#### Abstract

Subreflector misalignment of a large steerable radio telescope induces a pointing error and reduces the gain of the antenna system. To improve the antenna’s operational efficiency, it is necessary to measure and adjust the position and attitude of the subreflector in real time. In this paper, a method based on a position sensitive detector (PSD) and laser array without an optical system is proposed to measure the six degree-of-freedom (DOF) poses of the subreflector. The laser emitted by the laser module array ensures that the PSD can be covered as it moves with the subreflector, and the PSD can obtain more than three laser beams. These ensure the measurement of all attitude changes of a large-aperture antenna subreflector. The two-dimensional coordinates of the centroids of three laser spots are extracted using the PSD, and then the bursa model is established to complete the coordinate transformation. Finally, the 6-DOF attitude information of the antenna subreflector is obtained. The results of a 6.05 m measurement simulation show that it can obtain high 6-DOF PSD attitude information. The experimental results show that the 6-DOF position and attitude information of the subreflector at a distance of 5.78 m can be obtained within seconds. Moreover, the error of the translation is within 0.014 mm and the error of the rotation is within 0.37^{°}. This method can meet the pose measurement requirements of the subreflector.

#### 1. Introduction

Given the rapid development of science and technology along with a desire to understand the mystery of the universe, humans are constantly looking deeper into the universe. The world’s main space powers have invested large amounts of money and labor in space exploration. To observe the universe deeper and more clearly, large satellite antennas and large-aperture reflection antennas must be built [1–3].

In large-aperture reflection antennas, a large-caliber antenna subreflector adjustment mechanism is mounted on a support truss. During observation, it moves with the antenna’s main reflector. Because the subreflector is connected to the main reflector via a support structure, the distance from the subreflector to the main reflector is usually very long; the mechanical model can be approximated as a cantilever beam. When it is deformed by gravity, wind, and temperature, the subreflector position will change, which causes antenna pointing and phase errors and ultimately results in a significant decrease in antenna efficiency at high and low elevations. To improve the observation performance of such telescopes, subreflector compensation techniques have been widely used in recent years. The subreflector is actively adjusted through a six degree-of-freedom (DOF) parallel mechanism to correct the position deviation, ensuring that it is at the best position relative to the antenna main plane under the day’s working condition. Moreover, it has high spatial pose stability [4, 5].

Subreflector compensation has mainly been used in the American 100 m Green Bank Radio Telescope [6], German 100 m Effelsberg Radio Telescope, Italian 64 m Sardinia Radio Telescope (SRT), and the National Aeronautics and Space Administration’s Deep Space Net series of antennas [7]. The Chinese 13.7 m Delingha Millimeter-wave Telescope [8] and 65 m Shanghai Radio Telescope also use movable subreflector compensation technology [9]. The highly accurate position control of the antenna is one of the key ways to ensure the high pointing accuracy of the radio telescope antenna, and the accurate measurement of the position of the subreflector is of great importance for its position control.

To improve the adjustment accuracy of the subreflector position under all working conditions, the measurement should be conducted in real time. At present, laser tracker, radio holography, photogrammetry-based, and other mature techniques are adopted to measure the position of the subreflector. These measurement schemes are summarized in Table 1.

In the 65 m radio telescope of the Shanghai Astronomical Observatory (TM 65 m), Hou et al. [10] used a laser tracer to calibrate the 80 m adjustment mechanism to determine the position of the subreflector, as shown in Figure 1. After calibration, the position accuracy was 0.108 mm, the posture accuracy was 9.72, and the highest position and posture errors decreased by 87.75% and 46%, respectively.

Wang et al. [11] used very long baseline interferometry based on the Chinese She Shan 25 m antenna and TM 65 m antenna to obtain the amplitude and phase of the caliber plane while the TM 65 m antenna scanned the neighborhood around 3C84. Then, the subreflector orientation was separated from the obtained aperture plane phase.

As shown in Figure 2, the 64 m SRT adopts a multiple analysis measurement system to determine the subreflector position, in which the laser measurement system can measure the position changes in four DOFs of the subreflector, with a measurement accuracy of ±50 *μ*m and 1″ [12, 13].

