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Abstract
The laser tracker, as a new large-scale measuring instrument of combining conventional measurement technology and modern control technology, has the advantages of intelligence, portability, large measurement space, high measurement accuracy and short detection period. However, the laser tracker has strict requirements on the moving speed of the spherically mounted retroreflector. This deficiency not only limits the application of the measuring instrument in the field of high-velocity measurement, but also greatly reduces the measurement efficiency. In this work, we analyze the factors that affect the tracking velocity of the laser tracker, and propose for the first time to use the beam expander device to improve the transverse tracking measurement velocity of the instrument. The experimental results show that the laser tracker miss distance can reach 2.25 mm. The transverse tracking velocity and acceleration can reach 4.34 m/s and 2.4 g, respectively. Additionally, the acousto-optic modulator is used to increase the frequency difference between the reference beam and the measuring beam, so that the value is greater than 19 MHz. The radial tracking measurement velocity can reach 6.2 m/s. The high-velocity laser interference tracker developed by this new method can be used in the field of large-scale space precision measurement such as nuclear power, medical treatment and rail transit.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Precision measuring instruments are an important part of advanced manufacturing technology [1–6]. As a large size precision measuring has expanded, high-speed laser tracker design and research has become very critical and urgent.
Compared with the theodolite and the coordinate measuring machine, laser tracker has the advantages of intelligence, portability, large measuring space, high measuring accuracy, and short detection period [7]. In 1985, researchers R. Hocken and K. lan proposed the concept of laser tracker for the first time [8]. Subsequently, the scientific field is now changing to laser tracker, mainly including laser ranging technology, laser tracking control theory, coordinate calibration method, error analysis and compensation, and the application of laser trackers [9–12]. The tracking velocity is an important parameter in laser tracker. Improving the tracking velocity can make the laser tracker quickly lock the spherically mounted retroreflector (SMR), which provides a strong method for the normal operation of the instrument in the high-speed measurement environment. Therefore, it is of great significance to study the factors that affect the tracking velocity of the laser tracker and explore the methods to improve this parameter. The tracking velocity of the laser tracker can be divided into transverse and radial tracking velocities. The transverse tracking velocity of the laser tracker is proportional to the miss distance. The radial tracking velocity of the laser tracker is proportional to the frequency difference of the measuring system. The calculation of miss distance mainly depends on the overlapping area of the reference beam and the measuring beam. In fact, there have been some researches on measuring velocity and miss distance in previous papers. In 1998, Frank et al. made an interferometer which the measuring velocity is 2.1 m/s by use of the acousto-optic modulator (AOM) [13]. In 1999, Yokoyama et al. introduced a heterodyne interferometer without measuring velocity limit [14]. In 2001, Gao et al. developed a high-velocity measuring interferometer based on a dual-reflection film laser [15]. The frequency difference of the interferometer is 5 MHz and the measuring velocity is 1 m/s. In 2003, Konkola analyzed the power distribution in the overlap area of Gaussian beams in their Ph.D. dissertation [16]. In 2008, Cheng and Fan proposed to use the overlapping area of two interference beams to evaluate the signal strength [17,18]. They believe that the signal intensity of the interferometer is proportional to the overlapping area of two beams. In 2015, Dong et al. used position sensitive detector (PSD) to detect the miss distance of the laser tracker, and the miss distance was ± 2 mm, which satisfies greatly the need for fast tracking [19]. In recent years, scientists have also conducted research on femtosecond measurement [20–24]. In 2015, Yang et al. produced a femtosecond laser tracker with a tracking velocity of 2 m/s [25]. In 2016, Shang et al. pointed out in their study that grating rotation will cause multiple interference fringes in the overlapping area of two beams [26]. In 2022, Feng et al. proposed to increase the tracking velocity of the laser tracker by using the beam expander, but have not yet carried out an in-depth study [27].
