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Abstract
Precisely and efficiently measuring three-dimensional coordinates of key points on large-scale components in the manufacturing process of aircraft and ships is critically essential. This study presents a multi-target automatic positioning method based on rapid angle and distance measurement in parallel. The measurement processes for angles and distances are decoupled and, when executed simultaneously, aims to enhance the measurement efficiency and automation compared with conventional metrology systems. A cooperative target is devised to realize the rotary-laser scanning angle measurement and absolute distance measurement in parallel. The method of multi-target rough positioning based on rotary-laser scanning and then the precise coordinate measurement method introducing absolute distance constraint are detailed. Especially for the distance measurement, we propose a method to determine the internal zero length and compensate for the distance error caused by mirror offset. A real-site experiment is implemented to verify the method's feasibility and demonstrate that the 3D coordinate measurement accuracy is better than 0.17 mm compared with laser tracker.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Large-scale coordinate measurement technology and instruments are critical supporting elements for the manufacturing and assembly of airplanes, ships, wind turbines, and spacecraft [1,2]. They play a vital role in ensuring product quality and controlling equipment collaboration. Due to the huge size of these components, which can range from several meters to tens of meters, a significant number of control points are required to measure. Therefore, large-scale coordinate measurement systems need to possess capabilities of wide measurement range, high accuracy, and high efficiency to meet on-site requirements [3].
Current mainstream large-scale coordinate measurement systems can be categorized into two main types: distributed and centralized measurement systems [4]. Distributed measurement systems, such as theodolite [5], iGPS [6], close-range industrial photogrammetry [7], and laser tracer [8], rely on a multi-station measurement network that offers wide coverage and flexible configuration capability. Among them, iGPS is the only one that can realize automatic parallel measurement of multiple targets. Based on the principle of triangulation, due to the existence of angular uncertainty, the accuracy of coordinate measurement in x and y axes are significantly lower than that in z axis direction [9]. And this phenomenon becomes more obvious as the distance increases.
Unlike distributed measurement systems, centralized measurement systems can independently measure 3D coordinates using a single station. Laser tracker [10] and laser radar [11] are both examples that utilize the spherical coordinate measurement principle, relying on measurements of distance and angles.
Laser tracker has become the most widely used standardized instrument in large-scale equipment manufacturing, owing to its high accuracy, flexible deployment, and strong adaptability to the environment [12–14]. It relies on two high-precision circular gratings integrated into its horizontal and vertical rotating shafts to measure angles and provide tracking feedback. This mechanism ensures that the ranging beam aims at the center of the retroreflector, allowing for accurate distance measurements. Since angle and distance measurements closely rely on the same tracking and aiming mechanism, only two types of information can be obtained simultaneously. As a result, the laser tracker can only perform time-sharing, sequential, and point-by-point measurements. When conducting multi-target measurement tasks, target must be switched manually, which can be inefficient and lacks automation. Laser radar such as Nikon MV330 and Leica ATS600 shares a similar mechanical structure with laser tracker. They excel at high-precision scanning of surfaces without the need for targets or probes, leading to a significant boost in measurement efficiency. However, they are limited to measuring specific targets and areas, unable to concurrently position local measurement equipment, automated guided vehicles, and large components during assembly and docking processes. Consequently, there is a growing demand for a comprehensive large-scale coordinate measurement system capable of multi-target automatic measurements, which has become a hot topic in the manufacturing and measurement fields.
Hughes et al. [15] developed a multi-target coordinate metrology system based on multilateration. Multiple measurement units composed of cameras and spatial light modulators form a measurement network. The absolute distances are obtained through frequency-scanning interferometry technology simultaneously to determine the coordinates. However, the measurement range and efficiency of this method is still limited by the small field of view of the camera and spatial light modulator. Liu et al. [16] use a diffractive optical element to generate multiple beams to measure multiple targets based on multilateration. In practical applications, this method is limited by the measurement range and targets only can be arranged along the limited directions.