Target attitude measurement techniques are used in industrial applications in other fields. For instance, Liu et al. designed a noncontact optical measurement system [14, 15] that can simultaneously measure the geometric error of six DOFs of the rotation axis of a machine tool. The method can measure six DOFs at a distance of more than 200 mm, and the measurement accuracy can reach micron and arc-second level. Laser triangulation [16] and optical frequency comb precision measurement [17] can often achieve micron-level measurement accuracy at relatively short distances, but it is difficult to achieve high-precision measurements in harsh environments with a distance of tens of meter. However, the development of these techniques is still very informative for measurement methods for the antennas.

The QiTai 110 m Radio Telescope (QTT), currently under construction in China, is fully steerable [18]. The large-aperture antenna adopts an adjustable Gregorian optical system design, and it will become one of the largest and most advanced all-round rotatable radio telescopes in the world. During the movement of the antenna, the subreflector position will be severely affected by the environmental load. However, movable subreflector active compensation to improve the work efficiency of the antenna system is also a key part in the QTT project. This requires improving the adjustment accuracy of the subreflector under all working conditions, and its position needs to be detected in real time and with high-precision.

The existing measurement scheme does not meet the requirements of QTT position measurement. The photogrammetry-based method is used to calibrate the position of the system in the antenna construction stage because of the slow speed of image acquisition and calculation; hence, it cannot be used during the antenna’s operation. In addition, a laser tracker can only be used in certain measurement environments. It is not suitable for long-term continuous measurement, and hence it cannot be used for the real-time measurement of the large-caliber antenna. Using cooperative targets over the long-term also reduces the accuracy because of the influence of the external environment. The measurement system used in the joint test scheme on the SRT can measure only five DOFs, that is not sufficient. Therefore, for the high-precision measurement of the QTT subreflector position, a new rapid measurement scheme needs to be developed.

This paper presents a triangulation technique using laser array coverage based on a position sensitive detector (PSD). The main contents of this paper are as follows: in the second section, the measurement scheme using a laser array to cover the subreflector is introduced. The third section describes how the least squares method is used to fit the spatial straight line of the laser beam and the distance of the PSD two-dimensional laser spot triangular centroid is used to obtain the three-dimensional spatial coordinates of the spot centroid. The 6-DOF attitude information of the antenna subreflector is obtained by coordinate conversion through the model. The fourth section verifies the feasibility and accuracy of this scheme through experiments. The final section concludes the paper.

#### 2. Measurement Scheme

To detect the deformation of the QTT subreflector induced by factors in the environment such as wind, heat, and gravity in order to use the movable subreflector active compensation technique, it is necessary to accurately and quickly measure the position and attitude of the subreflector. QTT is a Gregorian antenna. The feed cabin is fixed relative to the origin of the central position of the main reflector. The subreflector is fixed by the support frame, which is supported by a driving mechanism for pose adjustment. The aperture of the subreflector is smaller than that of the antenna system, especially the main reflector, and hence the subreflector can be regarded as a rigid body when measuring the subreflector posture. Moreover, its posture can be represented by the posture of the PSD installed on the subreflector. In the method proposed in this paper, the laser array group is placed at the center of the main reflector, and PSD is installed to receive the laser beam in order to form a spot array on the subreflector and ensure that the laser beam emitted by the laser array group can cover the PSD, which moves with the subreflector, as shown in Figure 3. This scheme of laser coverage is suitable for a wide range of subreflector attitude change measurements.

By analyzing the two-dimensional position change of multiple light spots on the PSD, we can quickly obtain the three-dimensional coordinates of the spatial position of the subreflector relative to the center of the main reflector, realizing the closed-loop control of the position and attitude of the antenna subreflector. The deformation of the antenna subreflector caused by gravity, sunshine, and wind load can be corrected in real time, realizing the active compensation of the system.

After the laser module has adjusted its pose, it forms an emission array aimed at the PSD on the subreflector. The spatial information of the outgoing laser beam close to the subreflector measurement section is calibrated in advance to ensure that the laser beam array formed by multiple sets of lasers can cover the motion range of the PSD with the subreflector and the PSD can always observe enough laser spots, as shown in Figure 4.

