Open Access
Add to BibSonomy
Abstract
The rotary axis is the basis of rotational motion. The motion errors of a rotary axis have an extremely important impact on the accuracy of precision machining measuring equipment such as CNC machines, robot manipulators, and laser trackers. It is a difficult problem to realise the fast and precision simultaneous measurement of multi-degree-of-freedom motion errors of a rotary axis. Therefore, a novel method for the simultaneous measurement of six-degree-of-freedom motion errors of a rotary axis by a single-mode fiber coupled semiconductor laser is proposed in this paper. The corresponding system is developed, which has the advantages of high measurement efficiency, simple structure and low cost. A phase-solving method taking the advantages of both the eight-subdivision and the Cordic algorithm is proposed to solve the phase of interference signal, cannot only realize the high-resolution solving of the current signal phase but also quickly obtain high-precision interferometric results. A series of experiments were carried out on the developed system. An experimental system was built and a series of experiments were performed. The experimental results show that the standard deviation of stability for 1 hour of the six-degree-of-freedom measurement is 0.03 µm, 0.02 µm, 0.03 µm, 0.10$^{\prime\prime}$, 0.05$^{\prime\prime}$ and 0.03$^{\prime\prime}$, respectively. The repeatability deviation of measuring a rotary axis is ±0.16 µm, ± 0.29 µm, ± 0.25 µm, ± 0.65$^{\prime\prime}$, ± 0.62$^{\prime\prime}$ and ±13.42$^{\prime\prime}$, respectively. The maximum deviation of comparison with standard instruments is 0.46 µm, 1.00 µm, 0.49 µm, 1.06$^{\prime\prime}$, 1.53$^{\prime\prime}$ and 0.74$^{\prime\prime}$, respectively. It provides a low-cost and high-precision measurement method for simultaneous measurement of six-degree-of-freedom motion errors of rotary axis of precision machining and measuring equipment.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The rotary axis is the benchmark of rotational motion. The six Degrees of Freedom (DOF) motion errors of a rotary axis greatly affect the machining accuracy of CNC machine tools [1], the control accuracy of manipulator arms, satellites and radar antenna bases [2–5], the measuring accuracy of roundness instrument, 3D laser scanner and laser tracker [6–8], and the analysis results of grating spectrometer [9]. Therefore, it is a common problem to reduce the influence of motion errors of the rotary axis on the machining, measurement and control accuracy of the above equipment. There are two main ways to reduce it. One is to directly improve the manufacturing accuracy of the rotary axis and related equipment, but this method has problems such as high cost and long development cycle [10,11]. The other is the error compensation method, which generally includes three steps: (1) measuring the motion errors of the rotary axis, (2) establishing error models, (3) compensating the motion errors. It has the advantages of low cost and short development cycle, and has become the research focus worldwide [12]. The key to error compensation is to measure all kinds of motion errors on a rotary axis accurately and quickly.
According to the international standard ISO230-7 [13], when the axis rotates around the Z-axis, there are six DOF geometric motion errors, including the radial motion errors along the X-axis ${\delta _\textrm{x}}(\theta )$ and Y- axis ${\delta _\textrm{y}}(\theta )$, the axial motion error along the Z-axis ${\delta _\textrm{z}}(\theta )$, the tilt motion errors around the X-axis ${\varepsilon _\textrm{x}}(\theta )$ and Y-axis ${\varepsilon _\textrm{y}}(\theta )$, the angular positioning error around the Z-axis ${\varepsilon _\textrm{z}}(\theta )$.
