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Abstract

Error compensation technology as an economical and effective method for improving the machining accuracy of the machine tools has been widely applied in the manufacturing. How to quickly and accurately obtain machine tool error is an important prerequisite for achieving the above task. A laser tracker is adopted the multi-station and time-sharing measurement principle to achieve quick and accurate identification of the geometric errors of the machine tool. In this article, the error of single-station measurement of laser tracker is analyzed first, and then the principle and algorithm of multi-station and time-sharing measurement are given. Based on the kinematic model of the machine tool, the mapping relation equations between the space motion error and individual geometric error of the machine tool are established, and two different identification algorithms are derived, respectively. Meanwhile, the identification accuracy of two algorithms and the impact of different measurement areas and squareness errors on two algorithms are compared and analyzed in depth. The experiment results indicate that higher identification accuracy can be obtained by the second algorithm, and individual geometric error of the machine tool can be quickly and accurately separated by this algorithm.

Keywords

Laser tracker multi-station and time-sharing measurement geometric error kinematic model error identification

Introduction

With the development of modern manufacturing, the demand of machining accuracy of numerical control machine tool is increasing. The accuracy and stability of accuracy are important performance index of numerical control machine tools, so how to improve the machining accuracy is very important. Error compensation technology is an economical and effective method for improving the machining accuracy of machine tools,^1–5 which has been widely used in the manufacturing. Studies have shown that geometric error accounts for large proportion of machine tool error. Meanwhile, geometric error is relatively stable,^6–8 so it is easy to be compensated. Currently, compensation for the geometric errors of the machine tool has become an important way to improve the machining accuracy of machine tool.^9–11 How to quickly and accurately obtain machine tool error is an important prerequisite for achieving the above task, which directly affects the effectiveness of error compensation.

At present, there are many methods for detecting the geometric error of numerical control (NC) machine tool. According to the different measurement principles, these methods can be divided into direct measurement (single error detection) and indirect measurement (comprehensive error identification). For single error detection, the instrument is used to directly detect the machine tool error, such as detecting the position error of machine tool by laser interferometer and gauge block. Each error of machine tool can be obtained by direct measurement,^12,13 but the detection cycle is long. Meanwhile, different errors should be detected by different instruments, so the measurement cost is relatively high. For comprehensive error identification, the kinematic model of machine tool is used. The motion contour trajectory of machine tool or the volumetric position errors of measuring points during the working space of machine tool are measured, and then individual error can be separated according to machine tool error model, such as ball bar measurement,^14,15 plane grating measurement¹⁶ and one-dimensional (1D) ball array measurement.¹⁷ In recent years, laser interferometer as a high-precision measuring instruments has been widely used in machine error detection.^18,19 A variety of machine error detection methods based on the displacement measurement are developed, such as the 22-line method, 15-line method, 14-line method and 9-line method. In the above measurement methods, laser interferometer measurement is most commonly used. The laser wavelength is taken as the measurement reference, so this method has high accuracy. But the light adjustment process is more complex, and the detection cycle is longer, which directly affects the measurement efficiency. The other methods mentioned above also have some certain deficiencies in measurement efficiency and accuracy. Since 1980s, three-dimensional (3D) and dynamic tracking measurement has been developed rapidly to measure the motion of the robot and the large workpiece assembly.^20–22 This method has the advantages of fast, dynamic measurement,²³ and it has been widely used in industrial measurement field.^24–26 Compared with laser interferometer, a tracking mechanism is added to laser tracker to achieve the space coordinates’ measurement.²⁷ The light adjustment process is easy for this measurement, and the measurement cycle is shorter. Based on the advantage of this method, a laser tracker is adopted multi-station and time-sharing measurement principle to rapidly and accurately detect the geometric errors of machine tools in this article. The machine tool is controlled to move on the preset path in the 3D space, and a laser tracker is used to measure the same motion trajectory of the machine tool at different base stations. Then each position of base station and the space coordinates of measuring points can be determined according to Global Positioning System (GPS) principle. The mapping relation equations between the space motion error and individual geometric error of the machine tool are established, and individual error can be separated by solving the equations. According to whether the relative position deviations between different axes are considered in establishing the mapping model of error identification, two identification algorithms are derived, and the identification accuracy of two algorithms is compared and analyzed in depth. The experiment results indicate that higher identification accuracy can be obtained by the second algorithm, and individual geometric error of machine tool can be quickly and accurately separated by this algorithm.

Measurement principle of laser tracker

When the laser tracker is used, the measurement can be divided into single-station measurement and multi-station measurement according to the number of laser trackers. Single-station measurement adopts spherical coordinate measurement to calculate the space coordinates of a moving target as shown in equation (1)

{\begin{matrix} x = L \sin φ \cdot \cos θ \\ y = L \sin φ \cdot \sin θ \\ z = L \cos φ \end{matrix}

(1)

The error transfer function is

{\begin{matrix} δ_{x} = \frac{\partial x}{\partial L} \times δ_{L} + \frac{\partial x}{\partial φ} \times δ_{φ} + \frac{\partial x}{\partial θ} \times δ_{θ} \\ δ_{y} = \frac{\partial y}{\partial L} \times δ_{L} + \frac{\partial y}{\partial φ} \times δ_{φ} + \frac{\partial y}{\partial θ} \times δ_{θ} \\ δ_{z} = \frac{\partial z}{\partial L} \times δ_{L} + \frac{\partial z}{\partial φ} \times δ_{φ} + \frac{\partial z}{\partial θ} \times δ_{θ} \end{matrix}

(2)

$δ_{L}$ , $δ_{φ}$ and $δ_{θ}$ are defined as the distance measurement error, horizontal angle measurement error and elevation angle measurement error of the measuring point, respectively, in equation (2). For a measuring point $M$ , its space measurement error is given in equation (3)

e = \sqrt{δ_{x}^{2} + δ_{y}^{2} + δ_{z}^{2}} = \sqrt{δ_{L}^{2} + δ_{θ}^{2} L^{2} \sin φ^{2} + δ_{φ}^{2} L^{2}}

(3)

Then, the factors affecting the accuracy of single-station measurement of laser tracker can be analyzed as follows. In the measurement, the horizontal angle and elevation angle of a measuring point are assumed as $φ = π / 6$ and $θ = π / 12$ , respectively. The angle measurement error is taken as $1 ″$ , and the distance measurement error is taken as $0.6 μ m / m$ . Table 1 shows the space measurement errors of a measuring point caused by angle measurement and distance measurement with the measuring distance $L$ as 1, 3 and 5 m, respectively.