A group of laser beams are emitted from the laser array, of which at least three laser beams are irradiated onto the PSD. This enables the linear equation of the space laser where the three light spots on the PSD are located for identification. The current operation of the antenna attitude change shows that the attitude change of the antenna is mainly due to position error during pitch motion [19], and hence the beams can be identified by the movement of the three light spots on the shading plate. Then, the three-dimensional space coordinates of the light spots are calculated according to the distances between them, as captured by the PSD, and the pose of the secondary surface can be obtained, as shown in Figures 5 and 6. The test workflow of the laser measurement system is shown in Figure 7.

#### 3. Measurement Principle

In the method proposed in this paper, the motion range of the measured target point is covered by the laser array, and the attitude information of the point target is obtained using the principle of a spatial geometric coordinate transformation of the location where the spatial straight line intersects the target surface of the detector.

#### 4. Line Fitting of the Laser Beam

The three-dimensional information of the discrete points of the laser spot is obtained from the premeasured PSD translation information, the two-dimensional spot information is obtained by the actual measurement, and then the three spatial straight lines of the emitted laser beam are fitted by the principle of least squares [20].

The spatial linear equation can be expressed as follows:

Here, denotes the three-dimensional coordinates of the centroid space of the spot formed by the intersection of a laser beam and the PSD; are linear equation constants.

Equation (1) can also be expressed as follows:

Then, the coordinates of *N* discrete points are expressed as follows:

Finally, the following three linear equations of the emitted beam of the laser group are obtained:

Here, denotes the spatial coordinates of the centroid of the scattered light spot on the -th laser beam and are the constant parameters of the linear equation, where .

#### 5. Solving for Intersecting Spot Centroid Coordinates

The two-dimensional coordinates of the centroid of the detector spot before the subreflector pose change are defined as , , and .

The three-dimensional coordinates of the centroid of the spot before the subreflector pose change are defined as , , and .

The two-dimensional coordinates of the centroid of the detector spot after the subreflector pose change are defined as , , and .

Finally, the three-dimensional coordinates of the centroid of the spot after the subreflector pose change are defined as , , and .

Because the space line segments are equal in length, we have the following:

Solving Equations (6)–(11), the three-dimensional coordinates of the PSD intersection points before the pose change of the subreflector can be obtained from the following intersection point coordinates on the PSD: , , and .

Similarly, the spatial three-dimensional coordinates of the PSD intersection point after the subreflector pose change can also be obtained as , , and .

#### 6. PSD Pose Acquisition with Six DOFs

In this paper, the Bursa model [21] is employed to analyze the 7-parameter transformation from the two-dimensional coordinates of the PSD spot centroid to the three-dimensional coordinates of the antenna system space coordinate system. The translation and rotation data of the PSD reflect the 6-DOF attitude change information.

As shown in Figure 8, the two-dimensional coordinate points of the spot centroid on the PSD can be converted into three-dimensional spatial coordinate points under the antenna spatial coordinate system through translation and rotation as follows:

Here, denotes the three-dimensional coordinates of the light spot in the antenna coordinate system; is the translation vector ; is the scale change coefficient; is the rotation matrix; and denotes the coordinates of the light spot in the PSD coordinate system, where *Z* = 0.

Equation (12) can be transformed into the following:

The centroid coordinates of the three light spots in the spatial coordinate system of the antenna system are defined as . The coordinates of the corresponding three points in the PSD coordinate system are defined as . Then, equation (14) can be further expressed as follows:

Using the least squares method, the transformation matrix of multiple groups of points in different coordinate systems can be solved. The solution is defined as follows:

By solving the 6-DOF coordinate change in PSD in space, the change in the subreflector pose can finally be obtained.

#### 7. Simulation Example

To evaluate the measurement principle, a simulation of an example was performed. Three emission lasers were set to irradiate on a PSD target surface 6.05 m away. The PSD attitude was given a predetermined offset, and the change in attitude was calculated using the method proposed in the previous section, and the results were compared with the predetermined offset.

The initial conditions are shown in Table 2.