The angular positioning error is the most important motion error of a rotary axis. Early studies mainly focus on measuring the angular positioning errors, which are based on the interference principle and the auto-collimation principle [14–16]. However, with the angular positioning error alone, it is difficult to fully compensate the machining, measurement, and control errors caused by the six DOF motion errors of the rotary axis. In recent years, the multi-DOF motion errors measurement method of a rotary axis has become the research focus in this field. These measurement methods can basically be divided into contact and non-contact methods. The contact method mainly includes the piezo-electric ceramic method, the path test method, and the off-line detection method. Ahn et al. used a cylindrical capacitance sensor to measure the five DOF motion errors, which is small in size and easy to integrate into the rotary axis [17]. The path test method mainly includes the double ball-bar test and R-test method. Osei et al. obtained the six DOF motion errors of the rotary axis by decoupling the circular trajectories of the double ball-bar at different positions in the space of the rotary axis [18]. Pu et al. measured four points on the rotary table by R-test to obtain different circular trajectories, and decoupled the motion errors of the rotary axis according to the trajectories. [19]. The off-line detection method uses the standard workpiece which fixed in the rotary axis as the reference. The axis of rotation space coordinates can be using to measure the motion errors which reconstructed by measuring the standard workpiece with the probe. Erkan et al. used the method of detecting 3D ball array system with variable number and position to complete the measurement of five DOF motion errors except the angular positioning error [20]. Soichi et al. proposes a scheme to measure motion errors of a rotary axis by on-the-machine measurement of a test piece by using a contact-type touch-trigger probe installed on the machine's spindle [21]. Optical measurement has become the mainstream of research because of its advantages of non-contact, high accuracy and wide range. Including vision-based, laser R-test based method, diffraction-based, laser tracker-based, interferometry-based, and collimation-based. Hong et al. measured the six DOF motion errors of the rotary axis by laser R-test [22]. Liu et al. measured the six DOF motion errors using a reflective diffraction grating and PSD [23]. Zhang et al. proposes the three-point method based on laser tracker to measure the geometric error of the rotary axes of the machine tools. This method has the advantages of simplicity and speed [24]. Yang et al. measured the six DOF motion errors of the rotary axis by using multi-wavelength phase-shifting interferometry technology [25]. In recent years, the method based on collimation is widely studied by scholars all over the world. According to the change of laser spot position after reflection, transmission and splitting, the motion errors can be obtained [26–31].
The above methods still have problems such as low measurement accuracy and efficiency, complicated decoupling of errors and inability to measure all six DOF motion errors at the same time. For this reason, our research team proposed a method based on laser interferometry and collimation to simultaneously measure the six DOF motion errors [32]. However, this method uses a He-Ne laser which has some disadvantages of high cost, low power, not conducive to industrial online measurement and application. The radial motion error of the rotary axis is measured by laser interference principle in the simultaneous measurement system of the six DOF motion errors. When we measure the radial motion error of the rotary axis, the interferometric values will include two parts of values: the radial motion error of the rotary axis and the eccentricity caused by the coaxial installation error between the target rotary axis and the reference rotary axis. Their total error range is within ± 100 µm, which means that the coherent length and frequency stability of the laser are not demanding. Therefore, this paper proposes to replace He-Ne laser with semiconductor laser, and designs a new measurement optical path for single-frequency semiconductor laser interferometry, proposes signal processing circuit and phase solving algorithm, and finally realises the simultaneous measurement of six DOF motion errors of a rotary axis. Section 2 of this paper introduces measurement principles and main composition; section 3 introduces system design; and section 4 introduces experimental results.
2. Measurement principles
The proposed method for simultaneously measuring the six DOF motion errors of the rotary axis by a semiconductor laser is shown in Fig. 1. It is composed of a laser source and fiber coupling unit, a measurement unit, an error sensitive unit and a reference rotary axis.
Fig. 1. Measurement principle of six degrees of freedom (DOF) motion errors of a rotary axis based on semiconductor laser.
The laser source and fiber coupling unit consists of a semiconductor laser, a fiber-coupled lens and a single-mode fiber. The application of the single-mode fiber can not only improve the thermal stability by isolating the heat output of the laser source, but also significantly improve the spatial stability of laser beam. The reference rotary axis is a high-precision rotary axis installed between the sensitive unit and the target rotary axis. When the target rotary axis rotates, the reference rotary axis rotates in the opposite direction to ensure the continuity of interference beam. The error sensitive unit is mainly composed of two corner-cube retro-reflectors and a beam-splitting film. The sensitive unit is fixed on the reference rotary axis, and the reference rotary axis is installed coaxially with the target rotary axis. The laser beam emitted from the semiconductor laser is coupled through the coupling lens (C-Lens) into the single-mode fiber. The beam reflected by the sensitive unit carries the motion errors of the rotary axis. The six DOF motion errors of the rotary axis are calculated according to the change in spot position and energy of the beam returning to the measuring unit on the detector. The measurement unit is mainly consists two parts: (1) part of measuring the radial motion error along the X-axis, (2) part of measuring the other five DOF motion errors. Detail principles for measuring each DOF motion error are given below.