To enhance the measurement efficiency of laser tracker and take advantage of the well-established absolute distance measurement technology, vision methods are proposed and integrated for target recognition and automatic guidance [17]. One such example is the utilization of Leica's Power Lock. However, the camera's limited field of view and low recognition efficiency in complex environments restricts the effectiveness of visual methods. While they can perform tasks such as reconnection after light interruption, they cannot achieve multi-target parallel measurement. Yang et al. [18] study a method based on rotary-laser scanning measurement system to improve laser tracker’s automation capability. Both the above methods achieve automatic guidance of the tracker, but due to the large moment of inertia and highly coupled mechanical structure of the tracker itself, the measurement efficiency is still limited. To address these challenges, it is necessary to decouple the angle and distance measurement processes and implement them in parallel. By doing so, we can significantly improve the measurement efficiency of centralized measurement systems while reducing the correlation between angle and distance measurements.
The rotary-laser scanning angle measurement technology offers the unique advantage of enabling multi-target parallel measurement. With its large coverage and fast measurement speed [19–23], it has become the preferred choice for various applications, such as iGPS and wMPS measurement systems. In addition, the research [24] has demonstrated that coordinate measurement can be accomplished using only one laser transmitter through the use of a multi-sensor 3D auxiliary target and the principle of resection. Compared to other methods, rotary-laser scanning angle measurement technology is particularly well-suited for parallel measurement of large coverage with high precision.
According to the above analysis, we propose a centralized large-scale multi-target coordinate measurement system based on rotary-laser scanning angle measurement and absolute distance measurement (ADM) in this paper. Our system includes a cooperative target equipped with three photoelectric receivers positioned around a retroreflector. Rotary-laser scanning enables simultaneous angle measurements of multiple targets across the entire circumference, and offering guidance information for distance measurements. Utilizing a high-performance fast steering mirror (FSM), the ranging beam efficiently scans multiple targets, providing absolute distance data. This method enables the automatic identification and near-simultaneous measurement of multiple targets.
Compared with previous three-dimensional large-scale coordinate measurement systems and researches, main contributions of this study can be summarized into three aspects. Firstly, by decoupling the strong correlation between angle and distance measurement, both of information can be obtained in parallel. This departure from the traditional point-by-point measurement mode of the spherical coordinate system enables precise and efficient multi-target coordinate measurement using a single station. Secondly, a cooperative target is designed to fuse angle and distance information to obtain three-dimensional coordinates. Based on this target, the ranging beam is guided and the coordinates are accurately measured. Thirdly, the ranging error introduced by the structural error of the fast steering mirror is analyzed and numerically compensated, and an offline internal zero length determination method is proposed.
The rest part of this paper is organized as follows: Section 2 describes the concept and principle of the proposed measurement system, and provides a detailed explanation of the coordinate measurement method. In Section 3, we introduce the calibration method for rotary-laser scanning angle measurement and absolute distance measurement module. Meanwhile, we elaborate on the compensation method of the distance error introduced by the fast steering mirror and the determination of some key parameters in the absolute distance measurement module. The measurement uncertainty is still discussed in this section. Section 4 presents the experiment to verify the feasibility and accuracy of the system. Section 5 demonstrates the conclusions and possible improvements for the system.
2. Concept and principle
2.1. Principle and composition of the system
The schematic diagram of the measurement system is shown in Fig. 1. The coordinate system of the rotary-laser scanning angle measurement module is ${O_S} - {X_S}{Y_S}{Z_S}$ detailed in Section 2.2.1, which is defined as the measurement system's global coordinate system (GCS). ${O_L} - {X_L}{Y_L}{Z_L}$ (LCS) is the coordinate system of the absolute distance measurement module, which will be described in detail in Section 3.2. The origin of absolute distance measurement is ${O_L}$, and its coordinate $({{x_l},{y_l},{z_l}} )$ in GCS is obtained by precise calibration by multilateration. ${O_T} - {X_T}{Y_T}{Z_T}$ (TCS) is the coordinate system for cooperative target, engineered to fuse angle and distance to coordinate. The target consists of a retroreflector ${S_r}$ surrounded by three photoelectric receivers ${P_i}$ (i = 1,2,3) which are used to receive photoelectric scanning signals. They are adjusted to be coplanar so that coordinate measurement has a unique solution refer to [25]. And the high-precision instrument calibrates their relative positions as a system intrinsic parameter before measurement.