Table 1.

Space measurement errors of a measuring point (mm).

Measuring distance	Space measurement error caused by angle measurement	Space measurement error caused by distance measurement	Overall space measurement error
1	0.00500	0.00059	0.00504
3	0.01502	0.00179	0.01513
5	0.02503	0.00299	0.02521

It can be seen from Table 1 that the coordinate measurement error is mainly caused by angle measurement. Meanwhile, the angle measurement error will be further amplified with the increase in distance, which directly affects the overall accuracy of the spatial coordinates.²⁸ When the single-station measurement is used to detect the high-accuracy machine tool, the measurement accuracy cannot be guaranteed due to the impact of angle measurement error.

Principle and algorithm of multi-station and time-sharing measurement

In order to avoid the influence of angle measurement error,^29,30 the multi-station and time-sharing measurement is adopted in the article. A laser tracker is used to measure the target point at different base stations, and the spatial coordinates of measuring point can be determined by the laser ranging information measured at multiple base stations. Only distance is involved in the measurement, and the impact of angle measurement error is avoided. Therefore, this measurement has higher accuracy. Figure 1 shows the measurement principle of the machine tool by multi-station and time-sharing measurement. In the measurement, the machine tool is controlled to move on a preset path in the 3D space, and a laser tracker is used to measure the same motion trajectory of the machine tool at different base stations. The corresponding measurement algorithm is given as follows.

Figure 1.

Principle of multi-station and time-sharing measurement.

Taking the calibration of base station $P_{1}$ for example, $A_{0}$ is defined as the initial measuring point, and the theoretical coordinates of measuring point in the machine tool motion are assumed as $A_{i} (x_{i}, y_{i}, z_{i}) (i = 1, 2, \dots, n)$ . When the laser tracker is aimed at the initial measuring point $A_{0}$ , the distance reading of laser tracker is assumed to be set 0. The distance between base station $P_{1}$ and initial measuring point $A_{0}$ is defined as $L_{1}$ , and the distance variation between base station $P_{1}$ and measuring point $A_{i} (x_{i}, y_{i}, z_{i})$ is defined as $l_{1 i}$ . For base station $P_{1}$ , the following equations can be established

{\begin{matrix} \sqrt{{(x - x_{1})}^{2} + {(y - y_{1})}^{2} + {(z - z_{1})}^{2}} = L_{1} + l_{11} \\ \sqrt{{(x - x_{2})}^{2} + {(y - y_{2})}^{2} + {(z - z_{2})}^{2}} = L_{1} + l_{12} \\ ⋮ \\ \sqrt{{(x - x_{i})}^{2} + {(y - y_{i})}^{2} + {(z - z_{i})}^{2}} = L_{1} + l_{1 i} \end{matrix}

(4)

The least squares method can be used to solve the above nonlinear redundant equations.³¹ Let $C = x^{2} + y^{2} + z^{2} - L_{1}^{2}$ , the objective function is defined as

\begin{matrix} F (x, y, z, L_{1}, C) = \sum_{i = 1}^{N} (x_{i}^{2} + y_{i}^{2} + z_{i}^{2} - 2 x_{i} x \\ {- 2 y_{i} y - 2 z_{i} z + C - 2 L_{1} l_{1 i} - l_{1 i}^{2})}^{2} \end{matrix}

(5)

The analytical method is used to solve the least squares problem, then we can obtain

\begin{matrix} [\begin{matrix} 2 \sum_{i = 1}^{N} x_{i}^{2} & 2 \sum_{i = 1}^{N} x_{i} y_{i} & 2 \sum_{i = 1}^{N} x_{i} z_{i} & 2 \sum_{i = 1}^{N} x_{i} l_{1 i} & - \sum_{i = 1}^{N} x_{i} \\ 2 \sum_{i = 1}^{N} x_{i} y_{i} & 2 \sum_{i = 1}^{N} y_{i}^{2} & 2 \sum_{i = 1}^{N} y_{i} z_{i} & 2 \sum_{i = 1}^{N} y_{i} l_{1 i} & - \sum_{i = 1}^{N} y_{i} \\ 2 \sum_{i = 1}^{N} x_{i} z_{i} & 2 \sum_{i = 1}^{N} y_{i} z_{i} & 2 \sum_{i = 1}^{N} z_{i}^{2} & 2 \sum_{i = 1}^{N} z_{i} l_{1 i} & - \sum_{i = 1}^{N} z_{i} \\ 2 \sum_{i = 1}^{N} z_{i} l_{1 i} & 2 \sum_{i = 1}^{N} y_{i} l_{1 i} & 2 \sum_{i = 1}^{N} z_{i} l_{1 i} & 2 \sum_{i = 1}^{N} l_{1 i}^{2} & - \sum_{i = 1}^{N} l_{1 i} \\ - \sum_{i = 1}^{N} x_{i} & - \sum_{i = 1}^{N} y_{i} & - \sum_{i = 1}^{N} z_{i} & - \sum_{i = 1}^{N} l_{1 i} & \frac{N}{2} \end{matrix}] \\ [\begin{matrix} x \\ y \\ z \\ L_{1} \\ C \end{matrix}] = [\begin{matrix} \sum_{i = 1}^{N} x_{i} (x_{i}^{2} + y_{i}^{2} + z_{i}^{2} - l_{1 i}^{2}) \\ \sum_{i = 1}^{N} y_{i} (x_{i}^{2} + y_{i}^{2} + z_{i}^{2} - l_{1 i}^{2}) \\ \sum_{i = 1}^{N} z_{i} (x_{i}^{2} + y_{i}^{2} + z_{i}^{2} - l_{1 i}^{2}) \\ \sum_{i = 1}^{N} l_{i} (x_{i}^{2} + y_{i}^{2} + z_{i}^{2} - l_{1 i}^{2}) \\ - \frac{1}{2} \sum_{i = 1}^{N} (x_{i}^{2} + y_{i}^{2} + z_{i}^{2} - l_{1 i}^{2}) \end{matrix}] \end{matrix}