2.1 Measurement of the radial motion error along the X-axis based on interference principle
The measurement of the radial motion error along the X-axis is based on the measurement principle of laser interference, as shown in Fig. 2. The equal optical path layout of the reference beam and the measurement beam can ensure the visibility of the fringe pattern and reduce the influence of atmospheric refractive index change on measurement. The linearly polarized light after collimation by the fiber collimator passes through the half-wave plate HWP1 and then incident on the polarized beam splitter cube PBS1. Rotation of HWP1 can change the intensity ratio of transmitted beam and reflected beam in PBS1, so as to maximise the AC amplitude of the final interference signal and obtain the interference signal with the highest contrast. The transmitted beam, whose polarization direction is parallel to the paper surface is reflected by the corner-cube retro-reflector RR2 after passing through the quarter-wave plate QWP2, and then returns to PBS1 as the measurement beam. The polarization direction changes to be perpendicular to the paper surface and is reflected from PBS1 into BS1. The reference beam is reflected by PBS1, and the return beam is transmitted into BS1. The two linearly polarized beams reflected by BS1 go through the quarter-wave plate QWP3 and then become left and right circularly polarized light. They are separated by PBS to produce interference signals with a phase difference of 180° on photodetector PD3 and PD4, respectively. Similarly, the transmitted beam of BS1 produces an interference signal with a phase difference of 180° on PD1 and PD2. Because the reflected beam and transmitted beam of BS1 pass through quarter-wave plate QWP3 and half-wave plate HWP2, respectively, the phase of the interference signals output by the four photodetectors differs by 90° in turn. The effect of light source noise, environmental noise, and stray light can be eliminated by differentializing the two signals with opposite phases in the analog circuit. At the same time, the DC component in the interference signal can be eliminated, and two orthogonal signals with a difference of 90° can be obtained. Thus, the number N of interference fringes on the detector can be obtained, then ${\delta _x}$ can be calculated, and the radial motion error along the X-axis can be measured.
Where, λ is the wavelength of the laser.2.2 Five other DOF motion errors measurement and beam angle-drift monitoring based on collimation and autocollimation principle
Besides the radial motion error along the X-axis of the rotary axis, the other five DOF errors and beam angle-drift are measured based on the laser collimation and self-collimation principles, and the schematic diagram is shown in Fig. 3. The beam reflected by PBS4 is transmitted by QWP4 and then reflected by the beam-splitting film BS5. After the reflected beam is received by the quadrant detector QD4, the angular positioning error ${\varepsilon _\textrm{z}}$ and the tilt motion error around the Y-axis ${\varepsilon _\textrm{y}}$ can be measured. After the transmitted beam of BS5 is received by QD2, the radial motion error along the Y-axis ${\delta _\textrm{y}}$ and axial motion error ${\delta _\textrm{z}}$ can be measured. The radial motion error along the Y-axis and the axial motion error can also be measured by the reflected beam of retro-reflector RR2 received by QD1. According to the values of axial motion error obtained on QD1 and QD2, respectively, the tilt motion error around the X-axis in the can be obtained.
The beam incident on BS4 is transmitted and reflected into QD1 and QD3, respectively. Since the optical paths of reflected beam to QD3 and transmitted beam to QD1 are different, the beam angle-drift around the Y and Z axes can be obtained according to the difference between the changes in the beam spot positions on the two detectors as follows:
2.3 Expressions of the six DOF motion errors of the rotary axis
The measurement results of the six DOF motion errors of the target rotary axis are affected by the six DOF motion errors of the reference rotary axis, beam angle-drift, installation errors between components, manufacturing errors, and error-crosstalk. To this end, we established a mathematical error model to analyze and compensate for these influences. Due to the complexity of the modeling and compensation process, details will be presented to readers later. Here the expressions of the six DOF motion errors of the target rotary axis considering the beam angle-drift and the six DOF motion errors of the reference rotary axis are given as follows:
According to Formulas (4)∼(9), the six DOF motion errors of the reference rotary axis given by the production company can be directly used for real-time compensation of the six DOF motion errors of the target rotary axis. Similarly, the influence of beam angle-drift can also be compensated in real time using the results calculated by Formulas (2) and (3).
3. Design of our measurement system
Based on the measurement principle using a semiconductor laser proposed above, as shown in Fig. 1, the system design is carried out. The following is a brief introduction to the main parts of the system design.