When two laser planes pass through the receivers ${P_1}$, ${P_2}$ and ${P_3}$, the rough coordinate of the retroreflector ${S_r}$ in GCS is obtained by the geometric constraints of the receivers and laser plane constraints. By coordinate transformation, the azimuth and pitch angles of the retroreflector in LCS are obtained. To guide the ADM laser to the center of the retroreflector and measure the absolute distance between it and the origin ${O_L}$, we utilize a fast steering mirror as the beam deflection mechanism. Finally, by adding absolute distance constraints, high-precision 3D coordinates can be obtained.
To achieve simultaneous measurement of multiple targets in the measurement space, fast angle scanning enables rough coordinate measurement. Based on the initial guidance information obtained previously, the fast steering mirror guides the ranging beam to switch between multiple targets and quickly measure the absolute distance. With this approach, the accurate coordinates of every target can be acquired, resulting in automatic and simultaneous measurement.
2.2. Coordinate measurement
2.2.1. Rough coordinate measurement
The rotary-laser scanning angle measurement module consists of a rotary-laser transmitter, signal processor, and photoelectric receivers. Two line-lasers are installed on the rotating head, their optical axes perpendicular to each other. The tilt angles of the two laser beams relative to the rotating shaft are designed as +45° and -45°. Some pulsed lasers are uniformly mounted on the stationary base. When the transmitter is working, the rotating head spins at a constant speed in the anticlockwise direction, driving two laser planes to scan the entire measurement space. While the rotating head spins to the predefined zero position every circle, the pulsed lasers emit as the synchronization signal. Photoelectric receivers on the points to be measured capture the signal and record the time as ${t_0}$ at this time, serving as the initial time. When laser plane 1 sweeps through receiver i, the time is recorded as ${t_{1i}}$, which is the end of scanning time of laser plane 1. Similarly, when laser plane 2 sweeps through the same receiver, the time is recorded as ${t_{2i}}$. According to the rotational angular velocity ω of the rotating head, the angle of the laser plane rotates from the initial position to the receiver position can be obtained as follows:
According to the definition of GCS above, the laser plane equations at the initial time ${t_0}$ can be expressed as:
From Eq. (1), the rotation angles of the two laser planes around the ${Z_S}$-axis can be calculated, so the laser plane coefficient when it rotates to the receiver ${P_i}$ (i = 1, 2, 3) can be described as:
There are 12 unknowns in total in this method, which are the coordinates of three photoelectric receivers and a retroreflector on the cooperative target under GCS. The laser plane and structural constraints can constitute 12 constraint equations to solve the coordinates successfully. According to the above constraints, the objective function is constructed as follows:
2.2.2. Accurate coordinate measurement
After solving the rough coordinate of the retroreflector in GCS, the azimuth and pitch angle of it in LCS can be obtained through the coordinate system transformation. Using beam deflection mechanism to guide the laser and measure the distance ${L_r}$ from the retroreflector to the origin ${O_L}$. Thus, we get the absolute distance constraint:
where $({x_l},{y_l},{z_l})$ is the coordinate of origin ${O_L}$ in GCS. Combining with the 12 constraints in the rough positioning, there are now 13 constraints in total to establish an objective function in accurate coordinate measurement:3. System calibration and uncertainty analysis
In order to realize the transfer of the rough coordinate between the two coordinate systems and obtain the coordinate of origin ${O_L}$ in GCS, it is necessary to calibrate the measurement system and the two coordinate systems accurately. In this paper, the laser tracker is used as an intermediate frame to calibrate the two coordinate systems and to facilitate the comparison of coordinate measurement accuracy. In addition, we also analyze the uncertainty of the proposed method at the end of this section.
3.1. Angle measurement system calibration
A high-precision control field is constructed by laser tracker to calibrate the plane parameters of the two scanning laser planes at the initial position, as well as the rotation matrix ${\mathbf{R}}_{\mathbf{r}}^{\mathbf{g}}$ and the translation vector ${\mathbf{T}}_{\mathbf{r}}^{\mathbf{g}}$, which indicate the transformation from the coordinate system of laser tracker (RCS) to GCS.
There are two sets of coordinates $({x_{ri}},{y_{ri}},{z_{ri}})$ and $({x_{gi}},{y_{gi}},{z_{gi}})(i = 1,2,3\ldots k)$ where the subscript r refers to RCS and subscript g refers to GCS. When each control point captures the signal from the rotary-laser scanning module, the laser plane equations can be expressed as follows, like Eq. (4).