(6)

Repeating the above process, the position of other base stations can also be calibrated. When the positions of four base stations are obtained, the actual coordinates of each measuring point $A'_{i} (x'_{i}, y'_{i}, z'_{i})$ can be determined by the similar solving method. Then the space motion error $Δ A_{i} (Δ x_{i}, Δ y_{i}, Δ z_{i})$ of machine tool can be easily obtained at each measuring point.

Geometric error identification for machine tool

There exists six error components for each moving part of machine tool, and Figure 2 shows six geometric errors of $x - axis$ of machine tool.

Figure 2.

Six geometric errors of $x - axis$ .

There are 21 geometric errors for three-axis machine tool, and how to accurately identify each error is particularly important. Here, the comprehensive identification method is adopted. Based on the kinematic model of machine tool,^32,33 the mapping relation equations between the space motion error and each geometric error of machine tool are established, and each error can be separated by solving the equations. The error identification principle is shown in Figure 3. Based on the identification principle of nine-line method,³⁴ a identification algorithm by laser tracker is derived. For this algorithm, the total 21 geometric errors of three-axis machine tool cannot be identified simultaneously, and the squareness error is determined by the corresponding straight errors identified. To identify 21 geometric errors of machine tool simultaneously, a new identification algorithm is proposed considering the relative position deviations between different axes in establishing the mapping model of error identification. Two identification algorithms are deduced as follows.

Figure 3.

Error identification principle of machine tool.

First algorithm

When the machine tool moves the distance $x$ along the $x - axis$ from $A_{1} (x_{1}, y_{1}, z_{1})$ , the theoretical homogeneous transformation matrix is

T = [\begin{matrix} 1 & 0 & 0 & x \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

(7)

In the motion of machine tool, the homogeneous transformation matrixes caused by position error $δ_{x} (x)$ , straightness errors $δ_{y} (x)$ and $δ_{z} (x)$ , roll error $ε_{x} (x)$ , pitch error $ε_{y} (x)$ and yaw error $ε_{z} (x)$ are given, respectively, as follows³⁵

T_{1} = [\begin{matrix} 1 & 0 & 0 & δ_{x} (x) \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

(8)

T_{2} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & δ_{y} (x) \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

(9)

T_{3} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & δ_{z} (x) \\ 0 & 0 & 0 & 1 \end{matrix}]

(10)

T_{4} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos ε_{x} (x) & - \sin ε_{x} (x) & 0 \\ 0 & \sin ε_{x} (x) & \cos ε_{x} (x) & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

(11)

T_{5} = [\begin{matrix} \cos ε_{y} (x) & 0 & \sin ε_{y} (x) & 0 \\ 0 & 1 & 0 & 0 \\ - \sin ε_{y} (x) & 0 & \cos ε_{y} (x) & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

(12)

T_{6} = [\begin{matrix} \cos ε_{z} (x) & - \sin ε_{z} (x) & 0 & 0 \\ \sin ε_{z} (x) & \cos ε_{z} (x) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

(13)

So the total error transformation matrix is

P = T_{1} T_{2} T_{3} T_{4} T_{5} T_{6}

(14)

Roll error $ε_{x} (x)$ , pitch error $ε_{y} (x)$ and yaw error $ε_{z} (x)$ are very small in the motion of machine tool; the following simplifications can be obtained

\begin{matrix} \cos ε_{x} (x) \approx 1, \sin ε_{x} (x) \approx ε_{x} (x), \cos ε_{y} (x) \approx 1, \\ \sin ε_{y} (x) \approx ε_{y} (x), \cos ε_{z} (x) \approx 1, \sin ε_{z} (x) \approx ε_{z} (x) \end{matrix}

(15)

Then we can obtain

P = [\begin{matrix} 1 & - ε_{z} (x) & ε_{y} (x) & δ_{x} (x) \\ ε_{z} (x) & 1 & - ε_{x} (x) & δ_{y} (x) \\ - ε_{y} (x) & ε_{x} (x) & 1 & δ_{z} (x) \\ 0 & 0 & 0 & 1 \end{matrix}]

(16)

The total transformation matrix is

Q = TP = [\begin{matrix} 1 & - ε_{z} (x) & ε_{y} (x) & x + δ_{x} (x) \\ ε_{z} (x) & 1 & - ε_{x} (x) & δ_{y} (x) \\ - ε_{y} (x) & ε_{x} (x) & 1 & δ_{z} (x) \\ 0 & 0 & 0 & 1 \end{matrix}]

(17)

So the motion error of machine tool at measuring point $B_{1}$ can be determined from equation (18)

[\begin{matrix} Δ x_{1} \\ Δ y_{1} \\ Δ z_{1} \\ 1 \end{matrix}] = Q [\begin{matrix} x_{1} \\ y_{1} \\ z_{1} \\ 1 \end{matrix}] - [\begin{matrix} x_{1} + x \\ y_{1} \\ z_{1} \\ 1 \end{matrix}]

(18)