3.1 Laser source module
LPS-PM635 semiconductor laser from Thorlabs is used as the laser source module of the simultaneous measurement system. The polarization-maintaining pigtail can ensure the linear polarization direction remains unchanged and improve the coherent signal-to-noise ratio. The central wavelength λ is 636.4 nm, the output power of the laser source is 2.5 mw, and the mode field diameter DMFD of the fiber is 4.5 ± 0.5 µm. Thorlabs F220FC optical fiber collimator is applied, the focal length f is 10.99 mm, and the output beam diameter d can be approximated from:
The maximum waist distance Zmax (the furthest distance in order to maintain collimation) may be approximated by:3.2 Interferometric module
Interferometric module mainly includes the acquisition, processing, and phase-solving of interference signals. As shown in Fig. 4, the upper computer software can send control commands to the data acquisition processing and control circuit, which controls the beginning and ending of interferometry by controlling the phase resolving module.
When the four differential signals obtained from PD detectors enter the analog-to-digital conversion (ADC) in pairs, the first-order passive low-pass filter is used to filter out the high-frequency noise from the power supply and beam sources. The interference signal after ADC enters the FPGA module for data processing and phase solving. In order to further improve the signal-to-noise ratio, the finite impulse response (FIR) digital filtering process is carried out. Moreover, a nonlinear error correction module is designed to correct the nonlinear error of the interference signal and improve the phase resolution.
The existing interferometric phase resolution methods include the eight-subdivision method, the look-up table method, and the Cordic method, etc. The eight-subdivision method consumes the least hardware resources, but its resolution is only λ/16 [33]. The look-up table method is easy to implement, but it will consume a lot of memory cells [34]. The Cordic algorithm is the most commonly used method at present, which can resolve the phase angle of the interference signal quickly and accurately [35,36]. But only using the Cordic algorithm cannot obtain the interferometric results. Therefore, a method combining the eight-subdivision method and the modified Cordic algorithm (ESMCA) is proposed, which can not only achieve extremely high resolution, but also obtain interferometric results quickly.
The principle of our phase resolving method, ESMCA, is shown in Fig. 5. In the integer part, an eight-subdivision algorithm is used to accumulate the number of interference fringes. The vector circle composed of two orthogonal signals is evenly divided into eight parts, and the phase can be located in a certain interval according to the sign and the absolute values of the two signals. m is denoted as the number of eight-subdivision intervals passed by the vector circle of the interference signal, and the integer counting value N1 can be obtained:
This method can accurately calculate the changes of interference fringes in the measurement process, but it has a resolution of only λ/16. In order to improve the measurement resolution, the Cordic algorithm is used to achieve a phase subdivision of λ/16.The basic principle of the Cordic algorithm comes from Cartesian coordinate plane rotation, as shown in Fig. 5. The angle of rotation from the point (x1, y1) to the point (x2, y2) on the X-Y coordinate plane is θ′, and the phase angle α in the eight-subdivision intervals needs to be calculated.
The coordinates after rotation can be calculated according to the initial coordinates:
The Cordic algorithm here is to calculate the phase angle in the eight-subdivisions. First, according to the specific situation of the interference signal in different rotation directions and different intervals on the vector circle, the interference signal is converted to the applicable range of the Cordic algorithm. The phase angle θc from the interference signal to the positive direction of the Y coordinate axis is calculated by the Cordic algorithm. Then, each phase angle is converted into angle α within eight-subdivisions by angle conversion in the coordinate system. The method of calculating α at different intervals and in different rotation directions of a vector circle is shown in Table 1. α at the initial time is α0.
Table 1. The calculation method of vector circle of different rotation directions in different intervals
Therefore, the phase of the fractional part of the interference signal can be calculated and should be α-α0. The total fringe number N where the interference signal changes can be obtained from the cumulative fringe number based on the eight-subdivision algorithm and the fractional phase calculated based on the modified Cordic algorithm.
The radial motion error of the rotary axis along the X-axis can be obtained by Formula (1) and (15).Therefore, the ESMCA can not only realize the high-resolution solving of the current interferometric phase but also quickly obtain high-precision interferometric results.
3.3 Five other DOF motion errors measurement and beam angle-drift monitoring module
This module uses four QDs to collect changes in the position of relevant beam spots. The collected analog signal is filtered and converted to a digital signal. The data can be transmitted together with the interferometric data through serial communication to the computer software for calculating all the six DOF motion errors of the rotary axis.