This paper proposes an effective scheme to improve the tracking and measuring velocity of the laser tracker. Firstly, based on the classical model, we analyze the factors that affect the tracking measurement velocity in theory. Secondly, a beam expander device is added to the self-made dual-frequency laser interference tracking system (LITS) to improve the miss distance. Finally, an AOM is added to the optical path to make the tracking velocity reach 6.2 m/s. In addition, we built an experimental device to test the tracking measurement velocity and miss distance of the laser tracker. The experimental results show that the miss distance of the laser tracker can reach 2.25 mm, and the transverse tracking velocity is 4.34 m/s. The high-velocity laser tracker developed by this new method can be used in the field of large-scale space precision measurement such as nuclear power, medical treatment and rail transit.
2. Simulation of miss distance
High-quality signal strength is the premise for a dual-frequency interference tracking system to achieve high-precision measurement. As shown in Fig. 1(a), the position separation of the reference beam and the measuring beam is usually caused by the off-axis deviation between the optical axis and the central axis of the SMR. In most cases, the spatial energy distribution of the laser is usually Gaussian. Figure 1(b) shows the projection of two Gaussian beams on the two-dimensional plane [28]. In simple terms, the reference beam and the m mainly ensuring beam position are entirely coincident under ideal conditions. At the same time, the interference signal intensity reaches the highest level. In practice, the value of the beam overlapping area will fluctuate, which will lead to the change of the measuring signal strength. The transverse tracking velocity mainly depends on whether the interference signal will be distorted when the SMR is transverse translated. We describe the maximum distance value that the SMR can move as the miss distance of the laser interference tracking measurement system.
Fig. 1. (a) Beam shift of LITS; (b) Overlapping area of Gaussian beam; (c) Relationship between optical power and beam shift; (d) Zemax simulation of miss distance.
The wave equation of two beams with frequencies ${f_1}$ and ${f_2}$ can be expressed as
where E, A and $\textrm{t}$ represent the electric field strength, amplitude and time of the light wave, respectively. The light intensity of the interference part can be written asApplying Eqs. (1) through (3), the light intensity is expressed as
Because the photodetector (PD) cannot detect the signal with extremely high frequency, the light intensity expression can omit the second harmonic term and the sum of fundamental frequencies term. We can write it as
The amplitude of Gaussian beam is given by
Here, A is a constant proportional to the field magnitude. The variable $\omega $ is the radius of the Gaussian beam. The term r is the distance from the optical axis. Applying Eqs. (5) through (7), the light intensity is expressed as
Suppose that the offset of the two beams on the x-axis is $2\Delta x$, and the offset on the Y-axis is $2\Delta \textrm{y}$. ${r_1}$ and ${r_2}$ can be written as
where, x and y are the initial distance between the optical axes of the two beams in the X and Y directions. Applying Eqs. (8) through (10), the light intensity is expressed as${I_\textrm{o}}$ is the light intensity in the beam overlapping area. The frequency meter senses the zero crossing of an alternating current signal so only the term proportional to $({f_1} - {f_2})$ affects the measurement. Therefore, formula (11) can be simplified as
${P_0}$ is the total power in the beam given by the integral of the intensity.
The integration of Eq. (13) can be evaluated with the following identity obtained from Mathematica
Applying Eqs. (12) through (14), the optical power expression can be written as
The power of two Gaussian beams can be expressed as
This can be written in a more useful form using the following relations.
Applying Eqs. (16) through (19), the optical power expression can be written as
The separation between two beam centers is $\Delta d$ and the beam diameter is d.
According to Eq. (20), the relationship between the optical power in the overlapping area and the beam position shift is shown in Fig. 1(c). The simulation results show that the optical power of the measuring signal decays exponentially with the beam position shift. When the beam shift value is constant, the optical power attenuation rate is inversely proportional to the beam size. The allowable error range of the LITS can be improved by appropriately expanding the beam. Figure 1(d) shows the simulation graph of the miss distance by use ZEMAX when the Gaussian beam diameter is 2 mm. The interference image gradually becomes distorted when the beam shift changes from 0 to 0.5 mm. This can be used to determine the limit value of the miss distance of the LITS. These scientific bases can provide theoretical support for improving the transverse tracking velocity of the laser tracker.