Then we can obtain

{\begin{matrix} Δ x_{1} = δ_{x} (x) - ε_{z} (x) y_{1} + ε_{y} (x) z_{1} \\ Δ y_{1} = δ_{y} (x) + ε_{z} (x) x_{1} - ε_{x} (x) z_{1} \\ Δ z_{1} = δ_{z} (x) + ε_{x} (x) y_{1} - ε_{y} (x) x_{1} \end{matrix}

(19)

Similarly, the motion error equations at measuring point $B_{2}$ , $B_{3}$ and $B_{4}$ can be established, and then we can deduce

[\begin{matrix} Δ x_{1} \\ Δ y_{1} \\ Δ z_{1} \\ ⋮ \\ Δ x_{4} \\ Δ y_{4} \\ Δ z_{4} \end{matrix}] = [\begin{matrix} 1 & 0 & 0 & 0 & z_{1} & - y_{1} \\ 0 & 1 & 0 & - z_{1} & 0 & x_{1} \\ 0 & 0 & 1 & y_{1} & - x_{1} & 0 \\ ⋮ \\ 1 & 0 & 0 & 0 & z_{8} & - y_{8} \\ 0 & 1 & 0 & - z_{8} & 0 & x_{8} \\ 0 & 0 & 1 & y_{8} & - x_{8} & 0 \end{matrix}] [\begin{matrix} δ_{x} (x) \\ δ_{y} (x) \\ δ_{z} (x) \\ ε_{x} (x) \\ ε_{y} (x) \\ ε_{z} (x) \end{matrix}]

(20)

In equation (20), there are six unknown parameters to be identified, and the number of equations is 12. So the least squares method can be used to solve the equations using space motion error of the machine tool measured by laser tracker at each measuring point, and then each geometric error of $x - axis$ can be identified.

Using the same method, six geometric errors of $y - axis$ and $z - axis$ can be identified. Concerning the squareness error, taking identifying the squareness error $ε_{yz}$ between $y - axis$ and $z - axis$ , for example, the corresponding identification process is given as follows. Using the values of $δ_{z} (y)$ and $δ_{y} (z)$ obtained at each measuring point, two lines can be determined by least square fitting. The angles between the two fitting lines and $y - axis$ and $z - axis$ are assumed as $β_{1}$ and $β_{2}$ , respectively, and the squareness error $ε_{yz}$ is $β_{2} - β_{1}$ . Using a similar method, the squareness error $ε_{xy}$ and $ε_{xz}$ can also be obtained.

The squareness error is determined by the corresponding straight errors in the above error identification process. Currently, some methods adopt this identification principle. When the machine tool moves along $y - axis$ and $z - axis$ , there also exists squareness error between different axes except six geometric errors of each axis. In the above error identification of $y - axis$ and $z - axis$ , the squareness error is ignored in establishing the mapping relation equations between the space motion error and individual geometric error of machine tool, so there exist certain identification deviations. Meanwhile, the squareness error is indirectly obtained from the straightness error, so certain error transfer will occur, which will also impact the identification accuracy. Detecting the large machine tool for example, the relative position deviations between different axes have great effect on the overall motion accuracy of moving parts due to the large structure of machine tool. If the above identification algorithm is adopted, large identification deviations will be brought.

Second algorithm

The error identification of $x - axis$ of machine tool by the second algorithm is the same with that using the first algorithm. So we only give the error identification process of $y - axis$ and $z - axis$ by the second algorithm as follows.

Error identification of $y - axis$

When the machine tool moves the distance $y$ along the $y - axis$ from $A_{1} (x_{1}, y_{1}, z_{1})$ , the theoretical homogeneous transformation matrix is

M = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & y \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

(21)

There exist position error $δ_{y} (y)$ , straightness errors $δ_{x} (y)$ and $δ_{z} (y)$ , roll error $ε_{y} (y)$ , pitch error $ε_{x} (y)$ , yaw error $ε_{z} (y)$ and squareness error $α_{xy}$ in the motion. The homogeneous transformation matrixes caused by position error $δ_{y} (y)$ , straightness errors $δ_{x} (y)$ and $δ_{z} (y)$ , roll error $ε_{y} (y)$ , pitch error $ε_{x} (y)$ and yaw error $ε_{z} (y)$ are

M_{1} = [\begin{matrix} 1 & - ε_{z} (y) & ε_{y} (y) & δ_{x} (y) \\ ε_{z} (y) & 1 & - ε_{x} (y) & δ_{y} (y) \\ - ε_{y} (y) & ε_{x} (y) & 1 & δ_{z} (y) \\ 0 & 0 & 0 & 1 \end{matrix}]

(22)

The error caused by the squareness error $α_{xy}$ is shown in Figure 4. The motion errors in $x - axis$ and $y - axis$ caused by $α_{xy}$ are

Δ x = A' B = y \sin α_{xy} \approx y α_{xy}

(23)

Δ y = OA - OB = y - y \cos α_{xy} = y \sin \frac{α_{xy}^{2}}{2} \approx 0

(24)

Figure 4.

Motion error caused by squareness error $α_{xy}$ .