3.2. Distance measurement system calibration
The accurate coordinate measurement is based on acquiring a precise absolute distance between the retroreflector and the origin ${O_L}$. In order to deflect the beam to align the center of the target, a dual-axis fast steering mirror (Optotune, MR-15-30) is introduced into our system since it has the advantages of small moment of inertia, compact structure and fast response speed. In addition, it has a steering resolution of less than 5 µrad and a ± 50° optical deflection range from each axis. Based on the mirror, we built a beam deflection scheme. The laser beam for ADM is incident to the mirror obliquely with an angle of 45° from the collimator. The mirror is controlled by electronics to aim the beam and lock the center of the retroreflector. Because there is an offset between the mirror surface center and its rotation center, which is called mirror offset, the reflection point constantly changes with different mirror rotation angles. In addition, the distance measured by the system includes two parts: one is from the fiber end to the reflection point, and the other is between the latter and the cooperative target. However, our coordinate measurement model demands an exact distance of a fixed reflection point to the target. For this reason, we propose the method to determine the internal zero length and to compensate for the distance between the length origin and cooperative target. Meanwhile, we still describe the method to calibrate the optimized point as length origin.
3.2.1. Compensation for absolute distance
In order to analyze the distance measurement error caused by mirror offset, the simplified model is built as depicted in Fig. 2, which is of a top view of the system. Since the beam offset can be reduced to a small amount by precise adjustment, and secondly, the distance measurement error introduced by the non-orthogonality and offset of the mirror rotation axis is small [28], we take these two error sources into ignoration to simplify the model. The mirror coordinate system is chosen such that the intersection of two rotation axes coincides with the origin, and the mirror surface is parallel to the plane z = 0. Besides, the mirror offset d is restricted to the z-axis and lets the incident beam lie on the y-z plane. The incident beam passes through the mirror surface center with an angle to the mirror surface of ${\theta _{in}} = 45^\circ $ when it is at the initial position.
The mirror rotates along the x-axis and y-axis with angles of ${\theta _x}$ and ${\theta _y}$, respectively. The location of the mirror after rotation is determined by the mirror normal vector ${{\boldsymbol n}_{mr}}$ and point ${{\boldsymbol e}_{mr}}$ with
Therefore, the distance error caused by mirror offset is
where ${M_{in}}$ is the reference point for absolute distance measurement. In Eq. (15), the first two items represent the absolute distance L obtained by the rangefinder. The third item is ${L_0}$ defined as the distance between the reference point ${M_{in}}$ and length origin ${O_L}$ called internal zero length. Thus, the mirror offset means the difference between actual and ideal measurements. From Eq. (15), we get the distance error at a fixed point, therefore the distance obtained in real can be compensated by this model.In order to build a more accurate error model, it is essential to decide an exact length origin ${O_L}$. The origin should be the point to minimize the sum of squares of distance measurement errors for points within the measurement range. The objective function can be described as follows:
Based on the model, a simulation is implemented with points in a 5000 mm square cubic space about 4000 mm away from the origin of the mirror coordinate system. The results show that the optimal origin is on the center of the mirror surface with no rotation. Therefore, we determine the point ${O_L}$ as the length origin in the model to calculate the distance error. In order to verify the necessity of error compensation, another simulation is completed to display the distance error at different points.As shown in Fig. 3, the distance error is less than 40µm when points are on the plane x = 0 mm within the simulation area. The error increases as the point moves away from the horizontal plane. When points are on the plane x = 2000 mm, the maximum distance error is as significant as 120µm. Similarly, due to the increase in the deflection angle of the mirror, the farther the point is from y = 0, the greater the distance error.
3.2.2. Internal zero length determination
In our model, the laser beam can be regarded as emitted from the length origin. The more accurate the distance between the origin and the retroreflector is obtained, the more precise the target coordinate will be acquired. In the previous section, we analyze the length error due to mirror offset and build a model to compensate for it. However, the numerical compensation method should fit well with reality to achieve accurate compensation. The internal zero length ${L_0}$ is a crucial factor affecting the determination of the distance between the length origin and target as well as the compensation effect. To determine the internal zero length also is a necessity for calibrating the laser tracker [29], but there are some differences between the two systems. The laser tracker can be rotated 180° around its axis to determine the Birdbath distance, whereas the steering mirror we used can only rotate ±25° in mechanical range. To achieve high-precision determination, we proposed a method as depicted in Fig. 4.