3. Laser interference tracking measurement system
Laser tracker is a new type of measuring product developed in the past 40 years, which combines laser interferometric ranging technology, photoelectric detection technology, precision mechanical technology, computer and control technology, and modern numerical calculation theory. Figure 2(a) shows the measurement principle of the laser tracker. The laser tracker obtains the coordinate data $(X,\textrm{ Y, Z})$ by acquiring the azimuth angle $\alpha $, pitch angle $\beta $ and distance D of the target point P. The laser tracker uses the principle of spherical coordinate measurement, and the coordinate relationship can be expressed as
Fig. 2. (a) Measurement principle diagram of laser tracker; (b) Prototype photos of the interferometer; (c) The beam path of laser tracker; (d) The spot of reference beam; (e) The spot of measuring beam.
The laser tracking beam path is divided into two parts, one is the ranging unit, and another is the angle measuring unit. The function of the ranging unit is completed by a dual-frequency interferometer. Figure 2(b) shows the photo of the prototype of interferometer. The He-Ne laser (Model: Uniphse-1007) emits a laser with a wavelength of 632.8 nm, and the output power is 1.3 mW. The Zeeman effect occurs after the laser passes through the magnetic field, and the laser spectrum is divided into two beams of left-handed and right-handed polarized with different frequencies. After passing through the polarization beam splitter (PBS), the left-handed and the right-handed beam are divided into two linearly polarized beams whose vibrating planes are perpendicular to each other and whose phase difference is $\pi /2$. The frequency of the measuring beam is ${\nu _1}$, and its vibration direction and propagation direction are perpendicular to each other. The maximum modulation frequency of the AOM (Model: I-FS040-2S2E-1-GH66, Gooch & Housego) is 40 MHz, which can be used to improve the radial measurement speed of the LITS. A half wave plate (HWP) is used to adjust the output power of the LITS because the AOM changes the phase difference of beam by $\pi $. The expanded beam passes through the quarter wave plate (QWP) twice, and the phase change is $\pi $. Since the moving frequency of SMR is $\Delta \nu $, the frequency of the measuring beam becomes ${\nu _1}\textrm{ + }\Delta \nu $. The reference beam with frequency ${\nu _2}$ converges with the measuring beam through the optical wedge and BS to generate an interference signal. The moving distance of the SMR can be obtained from the following formula.
where t, $\lambda $ and N represent time, wavelength and the number of interference fringes, respectively. The PSD feeds back the moving distance of the beam to the motor, which realize the real-time tracking of the SMR. Figure 2(c) shows the beam path diagram of the laser tracker. As shown in Fig. 2(d) and 2(e), we use a laser beam profiler (Model: BC106N-VIS/M, Thorlabs) to detect the sizes of the reference beam and measuring beam. The dimensions of the reference beam and the measuring beam are 0.488 mm and 1.324 mm in the horizontal direction, respectively. The surface deformation of optical devices may be one of the main reasons for the irregular shape of gaussian beam.The principle of laser beam expander is shown in Fig. 3. The beam expander consists of a concave lens1 with -3.5 mm focal length and a convex lens2 with 75 mm focal length. The beam expansion magnification ${M_e}$ of the beam expander can be expressed by formula (25). After calculation, the beam expansion ratio is 21.43. The diameter of the measuring beam is expanded to 10.2 mm. In addition, the inverted beam expander reduces the beam diameter to within the measurement range of the PD.