The homogeneous transformation matrix caused by squareness error $α_{xy}$ is

M_{2} = [\begin{matrix} 1 & 0 & 0 & - y α_{xy} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

(25)

So the total error transformation matrix is

W = M_{1} M_{2} = [\begin{matrix} 1 & - ε_{z} (y) & ε_{y} (y) & δ_{x} (y) - y α_{xy} \\ ε_{z} (y) & 1 & - ε_{x} (y) & δ_{y} (y) \\ - ε_{y} (y) & ε_{x} (y) & 1 & δ_{z} (y) \\ 0 & 0 & 0 & 1 \end{matrix}]

(26)

The total transformation matrix with moving along $y - axis$ is

U = MW = [\begin{matrix} 1 & - ε_{z} (y) & ε_{y} (y) & δ_{x} (y) - y α_{xy} \\ ε_{z} (y) & 1 & - ε_{x} (y) & y + δ_{y} (y) \\ - ε_{y} (y) & ε_{x} (y) & 1 & δ_{z} (y) \\ 0 & 0 & 0 & 1 \end{matrix}]

(27)

So the motion errors of machine tool with moving the distance $y$ along the $y - axis$ can be determined from equation (28)

[\begin{matrix} Δ x_{5} \\ Δ y_{5} \\ Δ z_{5} \\ 1 \end{matrix}] = U [\begin{matrix} x_{5} \\ y_{5} \\ z_{5} \\ 1 \end{matrix}] - [\begin{matrix} x_{5} \\ y_{5} + y \\ z_{5} \\ 1 \end{matrix}]

(28)

Then we can obtain

{\begin{matrix} Δ x_{5} = δ_{x} (y) - ε_{z} (y) y_{5} + ε_{y} (y) z_{5} - y α_{xy} \\ Δ y_{5} = δ_{y} (y) + ε_{z} (y) x_{5} - ε_{x} (y) z_{5} \\ Δ z_{5} = δ_{z} (y) + ε_{x} (y) y_{5} - ε_{y} (y) x_{5} \end{matrix}

(29)

Similarly, the motion error equations with moving the distance $y$ along the $y - axis$ from $A_{2}$ , $A_{5}$ and $A_{6}$ can be established, and then we can deduce

[\begin{matrix} Δ x_{5} \\ Δ y_{5} \\ Δ z_{5} \\ ⋮ \\ Δ x_{8} \\ Δ x_{8} \\ Δ x_{8} \end{matrix}] = [\begin{matrix} 1 & 0 & 0 & 0 & z_{5} & - y_{5} & - y \\ 0 & 1 & 0 & - z_{5} & 0 & x_{5} & 0 \\ 0 & 0 & 1 & y_{5} & - x_{5} & 0 & 0 \\ ⋮ \\ 1 & 0 & 0 & 0 & z_{8} & - y_{8} & - y \\ 0 & 1 & 0 & - z_{8} & 0 & x_{8} & 0 \\ 0 & 0 & 1 & y_{8} & - x_{8} & 0 & 0 \end{matrix}] [\begin{matrix} δ_{x} (y) \\ δ_{y} (y) \\ δ_{z} (y) \\ ε_{x} (y) \\ ε_{y} (y) \\ ε_{z} (y) \\ α_{xy} \end{matrix}]

(30)

Six geometric errors of $y - axis$ and the squareness error $α_{xy}$ can be identified by solving the above equations.

Error identification of $z - axis$

When the machine tool moves the distance $z$ along the $z - axis$ from $A_{1} (x_{1}, y_{1}, z_{1})$ , the theoretical homogeneous transformation matrix is

R = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & z \\ 0 & 0 & 0 & 1 \end{matrix}]

(31)

There exist position error $δ_{z} (z)$ , straightness errors $δ_{x} (z)$ and $δ_{y} (z)$ , roll error $ε_{z} (z)$ , pitch error $ε_{x} (z)$ and $ε_{y} (z)$ , squareness error $α_{xz}$ and $α_{yz}$ in the motion. The homogeneous transformation matrix caused by position error $δ_{z} (z)$ , straightness errors $δ_{x} (z)$ and $δ_{y} (z)$ , roll error $ε_{z} (z)$ , pitch error $ε_{x} (z)$ and $ε_{y} (z)$ is

R_{1} = [\begin{matrix} 1 & - ε_{z} (z) & ε_{y} (z) & δ_{x} (z) \\ ε_{z} (z) & 1 & - ε_{x} (z) & δ_{y} (z) \\ - ε_{y} (z) & ε_{x} (z) & 1 & δ_{z} (z) \\ 0 & 0 & 0 & 1 \end{matrix}]

(32)

The homogeneous transformation matrixes caused by squareness error $α_{xz}$ and $α_{yz}$ are

R_{2} = [\begin{matrix} 1 & 0 & 0 & - z α_{xz} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

(33)

R_{3} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & - z α_{yz} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

(34)

The total error transformation matrix is

G = R_{1} R_{2} R_{3} = [\begin{matrix} 1 & - ε_{z} (z) & ε_{y} (z) & δ_{x} (z) - z α_{xz} \\ ε_{z} (z) & 1 & - ε_{x} (z) & δ_{y} (z) - z α_{yz} \\ - ε_{y} (z) & ε_{x} (z) & 1 & δ_{z} (z) \\ 0 & 0 & 0 & 1 \end{matrix}]

(35)

The total transformation matrix with moving along $z - axis$ is

V = RG = [\begin{matrix} 1 & - ε_{z} (z) & ε_{y} (z) & δ_{x} (z) - z α_{xz} \\ ε_{z} (z) & 1 & - ε_{x} (z) & δ_{y} (z) - z α_{yz} \\ - ε_{y} (z) & ε_{x} (z) & 1 & z + δ_{z} (z) \\ 0 & 0 & 0 & 1 \end{matrix}]

(36)

So the motion errors of machine tool with moving the distance $z$ along the $z - axis$ can be determined from equation (39)

[\begin{matrix} Δ x_{9} \\ Δ y_{9} \\ Δ z_{9} \\ 1 \end{matrix}] = V [\begin{matrix} x_{9} \\ y_{9} \\ z_{9} \\ 1 \end{matrix}] - [\begin{matrix} x_{9} \\ y_{9} \\ z_{9} + z \\ 1 \end{matrix}]

(37)

Then we can obtain

{\begin{matrix} Δ x_{9} = δ_{x} (z) - ε_{z} (z) y_{9} + ε_{y} (z) z_{9} - z α_{xy} \\ Δ y_{9} = δ_{y} (z) + ε_{z} (z) x_{9} - ε_{x} (z) z_{9} - z α_{yz} \\ Δ z_{9} = δ_{z} (z) + ε_{x} (z) y_{9} - ε_{y} (z) x_{9} \end{matrix}