The position relationship between the collimator and the steering mirror is consistent with the model in the previous section. The beam is incident at an angle of 45° to the mirror's center. A spherically mounted retroreflector (SMR) and a corner prism are mounted on a displacement stage. They move along the linear guides perpendicular to the outgoing beam. Firstly, when the steering mirror is not deflected, measuring the absolute distance from the fiber end to the SMR, denoted as d1. Since the mirror has no rotation now, there is no length error in the absolute distance between the length origin and the SMR, which is recorded as $d = {d_1} - {L_0}$. Secondly, the distance measured by the rangefinder is obtained when the mirror rotates an angle of ${\theta _r}/2$. Because the mirror offset introduces length error, the distance between the length origin and SMR after compensation is expressed as ${d_2} - {L_0} + {e_i}$. ${d_3}$ is the distance of the SMR before and after the mirror rotation measured by an interferometer. As a result, we can get
The internal zero length can be obtained by the formula. The interferometric distance and the absolute distance are inherently accurate. Different distances d and the optical deflection angle ${\theta _r}$ can bring different uncertainties to the measurement of ${L_0}$. Figure 5 shows the simulation of the uncertainty of ${L_0}$ with different d and ${\theta _r}$. As the rotation angle increases, the measurement uncertainty corresponding to different d decreases considerably and gradually converges to 6µm after the rotation angle reaches 30°. According to the above results, we choose to calibrate ${L_0}$ at ${\theta _r}$= 40° with the distance d determined according to the experimental environment.3.2.3. Length origin calibration
In the distance error compensation model described in Section 3.2.1, we regard the beam reflection point as the length origin when the mirror surface is not rotated. In practice, we need to calibrate the coordinate of the length origin in GCS. The high-precision control field constructed by the laser tracker is also used to complete it. Based on the calibration method of the rotary-laser scanning module in Section 3.1, the length origin in GCS is obtained by taking RCS as the intermediate coordinate system.
For each control point, the steering mirror is controlled to reflect the beam to the center of the SMR, at which point the measured distance obtained by the rangefinder is ${l_i}$. After subtracting the internal zero length and compensating for the corresponding length error, the distance between the length origin and the control point can be expressed as:
The distance between the unknown origin $(x_l^T,y_l^T,z_l^T)$ and the known control point $(x_i^T,y_i^T,z_i^T)$ is: The localization problem can be resolved by multilateration. The objective function is shown as follows: where n is the number of control points. After obtaining the coordinate of length origin in RCS through multilateration, it can be unified into the GCS through the transformation between them.3.3. Uncertainty analysis
Uncertainty is an important performance index to evaluate the performance of the metrology system, and it is also a key parameter when assigning the value to the measured object. Therefore, this section will analyze the uncertainty of the method proposed in this paper. The error sources of this method mainly come from three aspects: angle measurement error, distance measurement error and calibration error of cooperative target.
The angle measurement error has been confirmed by experiments and conforms to the Gaussian distribution $\Delta \theta \sim N(\begin{array}{{cc}} {0,}&{\sigma _\theta ^2} \end{array})$, and ${\sigma _\theta } = 1.5^{\prime\prime}$ (k = 1). The distance measurement error source is more complicated, mainly composed of three parts: the measurement error of the rangefinder, the error introduced by the fast steering mirror even after compensation, and the calibration error of the length origin. The rangefinder we used in the experiment in Section 4 has a measurement uncertainty of 0.25µm/m(k = 1). However, since we are unable to accurately estimate the magnitude of the distance measurement error introduced by the fast steering mirror after compensation, and the calibration error of length origin also includes the error introduced by the mirror, we cannot accurately quantify the latter two errors. Therefore, we combine the three errors to have an overall error of 2.5µm/m(k = 1), afterwards, the correctness of this estimate has been verified in experiment. The calibration error of the target is much less than the other two errors because the tracker calibrates the target at multiple stations, thereby reducing the introduced angle error. Thus, in the analysis of this paper, this error is ignored and only the first two errors are taken into consideration.