4. Experimental results and analysis
Based on the design of the above system, a simultaneous measuring system for six DOF motion errors of the rotary axis was developed. The main photosensors in the measuring system are four PDs (Beijing Lightsensing Technologies, LSSPD-U3.2, Response time of 50 ns, Responsivity (λ=635 nm) of 0.41 mA/mW) and four QDs (Pacific Silicon Sensor, QP50-6SD2-DIAG, resolution of 0.08 µm, range of ± 500 µm). The sensitive unit include two retro-reflectors (Thorlabs, N-BK7, beam deviation of 3$^{\prime\prime}$) and a beam-splitting film coated on the surface of the retro-reflectors. The reference rotary axis is Aerotech ANT95R-360, whose resolution is 0.01$^{\prime\prime}$, the angular positioning error is 10$^{\prime\prime}$, and unidirectional repeatability is ± 0.5$^{\prime\prime}$. The target rotary axis was the SKQ-12200 produced by KEOLEA. The angular positioning error was about 40$^{\prime\prime}$, and the repeatability value was about ± 20$^{\prime\prime}$.
A lot of experimental research and validation have been carried out on the developed measuring system. In this experiment, an autocollimator (AcroBeam Collapex STD-3020, range of ± 1800$^{\prime\prime}$, accuracy of ± 0.15$^{\prime\prime}$) and a laser displacement sensor (Micro-Epsilon ILD2300-2LL, range of 2 mm, accuracy of 0.6 µm) were used. A precision polyhedral prism, which was mounted coaxially with the target rotary axis was used as the sensitive unit for error comparison experiment, as shown in Fig. 7.
4.1 Calibration experiments
Measurement resolution: The measurement resolution of the radial motion error along the X-axis depends on the number of iterations of the Cordic algorithm. The measurement resolution is 0.4 nm when the Cordic algorithm is used for seven iterations. The measurement resolution of the other five DOF motion errors is determined by the resolution and spatial position of the QD detectors. The distance L from the beam-splitting film to QD4 is 190 mm, and the resolution of the angular positioning error and the tilt motion error around the Y-axis is 0.043$^{\prime\prime}$. The resolution of the radial motion error along the Y-axis and axial motion error is half the resolution of the QD detector, namely 0.04 µm. The larger the center distance LRR23 between the two retro-reflectors in the sensitive unit is, the higher the resolution of the tilt motion error around the X-axis is, but the smaller the measurement range is. LRR23 in the measuring system is 30 mm, and resolution of the tilt motion error around the X-axis is 0.27$^{\prime\prime}$.
Calibration experiments: The errors measured by our system can be classified into three categories: the radial motion error along the X-axis measured by the PDs, displacement errors and angular errors measured by the QDs. A grating ruler (LG-50, accuracy ± 0.1 µm, resolution ± 0.05 µm) was used to calibrate the three linear displacement values, and the three angular values were calibrated by the auto-collimator.
The calibration results are shown in Fig. 8. In the range of ± 90 µm, the linear-fit determination coefficient between the grating ruler and the QDs is up to 0.9999, the standard deviation is 0.56 µm, and the range of fluctuation is from −1.52 µm to 0.6 µm. In the range of ± 120$^{\prime\prime}$, the linear-fit determination coefficient between the auto-collimator and the QDs is up to 0.9999, the standard deviation is 0.82$^{\prime\prime}$, and the range of fluctuation is from −1.2$^{\prime\prime}$ to 1.18$^{\prime\prime}$.
Fig. 8. Calibration results of the QDs for measuring displacement errors (left) and angular errors (right).
4.2 Stability experiments
The stability experiments were performed under laboratory conditions, and the environmental temperature range was about 25 ± 1°C. The experimental system was fixed on the optical platform, and the distance between the measurement unit and the sensitive unit was 130 mm. The experimental results were auto-recorded in computer software and extracted every 0.5 seconds for 60 minutes. The standard deviation of the radial motion error along the X-axis is 0.03 µm. The standard deviation of the angular positioning error and the tilt motion error around the Y-axis are 0.03$^{\prime\prime}$ and 0.05$^{\prime\prime}$, respectively. The standard deviation of the tilt motion error around the X-axis is 0.1$^{\prime\prime}$. The standard deviation of the radial motion error along the Y-axis and the axial motion error are, respectively, 0.02 µm and 0.03 µm.
