and angle errors of the B-axis were 1.7 μm and 3.5″ for the measured stroke, respectively. The uncertainty of the measurement and identification results was smaller than the repeatable positioning accuracy of the motion axes; thus, no significant difference existed among multiple measurements, and the measurement and identification accuracy had satisfactory consistency.

Geometric error compensation and accuracy comparison

The identified PDGEs can be compensated by generating new numerical control (NC) codes with inverse kinematic operation or post-processing software based on a well-defined compensation method,^3,31 and the same measurements were performed based on the five DBB setups. Then, the identified PDGEs were determined after compensation. As shown in Figures 7(a) and (b), for the linear error of the C-axis, the average compensation rate was greater than 51.5%, and the maximum compensation rate was 74.6%. For the angle error of the C-axis, the average compensation rate was greater than 52.7%, and the maximum compensation rate was 80.9%.

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Figure 7.

Identified PDGEs results with compensation: (a) Linear errors of the C-axis, (b) Angle errors of the C-axis, (c) Linear errors of the B-axis, (d) Angle errors of the B-axis.

As shown in Figures 7(c) and (d), for the linear error of the B-axis, the average compensation rate was greater than 68.4%, and the maximum compensation rate was 82.8%. For the angle error of the B-axis, the average compensation rate was greater than 64.6%, and the maximum compensation rate was 73.8%.

To further verify the identification accuracy of the method proposed in this paper, an experiment was carried out to compare the results from the proposed method with those from other common measurement methods; moreover, the compensation effects were analysed. ISO 10791-6:2014 ³² specifies the measurement trajectory of an equivalent cone frustum with synchronous motion of the linear axes and rotary axes, as shown in Figure 8(a). The experimental setup of a DBB on a five-axis machine tool with a tilting head and a rotary table is shown in Figure 8(b).

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Figure 8.

Cone frustum trajectory test setup: (a) Schematic of measurement trajectory and (b) Experimental DBB setup.

In the measurement stage, the inclined angle between the base of the measurement circle and the table surface is 30°, the apex angle of the cone frustum is 10°, the centre offset between the centre mount and the axis of the rotary table is 50 mm, and the feeding speed is 1000 mm/min, following the guidelines in ISO 10791-6:2014. Three measurements are conducted under the same experimental conditions and different compensation parameters.

A comparison of the equivalent cone frustum test results from the proposed method and the method in Jiang et al. ³³ is shown in Figure 9.

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Figure 9.

Comparison of measured trajectories.

As shown in Figure 9, the average radius deviation of the circular trajectory is 17.0 μm before PDGEs compensation, and the circular deviation of the circular trajectory is 25.3 µm. After the geometric error is compensated with the method in Jiang et al., ³³ the average radius deviation of the circular trajectory is 11.4 μm, and the circular deviation of the circular trajectory is 14.0 µm. After error compensation is carried out with the method proposed in this paper, the average radius deviation and circular deviation of the circular trajectory are 6.4 µm and 8.6 µm, respectively.

After error compensation by means of the values of multiple measurements, the radius deviation and circular deviation are reduced by 44.7% and 32.9%, respectively. After compensating for the error based on five measurements and identification with the adaptive LASSO method, the radius deviation and circular deviation are reduced by 66.0% and 62.3%, respectively.