(38)

Similarly, the motion error equations with moving the distance $z$ along the $z - axis$ from $A_{3}$ , $A_{4}$ and $A_{2}$ can be established, and then we can deduce

\begin{matrix} [\begin{matrix} Δ x_{9} \\ Δ y_{9} \\ Δ z_{9} \\ ⋮ \\ Δ x_{12} \\ Δ y_{12} \\ Δ z_{12} \end{matrix}] \\ = [\begin{matrix} 1 & 0 & 0 & 0 & z_{9} & - y_{9} & - z & 0 \\ 0 & 1 & 0 & - z_{9} & 0 & x_{9} & 0 & - z \\ 0 & 0 & 1 & y_{9} & - x_{9} & 0 & 0 & 0 \\ ⋮ \\ 1 & 0 & 0 & 0 & z_{12} & - y_{12} & - z & 0 \\ 0 & 1 & 0 & - z_{12} & 0 & x_{12} & 0 & - z \\ 0 & 0 & 1 & y_{12} & - x_{12} & 0 & 0 & 0 \end{matrix}] [\begin{matrix} δ_{x} (z) \\ δ_{y} (z) \\ δ_{z} (z) \\ ε_{x} (z) \\ ε_{y} (z) \\ ε_{z} (z) \\ α_{xz} \\ α_{yz} \end{matrix}] \end{matrix}

(39)

By solving the above equations, six geometric errors of $z - axis$ and the squareness error $α_{xz}$ and $α_{yz}$ can be identified. In the above error identification of $y - axis$ and $z - axis$ , the squareness error is considered in establishing the mapping relation equations between the space motion error and each geometric error of machine tool. So six geometric errors of each axis and the corresponding squareness error can be directly identified, and the squareness error does not need to be obtained from the straight errors identified.

Simulation for error identification of machine tool

Taking identifying the error of $y - axis$ of machine tool for example, the simulations are carried out with two identification algorithms deduced. According to whether the effect of random error during machine tool motion is considered, the simulations are carried out under two different conditions.

Without considering the effect of random error during machine tool motion

In order to analyze the impact of different measurement areas and different squareness errors on two identification algorithms of $y - axis$ , the simulations are conducted as follows:

Under different measurement areas

In the simulations, the motion areas of the machine tool are defined by 1000 mm × 1000 mm × 1000 mm, 2000 mm × 2000 mm × 2000 mm and 3000 mm × 3000 mm × 3000 mm, respectively. Meanwhile, nine measuring points are uniformly distributed on each edge of the cube. The squareness errors between $x - axis$ and $y - axis$ , $x - axis$ and $z - axis$ , and $y - axis$ and $z - axis$ are assumed to be $α_{xy} = 1.8 \times 10^{- 5} rad$ , $α_{xz} = 2.2 \times 10^{- 5} rad$ and $α_{yz} = 2.6 \times 10^{- 5} rad$ , respectively. In order to simply the calculation, six geometric errors of $y - axis$ are assumed to be linearly distributed, and the maximum values of each individual geometric error under different measurement areas are given in Table 2.

Table 2.

Maximum value of each individual geometric error of $y - axis$ under different measurement areas.

Geometric error	Measurement area 1	Measurement area 2	Measurement area 3
$δ_{x} (y) (mm)$	$2 \times 10^{- 2}$	$4 \times 10^{- 2}$	$6 \times 10^{- 2}$
$δ_{y} (y) (mm)$	$1.5 \times 10^{- 2}$	$3 \times 10^{- 2}$	$4.5 \times 10^{- 2}$
$δ_{z} (y) (mm)$	$3 \times 10^{- 2}$	$6 \times 10^{- 2}$	$9 \times 10^{- 2}$
$ε_{x} (y) (rad)$	$1 \times 10^{- 5}$	$2 \times 10^{- 5}$	$3 \times 10^{- 5}$
$ε_{y} (y) (rad)$	$1.5 \times 10^{- 5}$	$3 \times 10^{- 5}$	$4.5 \times 10^{- 5}$
$ε_{z} (y) (rad)$	$3 \times 10^{- 5}$	$6 \times 10^{- 5}$	$9 \times 10^{- 5}$

According to the volumetric error model of three-axis machine tool,³⁵ the space motion error $A_{i} (Δ x_{i}, Δ y_{i}, Δ z_{i})$ of machine tool at each measuring point can be determined. Then the geometric errors of $y - axis$ can be separated by error identification algorithms deduced.

The simulations results indicate that there exists certain identification deviations using the first algorithm, and Table 3 shows the identification deviations of some geometric errors of $y - axis$ at different positions.

Table 3.

Identification deviations of some geometric errors of $y - axis$ with the first algorithm.

Position	Identification deviations
Position	$Δ δ_{x} (y) (mm)$	$Δ δ_{y} (y) (mm)$	$Δ δ_{z} (y) (rad)$
y = 100	−0.00540	−0.00360	0.0000072
y = 300	−0.00810	−0.00090	0.0000018
y = 700	−0.01080	0.00180	−0.0000036
y = 900	−0.01260	0.00360	−0.0000072

Each geometric error of $y - axis$ can be accurately identified using the second algorithm, and Table 4 shows the identification deviations of some geometric errors at different positions of $y - axis$ . It can be seen from Tables 3 and 4 that there exists certain identification deviations due to ignoring the squareness error in establishing the error identification model, which is consistent with the former analysis.

Table 4.

Identification deviations of some geometric errors of $y - axis$ with the second algorithm.