4.3 Repeatability and comparison experiments
Repeatability experiments: The measurement of the six DOF motion errors of the rotary axis were repeated ten times consecutively. The results after error compensation are shown in Fig. 9. The repeatability of each measurement point is half the peak-to-peak value of the ten measurements, and the repeatability of the motion error of a certain DOF is the maximum value of the repeatability of all measurement points. The measurement results before and after the compensation and the repeatability values after the compensation are shown in Table 2. The measured result of motion error for each DOF is the maximum absolute value of motion errors for all measurement points.
Table 2. The measurement result before and after the compensation and the repeatability values after the compensation.
Comparison experiments: A auto-collimator and a laser displacement sensor were used as standard instruments for comparison experiments, and the sensitive unit was a metal 12-face prism installed coaxially with the rotary axis to be measured [37], as shown in Fig. 7. The two measurement systems simultaneously completed ten continuous measurements of the six DOF motion errors of the rotary axis. The maximum comparison error results are shown in Fig. 10. The maximum comparison error values for each DOF are the maximum absolute values of all measurement points. The maximum comparison error of the radial motion error along the X-axis is 0.46 µm. The maximum comparison error of the radial motion error along the Y-axis and axial motion error are 1.00 µ m and 0.49 µm, respectively. The maximum comparison error of the angular positioning error is 0.74$^{\prime\prime}$. The maximum comparison error of the tilt motion error around the X-axis and Y-axis are 1.06$^{\prime\prime}$ and 1.53$^{\prime\prime}$, respectively.
Fig. 10. Results of the maximum comparison error for measuring angular errors (left) and for measuring displacement errors (right).
Analysis of the results: The system can simultaneously measure the six DOF motion errors of the rotary axis, and all the other five DOF motion errors have good repeatability except the angular positioning error. According to the results of the comparison experiment, the six DOF motion errors of the rotary axis measured by the six DOF measuring system are close to (basically coincide with) the standard instrument. The maximum comparison error of the radial motion error along the X-axis based on interference is only 0.46 µm, the maximum comparison error of angular error measurement based on auto-collimation principles was 1.53$^{\prime\prime}$, and the maximum comparison error of displacement errors measurement based on collimation was 1 µm. The high precision measurement of six DOF motion errors of the rotary axis is realized. In addition, the comparison error of the angular positioning error of the rotary axis was 0.74$^{\prime\prime}$, which proves that the large repeatability value of the angular positioning error is due to the large repeatability value of the rotary axis itself. The measurement system in this paper has similar stability and repeatability values compared with our previous measurement system using a He-Ne laser [32], which proves that the semiconductor laser can replace the He-Ne laser to complete the measurement of the six DOF motion errors of the rotary axis. Besides, it has the advantages of low cost and miniaturization.
5. Conclusions
In this paper, a novel method based on a single-mode fiber coupled semiconductor laser is proposed to simultaneously measure the six DOF motion errors of a rotary axis. The application of the single-mode fiber coupled semiconductor laser greatly reduces the volume, energy consumption, and cost of the measuring instrument. Moreover, it has the advantages of simple structure and so on. An interferometric phase solution method ESMCA is proposed, which can measure the radial motion error of a rotary axis with high precision and has the characteristics of resolving phase quickly. The six DOF motion errors measuring system was developed, and a series of experiments including calibration experiments, stability experiments, repeatability experiments, and comparison experiments were performed. The experimental results show that the proposed measuring method and the developed measuring system have good repeatability and stability. The repeatability of the radial motion error along the X-axis ${\delta _\textrm{x}}$ was ± 0.16 µm. The repeatability of the other five DOF motion errors ${\delta _\textrm{y}}$, ${\delta _\textrm{z}}$, ${\varepsilon _\textrm{x}}$, ${\varepsilon _\textrm{y}}$ and ${\varepsilon _\textrm{z}}$ were ± 0.29 µm, ± 0.25 µm, ± 0.65$^{\prime\prime}$, ± 0.62$^{\prime\prime}$ and ± 13.42$^{\prime\prime}$, respectively. The results of simultaneous comparative experiments with standard instruments show that the measuring system has high precision, and the maximum comparison errors of the six DOF motion errors were 0.46 µm, 1.00 µm, 0.49 µm, 1.06$^{\prime\prime}$, 1.53$^{\prime\prime}$ and 0.74$^{\prime\prime}$, respectively. Thus, it provides a low cost and high precision method for measuring the six DOF motion errors of the rotary axis and related high-precision machining measurement and control equipment.
Funding
National Natural Science Foundation of China (No.51905030); Ministry of Science and Technology of the People's Republic of China (No.2022XAGG0200); Fundamental Research Funds for the Central Universities (2023JBZX004).