4. Detection of miss distance
Figure 4(a) shows the lab setup of the miss distance detection. The output beam of the laser tracker passes through the SMR and returns to the original path. The multimeter (Model: Fluke 15B+) is used to monitor the signal voltage amplitude. As shown in Fig. 4(b), the coordinate of the micrometer screw is (8.913 mm, 11.401 mm). The variation curve of voltage amplitude with miss distance is Gaussian-like, which is like Fig. 1(b). The minimum value of miss distance is 2.25 mm, which can meet the demand of transverse tracking velocity. The oscilloscope (Model: Agilent DSO-X 3034A) is used to detect the quality of the measured signal waveform. Figure 4(c) shows the evolution curve of signal waveform with miss distance. It can be seen from the figure that with the increase of miss distance, the interference signal amplitude becomes lower and lower.
Fig. 4. (a) Miss distance detection device; (b) Miss distance measuring curve; (c) Interference signal waveform; (d) PSD detection principle; (e) X-axis calibration of PSD (f) Y-axis calibration of PSD.
Figure 4(d) shows the schematic diagram of PSD. Based on the working principle of PSD, the relationship between the coordinate of the beam spot and the output current can be expressed as
5. Transverse velocity measurement of laser tracker
With the increasing application of laser tracker in dynamic measurement, the dynamic calibration technology of the instrument has also become an important part of the research in the field of measurement. It is an important requirement of LITS to maintain high signal strength within a large range of acceptable error. The method to verify the transverse tracking velocity is to use laser tracker measuring the standard ball bar rotating at a certain speed. We compare the measured results with the standard parameters to obtain the error value of the instrument.
In Fig. 5(a), a standard circular trajectory generator (Model: R500-AXD200-65, The Beijing Control Technology Co., Ltd.) with a radius of 0.8 m is used to test the dynamically performance of laser tracker. The relationship between the linear velocity ${\upsilon _L}$ and the rotational speed ${N_L}$ can be written as
The acceleration of circular motion can be expressed as
Fig. 5. (a) The detection device of transverse velocity; (b) The experiment setup of velocity detection; (c) Transverse velocity curve; (d) Ranging value of laser tracker; (e) X-axis value curve of PSD; (f) Y-axis value curve of PSD.
Figure 5(b) shows the experiment setup of velocity detection. The transverse velocity value change curve of the standard instrument is shown Fig. 5(c). The SMR is fixed on the circular trajectory generator and rotates at a linear speed of 4.34 m/s. The transverse tracking acceleration can reach 2.4 g. In Fig. 5(d), we show the change curve of laser tracker ranging data during the experiment. The tracker is placed 4.459 m away from the SMR. Figure 5(e) shows the fluctuation of PSD X-axis value during the test, and the maximum fluctuation is 0.56 mm. It can be seen from Fig. 5(f) that the maximum fluctuation of PSD Y-axis value is 0.65 mm.
6. Radial velocity of laser tracker
We add the AOM to the LITS to improve the radial measurement velocity. The working principle of the AOM is shown in Fig. 6(a). The reason for the appearance of the ultrasonic wave is the mechanical vibration of piezoelectric transducer in the electric field. When the ultrasonic wave with wavelength ${\lambda _{_\textrm{s}}}$ propagates in the acousto-optic medium, the medium can be regarded as a moving grating. The grating follows the Bragg diffraction formula.
where $k = 1$ represents that the output beam is first-order diffraction light. ${\lambda _c}$ represents the wavelength of the light in the vacuum. n represents the refractive index of the medium. The Bragg diffraction angle ${\theta _B}$ is generally small, which can be written as where ${\upsilon _s}$ and ${f_s}$ are the velocity and frequency of the acoustic wave, respectively. ${\theta _B}$ is proportional to the ${f_s}$. The angle $\theta$ between incident light and diffracted light can be adjusted by changing the frequency of acoustic wave. According to the Bragg condition, $\theta$ is twice times the value of ${\theta _B}$.Fig. 6. (a) Schematic diagram of AOM; (b) The detection device of radial velocity; (c) Running speed curve of guide rail; (d) Ranging error of tracker; (e) X-axis numerical curve of PSD; (f) Y axis value curve of PSD.