Position	Identification deviations
Position	$Δ δ_{x} (y) (mm)$	$Δ δ_{y} (y) (mm)$	$Δ ε_{z} (y) (rad)$
y = 100	$- 5.8546 \times 10^{- 18}$	$8.6736 \times 10^{- 18}$	$- 3.3881 \times 10^{- 18}$
y = 300	$- 1.3877 \times 10^{- 17}$	$1.7347 \times 10^{- 18}$	$2.7105 \times 10^{- 20}$
y = 700	$4.1633 \times 10^{- 17}$	$2.5643 \times 10^{- 18}$	$3.6719 \times 10^{- 20}$
y = 900	$- 1.0408 \times 10^{- 18}$	$6.9388 \times 10^{- 18}$	$1.3552 \times 10^{- 20}$

Taking identifying the straightness error $δ_{x} (y)$ for example, the impact of different measurement areas on two identification algorithms is analyzed. Tables 5 and 6 show the identification deviations with two algorithms under different measurement areas.

Table 5.

Identification deviations of straightness error $δ_{x} (y)$ with the first algorithm under different measurement areas (mm).

Position	Identification deviations of straightness error
Position	Measurement area 1	Measurement area 2	Measurement area 3
y = 100	−0.00540	−0.01080	−0.01620
y = 300	−0.00720	−0.01440	−0.02160
y = 700	−0.01080	−0.02160	−0.03240
y = 900	−0.01260	−0.02520	−0.03780

Table 6.

Identification deviations of straightness error $δ_{x} (y)$ with the second algorithm under different measurement areas (mm).

Position	Identification deviations of straightness error
Position	Measurement area 1	Measurement area 2	Measurement area 3
y = 100	$- 5.8546 \times 10^{- 18}$	$- 1.6263 \times 10^{- 17}$	$- 2.7756 \times 10^{- 17}$
y = 300	$- 1.3877 \times 10^{- 17}$	$- 6.0715 \times 10^{- 17}$	$- 2.0816 \times 10^{- 17}$
y = 700	$4.1633 \times 10^{- 17}$	$- 8.8471 \times 10^{- 17}$	$1.3531 \times 10^{- 15}$
y = 900	$1.4322 \times 10^{- 17}$	$- 1.3011 \times 10^{- 16}$	$1.1796 \times 10^{- 16}$

From Table 5, there exist certain identification deviations using the first algorithm. Meanwhile, the identification deviations are becoming larger with an increase in the measurement areas. From Table 6, individual geometric error of $y - axis$ can be accurately identified using the second algorithm under different measurement areas. So we can infer that when the first algorithm is used to identify the geometric error of large machine tool, large identification deviations will occur.

Under different squareness errors

The motion areas of the machine tool are defined by 1000 mm × 1000 mm × 1000 mm; six geometric errors of $y - axis$ are assumed to be linearly distributed. The corresponding error valves are given: the maximum position error $δ_{y} (y)$ is assumed to be 15 μm; the maximum straightness errors $δ_{x} (y)$ and $δ_{z} (y)$ are assumed to be 20 and 30 μm, respectively; the maximum pitch error $ε_{x} (y)$ is assumed to be $1 \times 10^{- 5} rad$ ; the maximum roll error $ε_{y} (y)$ is assumed to be $1.5 \times 10^{- 5} rad$ and the maximum yaw error $ε_{z} (y)$ is assumed to be $3 \times 10^{- 5} rad$ . According to different squareness errors, the simulations are carried out under three conditions; the corresponding squareness errors are given in Table 7.

Table 7.

Different squareness errors under three conditions.

Squareness error	Condition 1	Condition 2	Condition 3
$α_{xy} (rad)$	$1.8 \times 10^{- 5}$	$3.6 \times 10^{- 5}$	$5.4 \times 10^{- 5}$
$α_{xz} (rad)$	$2.2 \times 10^{- 5}$	$4.4 \times 10^{- 5}$	$6.6 \times 10^{- 5}$
$α_{yz} (rad)$	$2.6 \times 10^{- 5}$	$5.2 \times 10^{- 5}$	$7.8 \times 10^{- 5}$

The simulations show that individual geometric error of $y - axis$ can be accurately identified using the second algorithm under different squareness errors, and there exist certain identification deviations with the first algorithm. Taking identifying the straightness error $δ_{x} (y)$ for example, Tables 8 and 9 show the identification deviations with two algorithms under different squareness errors.

Table 8.

Identification deviations of straightness error $δ_{x} (y)$ with the first algorithm under different squareness errors (mm).

Position	Identification deviations of straightness error
Position	Squareness error 1	Squareness error 2	Squareness error 3
y = 100	−0.00540	−0.00810	−0.01350
y = 300	−0.00720	−0.01080	−0.01800
y = 700	−0.01080	−0.01620	−0.02700
y = 900	−0.01260	−0.02520	−0.03150

Table 9.

Identification deviations of straightness error $δ_{x} (y)$ with the second algorithm under different squareness errors (mm).

Position	Identification deviations of straightness error
Position	Squareness error 1	Squareness error 2	Squareness error 3
y = 100	$- 5.8546 \times 10^{- 18}$	$5.6378 \times 10^{- 18}$	$1.3010 \times 10^{- 18}$
y = 300	$- 1.3877 \times 10^{- 17}$	$- 2.0816 \times 10^{- 17}$	$- 4.3368 \times 10^{- 17}$
y = 700	$4.1633 \times 10^{- 17}$	$- 7.2858 \times 10^{- 17}$	$5.5511 \times 10^{- 17}$
y = 900	$1.4322 \times 10^{- 17}$	$6.2450 \times 10^{- 17}$	$5.8980 \times 10^{- 17}$

It can be seen from Tables 8 and 9 that individual geometric error of $y - axis$ can be accurately identified using the second algorithm under different squareness errors. There exist certain identification deviations with the first algorithm; moreover, the identification deviations are becoming larger with an increase in squareness error. So we can infer that when the squareness errors between different axes of the machine tool are larger, adopting the first algorithm will bring larger identification deviations.