Disclosures
The authors declare that there are no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. C. Hong, S. Ibaraki, and A. Matsubara, “Influence of position-dependent geometric errors of rotary axes on a machining test of cone frustum by five-axis machine tools,” Precis. Eng. 35(1), 1–11 (2011). [CrossRef]
2. N. A. Lawson, “Control system for an articulated manipulator arm,” U.S. Patent application 9,102,052 (11 Aug 2015).
3. Y. Ohnishi, S. Ekuni, and M. Kondo, “Radar antenna device and method for controlling electric power source thereof,” Patent 9,684,062 (20 Jun 2017).
4. Y. S. Zeng, S. W. Qu, and J. W. Wu, “Polarization-division and spatial-division shared-aperture nanopatch antenna arrays for wide-angle optical beam scanning,” Opt. Express 28(9), 12805–12826 (2020). [CrossRef]
5. J. Das and P. Venkateswaran, “Design and development of multiple antenna based system for RADAR applications,” IEEE 143–146 (2015), MTT-S International Microwave and RF Conference (IMaRC).
6. M. Melichar and J. Kutlwašer, “The Issue of Contactless Setup before Measuring Process,” Procedia. Eng. 69, 1088–1093 (2014). [CrossRef]
7. Z. Yang, J. Hong, J. H. Zhang, and M. Y. Wang, “Research on the rotational accuracy measurement of an aerostatic spindle in a rolling bearing performance analysis instrument,” Int. J. Precis. Eng. Manuf. 15(7), 1293–1302 (2014). [CrossRef]
8. J. Kang, B. Wu, Z. Sun, and J. Wang, “Calibration method of a laser beam based on discrete point interpolation for 3D precision measurement,” Opt. Express 28(19), 27588–27599 (2020). [CrossRef]
9. M. Holler and J. Raabe, “Error motion compensating tracking interferometer for the position measurement of objects with rotational degree of freedom,” Opt. Eng. 54(5), 054101 (2015). [CrossRef]
10. S. Zhu, G. Ding, S. Qin, J. Lei, L. Zhuang, and K. Yan, “Integrated geometric error modeling, identification and compensation of CNC machine tools – ScienceDirect,” Int. J. Mach. Tool. Manuf. 52(1), 24–29 (2012). [CrossRef]
11. X. Liu, X. Zhang, F. Fang, and S. Liu, “Identification and compensation of main machining errors on surface form accuracy in ultra-precision diamond turning,” Int. J. Mach. Tool. Manuf. 105, 45–57 (2016). [CrossRef]
12. Y. Y. Hsu and S. S. Wang, “A new compensation method for geometry errors of five-axis machine tools,” Int. J. Mach. Tool. Manuf. 47(2), 352–360 (2007). [CrossRef]
13. ISO, “Test code for machine tools. Part 7: Geometric accuracy of axes of rotation,” ISO230. 7, 3–4 (2006).
14. S. F. Wang, M. H. Chiu, C. W. Lai, and R. S. Chang, “High-sensitivity small-angle sensor based on surface plasmon resonance technology and heterodyne interferometry,” Appl. Opt. 45(26), 6702–6707 (2006). [CrossRef]
15. S. T. Lin and W. J. Syu, “Heterodyne angular interferometer using a square prism,” Opt. Laser. Eng. 47(1), 80–83 (2009). [CrossRef]
16. ISO, “Test code for machine tools. Part 1: Geometric accuracy of machines operating under no-load or quasistatic conditions,” ISO230. 1, 65–67 (2012).