The change of deflection angle is $\Delta \theta$ which can be written as
The Doppler shift generated by the beam passing through the grating moving at velocity ${\upsilon _s}$ is
where ${f_{in}}$ is the frequency of the incident light wave. ${\upsilon _s}_{/{/}}$ is the projection of the velocity vector ${\upsilon _s}_{}$ of the acoustic wave in the wave propagation direction.Applying Eqs. (34) through (35), the Doppler shift expression can be written as
The frequency of reflected light is equal to the frequency of incident light plus the frequency of acoustic wave. The illustration in Fig. 6(a) shows the quantum model of acousto-optic modulation. When sound waves propagate upwards, photon and phonon collide, which will consume a photon with frequency ${f_{in}}$, absorb a phonon with frequency ${f_s}$, and diffract a new photon with frequency ${f_d} = {f_{in}} + {f_s}$. On the contrary, when sound waves propagate downwards, photon and phonon collide, which will release a phonon with frequency ${f_s}$, and diffract a new photon with frequency ${f_d} = {f_{in}} - {f_s}$. k, K and ${k^{\prime}}$ represent the propagation direction of incident light, acoustic wave and diffracted light, respectively. The frequency ${f_d}$ of diffracted light can be written as
The diffraction process follows the law of conservation of energy, which can be expressed as
$h$ is the Planck constant. The frequency of the output laser varies with the frequency of the acoustic wave. The sound absorption device is placed at the end of the AOM to eliminate the interference of the echo to the device [29].Heterodyne interferometry has the advantages of good anti-interference, high measurement accuracy, good dynamic performance, and allowing multi-channel measurement [30]. However, the measurement speed of heterodyne interference system is limited by the beat frequency difference. The Doppler shift $\Delta \nu $ caused by the SMR movement shall not be higher than the beat frequency difference ${f_d} - {f_{in}}$, and the relationship can be written as
where $\upsilon $ represents the velocity of SMR movement. ${f_{in}}$ represents the frequency of incident light. ${f_d}$ represents the frequency of diffracted light. When the frequency difference of the heterodyne interferometer is higher than 20 MHz, the SMR with a velocity of 6.328 m/s can be measured. In this paper, we add an AOM to the LITS to make the frequency difference reach 19 MHz to meet the needs of high-speed measurement. As shown in Fig. 6(b), we use the high-speed guide rail (Model: TBD-2M-7MS-3 G, The Beijing Control Technology Co., Ltd.) to verify the measurement status of the laser tracker when the SMR speed is within 6 m/s. Figure 6(c) shows the velocity curve of the guide rail, and the maximum velocity of 6.2 m/s. Figure 6(d) shows the ranging repeatability accuracy of the LITS. The maximum value of repeatability accuracy is 9.1 µm, which can see that the tracker can work normally when the measuring velocity is 6.2 m/s. The ranging error of laser tracker is caused by the positioning error of the high-speed guide rail. Figure 6(e) and 6(f) show the PSD change curve during the test. The maximum variation on the X axis and Y axis are 0.06 mm and 0.23 mm, respectively. The 2.25 mm miss distance ensures that the LITS is in a normal tracking state during radial velocity testing. Table 1 compares the measurement velocities of several different types of tracking instruments. It is obvious that the measurement velocity of the instrument mentioned in this article has reached a good level.
Table 1. The measurement velocities of tracking instruments
7. Conclusions
This paper proposes a solution to improve the transverse and radial measurement speed of the laser tracker. We theoretically analyzed the influence of the internal components of the LITS on the transverse and radial tracking velocities. In addition, a LITS for high-velocity measurement was made, and an experimental platform was built to verify its measurement velocity. The experimental results show that the transverse and radial tracking velocities of the self-made laser tracker are greater than 6.2 m/s and 4.34 m/s, respectively. The miss distance can reach 2.25 mm. The maximum tracking acceleration can reach 2.4 g, which can meet the requirements of high-velocity measurement.
Funding
Ministry of Industry and Information Technology of China (TC220505T).
Disclosures
The authors declare no conflicts of interest.
Data availability
All data that support the findings of this study are included within the article.
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