Considering the effect of random error during machine tool motion

Under this condition, a random error is added to the theoretical motion error at each measuring point. To simplify the calculation, the random error during machine tool motion obeys a random normal distribution within [0, 3 μm], [0, 5 μm] and [0, 10 μm], respectively. Tables 10 and 11 show the identification deviations of individual geometric error of $y = 100$ using two algorithms under different random error distributions.

Table 10.

Identification deviations of individual geometric error with the first algorithm under different random error distributions.

Geometric error	Identification deviations of individual geometric error
Geometric error	Distribution 1	Distribution 2	Distribution 3
$Δ δ_{x} (y) (mm)$	−0.00622	−0.01037	−0.02073
$Δ δ_{y} (y) (mm)$	−0.00417	−0.00695	−0.01390
$Δ δ_{z} (y) (mm)$	−0.00325	−0.00542	−0.01083
$Δ ε_{x} (y) (rad)$	0.000009	0.000015	0.000030
$Δ ε_{y} (y) (rad)$	0.000013	0.000022	0.000043
$Δ ε_{z} (y) (rad)$	0.000033	0.000055	0.000011

Table 11.

Identification deviations of individual geometric error with the second algorithm under different random error distributions.

Geometric error	Identification deviations of individual geometric error
Geometric error	Distribution 1	Distribution 2	Distribution 3
$Δ δ_{x} (y) (mm)$	0.00047	0.00079	0.00157
$Δ δ_{y} (y) (mm)$	0.00039	0.00065	0.00130
$Δ δ_{z} (y) (mm)$	−0.00061	−0.00101	−0.00203
$Δ ε_{x} (y) (rad)$	0.000008	0.000013	0.000027
$Δ ε_{y} (y) (rad)$	0.000011	0.000018	0.000037
$Δ ε_{z} (y) (rad)$	0.000026	0.000044	0.000087

It can be seen from Tables 10 and 11 that the identification deviations with second algorithm is significantly smaller than the identification deviations obtained by the first algorithm. For the second algorithm, the maximum identification deviation is −0.00061 mm for $Δ δ_{z} (y)$ on the first occasion; the maximum identification deviation is −0.00101 mm for $Δ δ_{z} (y)$ on the second occasion; the maximum identification deviation is −0.00203 mm for $Δ δ_{z} (y)$ on the third occasion. The deviation is small, so this algorithm is feasible. In practical measurement, the motion trajectory of machine tool will be measured for multiple times, and the impact of random error on the measurement results can be reduced. So the identification accuracy of each error will be further improved.

Experimental verification

A laser tracker is adopted multi-station and time-sharing measurement to identify the geometric errors of a gear grinding machine as shown in Figure 5. The measurement area is 800 mm × 400 mm × 500 mm, and a laser tracker is used to measure the same motion trajectory of the gear grinding machine at four base stations. In order to reduce the influence of random errors of the gear grinding machine on the measurement results, the motion trajectory of the gear grinding machine is measured for three times at each base station. The total measuring time is about 5 h, and 21 geometric errors of the gear grinding machine can be identified using the measurement algorithm and identification algorithm mentioned above. Figures 6 and 7 show the part error curves of $y - axis$ of gear grinding machine identified with the second algorithm.

Figure 5.

Geometric error detection of gear grinding machine.

Figure 6.

Straightness error of $y - axis$ .

Figure 7.

Roll error of $y - axis$ .

To compare the identification accuracy of two algorithms derived, the position error of $y - axis$ of machine tool identified by multi-station and time-sharing measurement with the first and second identification algorithms is compared with that detected by laser interferometer as shown in Figure 8.

Figure 8.

Comparison of two identification algorithms.

Comparing with the identification result of the first algorithm in Figure 6, the position error of $y - axis$ identified with the second algorithm is closer to that detected using laser interferometer, which indicates higher identification accuracy can be obtained by the second algorithm. So the simulation results and experimental results are consistent. Meanwhile, the maximum deviation between the laser interferometer measurement and laser tracker measurement with the second algorithm is small, so we can infer that the geometric errors of machine tool can be accurately identified with the second algorithm. In addition, the experimental result also indicates that adopting multi-station and time-sharing measurement to detect the geometric errors of machine tool is feasible. In the measurement, the surrounding environment factors especially the vibration factor has a certain impact on the measurement results. To maintain measurement accuracy, the vibration source around the gear grinding machine should be reduced as much as possible, and the multi-station and time-sharing measurement is carried out in relatively quiet environment.

Conclusion

The factors affecting the accuracy of single-station measurement of laser tracker are analyzed. To avoid the influence of angle measurement error, a laser tracker is adopted the multi-station and time-sharing measurement to identify the geometric errors of machine tool.

Two identification algorithms of geometric error of machine tool by laser tracker are deduced. For the first algorithm, the total 21 geometric errors of three-axis machine tool cannot be identified simultaneously. For the second algorithm, the total 21 geometric errors of three-axis machine tool can be identified simultaneously.

The identification accuracy of two algorithms and the impact of different measurement areas and different squareness errors on two algorithms are compared and analyzed in depth. Simulations and experiments indicate that the identification accuracy of the second algorithm is higher than the identification accuracy of the first algorithm, and individual geometric error of the machine tool can be accurately separated by the second algorithm.

Footnotes

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This research was supported by the National Natural Science Foundation of China (Grant No. 51305370) and the Fundamental Research Funds for the Central Universities (Grant No. 2682014BR017).

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Geometric error identification algorithm of numerical control machine tool using a laser tracker

Abstract

Keywords

Introduction

Measurement principle of laser tracker

Principle and algorithm of multi-station and time-sharing measurement

Geometric error identification for machine tool

First algorithm

Second algorithm

Error identification of y - axis

Error identification of z - axis

Simulation for error identification of machine tool

Without considering the effect of random error during machine tool motion

Under different measurement areas

Under different squareness errors

Considering the effect of random error during machine tool motion

Experimental verification

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References

Error identification of $y - axis$

Error identification of $z - axis$