17. H. J. Ahn, D. C. Han, and I. S. Hwang, “A built-in bearing sensor to measure the shaft motion of a small rotary compressor for air conditioning,” Tribol. Int. 36(8), 561–572 (2003). [CrossRef]
18. S. Osei, W. Wang, and Q. Ding, “A new method to identify the position-independent geometric errors in the rotary axes of five-axis machine tools,” J. Manuf. Process 87, 46–53 (2023). [CrossRef]
19. Y. Pu, L. Wang, M. Yin, G. Yin, and L. Xie, “Modeling, identification, and measurement of geometric errors for a rotary axis of a machine tool using a new R-test,” Int. J Adv. Manuf. Technol. 117(5-6), 1491–1503 (2021). [CrossRef]
20. T. Erkan, J. R. R. Mayer, and Y. Dupont, “Volumetric distortion assessment of a five-axis machine by probing a 3D reconfigurable uncalibrated master ball artefact,” Precis. Eng. 35(1), 116–125 (2011). [CrossRef]
21. S. Ibaraki, T. Iritani, and T. Matsushita, “Calibration of location errors of rotary axes on five-axis machine tools by on-the-machine measurement using a touch-trigger probe,” Int. J. Mach. Tool. Manuf. 58(1), 44–53 (2012). [CrossRef]
22. C. Hong and S. Ibaraki, “Non-contact R-test with laser displacement sensors for error calibration of five-axis machine tools,” Precis. Eng. 37(1), 159–171 (2013). [CrossRef]
23. C. H. Liu, W. Y. Jywe, L. H. Shyu, and C. J. Chen, “Application of a diffraction grating and position sensitive detectors to the measurement of error motion and angular indexing of an indexing table,” Precis. Eng. 29(4), 440–448 (2005). [CrossRef]
24. Z. J. Zhang and H. Hu, “Three-point method for measuring the geometric error components of linear and rotary axes based on sequential multilateration,” J. Mech. Sci. Technol. 27(9), 2801–2811 (2013). [CrossRef]
25. W. Yang, X. Liu, X. Guo, W. Lu, Z. Yao, and Z. Lei, “A method for simultaneously measuring 6DOF geometric motion errors of a precision rotary stage based on absolute position-distance measurement,” Opt. Laser. Eng. 138, 106420 (2021). [CrossRef]
26. H. Murakami, A. Katsuki, and T. Sajima, “Simple and simultaneous measurement of five-degrees-of-freedom error motions of high-speed microspindle: Error analysis,” Precis. Eng. 38(2), 249–256 (2014). [CrossRef]
27. S. R. Park, T. K. Hoang, and S. H. Yang, “A new optical measurement system for determining the geometrical errors of rotary axis of a 5-axis miniaturized machine tool,” J. Mech. Sci. Technol. 24(1), 175–179 (2010). [CrossRef]
28. C. S. Liu, H. C. Hsu, and Y. X. Lin, “Design of a six-degree-of-freedom geometric errors measurement system for a rotary axis of a machine tool,” Opt. Laser. Eng. 127, 105949 (2020). [CrossRef]
29. C. J. Chen, P. D. Lin, and W. Y. Jywe, “An optoelectronic measurement system for measuring 6-degree-of-freedom motion error of rotary parts,” Opt. Express 15(22), 14601–14617 (2007). [CrossRef]
30. J. Li, Q. Feng, C. Bao, and Y. Zhao, “Method for simultaneous measurement of five DOF motion errors of a rotary axis using a single-mode fiber-coupled laser,” Opt. Express 26(3), 2535–2545 (2018). [CrossRef]
31. C. Sun, S. Cai, Y. Liu, and Y. Qiao, “Compact laser collimation system for simultaneous measurement of five-degree-of-freedom motion errors,” Appl. Sci 10(15), 5057 (2020). [CrossRef]
32. J. K. Li, Q. B. Feng, C. C. Bao, and B. Zhang, “Method for simultaneously and directly measuring all six-DOF motion errors of a rotary axis,” Chin. Opt. Lett. 17(1), 011203 (2019). [CrossRef]
33. S. Zhao, S. Zhu, and D. Wen, “Research on Nonlinear Error Compensation Method for Interferometers Based on Tracker,” J. Comput. Inf. Sci Eng 10(11), 3285–3293 (2013). [CrossRef]
34. J. Wang and J. Yang, “Research of a DSP-based Real–time Demodulation Method of One-frequency Laser Interferometer,” J. Eng. Heilongjiang. University 2(1), 77–82 (2011).
35. J. E. Volder, “The CORDIC trigonometric computing technique,” IRE Trans. Electron. Comput. EC-8(3), 330–334 (1959). [CrossRef]
36. S. Bhuria and P. Muralidhar, “FPGA implementation of sine and cosine value generators using Cordic Algorithm for Satellite Attitude Determination and calculators,” IEEE, 1–5, (2010), International Conference on Power, Control and Embedded Systems.
37. D. Ma, J. Li, Q. Feng, Q. He, Y. Ding, and J. Cui, “Simultaneous Measurement Method and Error Analysis of Six Degrees of Freedom Motion Errors of a Rotary Axis Based on Polyhedral Prism,” Appl. Sci. 11(9), 3960 (2021). [CrossRef]