A Monte Carlo simulation (MCS) is performed based on the above analysis to determine the uncertainty of points within the system coverage. At the simulated point, the target is always facing the origin of GCS, and the angle and distance measurement errors are added to the true value. Then the optimization shown in Eq. (8) is performed to determine the point coordinates. This process is repeated 100 times, and the standard deviations are regarded as the coordinate uncertainty(k = 1) [30]. A simulation result is shown in Fig. 6, in which the simulation area is located on the XOY plane in GCS where x is from -10 m to -3 m and y is from -5 m to +5 m with steps of 400 mm in every axis. The result shows that the standard deviations in every axis are less than 0.065 mm, and the point errors are less than 0.065 mm, too. When calculating the uncertainty for a specific point, input all measured values associated with that point into the MCS. Introduce errors to propagate through the measurement model, generating numerous point coordinates with corresponding errors. Subsequently, calculate the coordinate covariance matrix based on these points, and derive the point error as uncertainty. The uncertainty incorporates an expansion factor k = 1, with a confidence of 68%.
Fig. 6. The Monte Carlo simulation result of coordinate uncertainty: (a) x-axis (b) y-axis (c) z-axis standard deviation and (d) the point errors
4. Experiments
A real-site points measurement experiment is performed to validate the reliability and effectiveness of the proposed coordinate measurement method. In the experiment, we compared the measurement result of rough coordinates and accurate coordinates to verify the feasibility of the guidance mechanism of this method. Moreover, the position error using the laser tracker measurement results as the reference value is clearly shown, indicating that this method has high accuracy. A simulation verification experiment is also conducted to demonstrate the feasibility of the MCS method for uncertainty analysis.
4.1. Setup of experiment platform
The setup of experiment platform is shown in Fig. 7. The coordinate measurement system comprises a rotary-laser scanning module and an absolute distance measurement module. The rotary-laser scanning module has an angle measurement uncertainty of 1.5”. The Multiline (Etalon) is used to measure the absolute distance, which has a measurement uncertainty of 0.25µm/m within the range of 30 m. The verification experiment used a laser tracker (AT901, Leica) to calibrate the measurement system. Because the measurement uncertainty of the laser tracker is 15µm + 6µm/m(k = 2), it is also used as the reference for comparison. The high-precision control field is composed of 30 control points, which are measured by laser tracker at multiple different stations to reduce the angle measurement error.
Through the calibration method refer to Section 3.1, the laser plane parameters of the rotary-laser scanning module are obtained as follows:
The laser tracker calibrates the cooperative target in different stations before the experiment, and the calibration result is shown in Table 1. The receivers and the retroreflector are nearly coplanar by complex adjustments, even though they are still out of the plane by a few microns.
Table 1. Calibration result of receivers and retroreflector in TCS
4.2. Points measurement experiment
The verification experiment for point coordinate measurement of the proposed method is carried out in free space and compared with the measurement results of the laser tracker. The cooperative target mounted on a tripod is placed at 23 points evenly distributed in the measurement space. The range of measurement space is about 3.5m × 4m × 1 m, which is 4 m away from the coordinate measurement system. The distribution of points and measurement system is shown in Fig. 8. The 23 points are roughly at three different heights, with the first nine points having z-axis coordinates of about z = -640 mm and being at the bottom. The last six points are located at the top height, with a z-axis coordinate of about 150 mm. The other seven points locate in the middle layer, and the z-axis coordinate is about z = -230 mm.
According to the proposed method's measurement process, the rough coordinate of the target is obtained by using the laser plane constraints through fast scanning of angle. Afterward, the two-axis rotation angle of the fast steering mirror is obtained and translated from GCS to LCS. The fast steering mirror controls the ranging beam to point and aim at the target through the host computer software. In the end, the rangefinder measures the absolute distance, and the accurate coordinate is solved. When the target is moved to any point, the measurement system performs coordinate measurement according to the above process. We compared the measurement results of rough and accurate coordinates in Table 2. It is obviously that the maximum deviation between rough and accurate coordinates is 12.937 mm, and the error is mainly concentrated in the x-axis along the depth direction, which is much larger than the errors in the other two directions. That verifies the feasibility of the method to guide the ranging beam in a short time when we apply a fast search and alignment algorithm.
Table 2. Results of accurate and rough measurement and their deviations
At each measuring position, the coordinate of the point is first measured by the proposed method and then by the laser tracker. In this process, we will not move the cooperative target or rotate the SMR but drive the tracker to align the measured point by software. The coordinates measured by the laser tracker are converted to GCS as the reference values by the transformation obtained by the control field calibration. It is important to emphasize that using the laser tracker as the reference value of the comparison introduces two aspects of error. On the one hand, the measurement error exists in the tracker itself, which is determined by the angle and distance errors, and the angle errors are much larger than the distance errors. On the other hand, the transformation error introduced by the calibration procedures using the control field is also a significant error source.
The measurement result is shown in Fig. 9, the brown, green, and purple histograms in the figure represent the errors in three axes, respectively. In particularly, they mainly distribute within the range of ±0.1 mm. The error bars denote the standard deviation in that axis from 20 times different measurements. The measurement standard deviation of all points is less than 0.08 mm. More precisely, 91% of the standard deviations are less than 0.03 mm, which shows the high precision of the proposed method. Point deviations of the coordinates measured by the proposed method and laser tracker are plotted in Fig. 10. The maximum deviation is 0.17 mm at Point 10, and the point deviation of 20 out of 23 points are less than 0.15 mm. The result shows that the measurement system has high accuracy to meet the measurement requirements of most equipment manufacturing.
4.3. Simulation verification experiment
The experiment implemented in the previous section shows that the point error is much larger than the simulation result. According to our analysis, this difference mainly comes from two aspects. The first is that only two primary error sources are considered in the simulation, and some else is ignored. Secondly, the point error in the experiment also includes calibration errors between the tracker and our system. Therefore, we simulated the measurement uncertainty of the experimental points in Section 4.2. The comparison between the experiment and the simulation for point uncertainty(k = 1) is shown in Fig. 11. The results show good consistency in all three axes and validate the effectiveness of the MCS approach.
Fig. 11. Comparison between experiment and simulation for uncertainty in (a) x-axis, (b) y-axis and (c) z-axis.
5. Conclusion
A large-scale multi-target automatic positioning method based on angle and distance parallel measurement is proposed in this paper. Compared to the widely used centralized instrument, the laser tracker, we introduce the method that enables concurrent angle and distance measurements through decoupling the strong correlation between them. This approach allows for multi-target automatic coordinate measurements, leading to a significant enhancement in measurement efficiency.
A cooperative multi-sensor target is designed on which three photoelectric receivers surround a retroreflector. The rough coordinate of the retroreflector is solved by scanning laser plane constraints and geometric constraints of the receivers. With the precise calibration for rotary-laser scanning measurement module and absolute distance measurement module, the fast steering mirror guides the ranging beam to the retroreflector and measures the distance. By adding the distance constraints, we use nonlinear optimization methods to obtain the accurate coordinate of the target.
A dual-axis fast steering mirror is introduced into our system to deflect the beam and align the center of the target. In order to obtain a more accurate absolute distance, we propose the method to determine the internal zero length and to compensate for the distance between the length origin and cooperative target. Meanwhile, we still describe the method to calibrate the optimized point as length origin.
The verification experiment is conducted with reference values measured by a laser tracker as a comparison. In the measurement space of 3.5m × 4m × 1 m which is 4 m away from the measurement system, the point deviation is less than 0.17 mm and only three points larger than 0.15 mm. The result of the experiment shows the feasibility, validity, and high accuracy of the proposed method. Therefore, our proposed method represents an advancement in coordinate measurement systems and holds the potential for future applications in a range of fields.
However, some areas of the method proposed in this paper still need improvement and detailed work to be done. One point is to improve the accuracy of rough coordinate measurement to reduce the aim time when the ranging beam points to the retroreflector, eventually improving the measurement efficiency. The other is to promote the research on the uncertainty analysis of coordinate measurement of this method. On this basis, improve the accuracy of the final coordinate measurement to a higher level.
Funding
National Natural Science Foundation of China (52127810, 51975408, 52205572).
Disclosures
The authors declare 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.
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