Using the spatial coordinates of two points of any laser beam, three laser line equations can be obtained. Then, the two-dimensional coordinates of the three light spots can be obtained directly by the PSD and the three-dimensional coordinates of three light spots can be obtained by Equations (9)–(11). Finally, the bursa model is established, and the change in attitude of the PSD is obtained by Equation (17). The results are shown in Table 3. A schematic diagram of interaction between the PSD and laser beams before and after the attitude changes is shown in Figure 9.

The results in Table 3 show that the proposed method in the simulation example obtains the three-dimensional coordinates of spatial laser spots with high accuracy and obtains the final 6-DOF change in attitude of the PSD with high accuracy.

#### 8. Experiment

To further evaluate the measurement scheme, a prototype measurement system was built to simulate the attitude adjustment of the antenna. It mainly consists of a laser emission module array, PSD system, and two-dimensional translation platform. The laser module array is placed on the aluminum support frame, and the PSD system is placed on the two-dimensional movable table. The experimental equipment settings are shown in Figures 10 and 11. The main equipment parameters are shown in Tables 4 and 5.

The 41 groups of data obtained from the experiment are shown in Figure 12, and the corresponding 41 groups of translation in *X* direction and *Z* direction are shown in Figure 13.

The first 31 sets of data were used to fit the laser beam straight lines as shown in Figure 14, and the last 10 sets of data were used to test and evaluate the system.

The first 31 sets of data obtained were processed using Equations (5)–(8), and the three space linear equations were fitted as follows:

Taking the two-dimensional information obtained from the last 10 groups data as variables, Equations (18)–(20) and (9)–(11) were used to solve the space centroid coordinates of the spot where the PSD and lasers intersected. The results are shown in Figure 15. The selected high-precision two-dimensional translation table is equipped with a high-precision RH100 × 30D05 A grating ruler (Renishaw, UK), with an accuracy of 1 *μ*m. Therefore, the data from the motion of the two-dimensional translation platform can be used as the actual translation of the PSD, and the measurement accuracy can be obtained by comparing it with the solved change in attitude. Finally, the 6-DOF change in the subreflector position was obtained from Equations (15)–(17) as shown in Figures 16 and 17.

The change in PSD attitude caused by the two-dimensional translation table consisted of translation in the *X* and *Z* directions without rotation. The solved maximum translation root mean squared (RMS) error was 0.014 mm, and the maximum rotation RMS error was 0.370°, as shown in Figures 18 and 19.

In this experiment, the two-dimensional coordinate output operating frequency of PSD was 10 Hz, and the runtime of the measurement system’s algorithm was on the order of milliseconds, and hence the time needed for system measurement can be guaranteed to be within several seconds. The results of this verification experiment prove that the measurement scheme proposed in this paper is feasible and can be used for rapid and high-precision subreflector position measurement.

#### 9. Conclusion

This paper focused on the fast and high-precision pose measurements of a large-caliber antenna subreflector and proposed a PSD-based triangulation technique for laser array coverage. The precalibrated laser array is mounted on the subreflector so that the PSD always observes at least three different laser beams and at least three complete laser spots are formed. The PSD can rapidly acquire the two-dimensional centroid coordinates of the light spots by solving the spot centroid space coordinates using the three two-dimensional point distances and the laser beam line equation. Then, the coordinate transformation matrix can be obtained by the bursa model. Finally, the 3-DOF translation information and 3-DOF rotation information of the PSD can be obtained. Experimental results showed that a translation accuracy of 0.014 mm and a rotation accuracy of 0.37° can be achieved. In the future, we will devote ourselves to using detector array for research to further improve the measurement accuracy. Moreover, the proposed method can also meet the requirements of quasi-real-time subreflector position measurement. Using this technique, the antenna subreflector can be moved to the optimal position to improve antenna electrical performance and observation efficiency. This will be beneficial for obtaining high quality observation data and help humans explore the mystery of the universe.

#### 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.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest.

#### Acknowledgments

This study was supported by National Key Research and Development Program of China (grant no. 2021YFC2203501) and National Natural Science Foundation of China (NSFC) (grant no. 11803075).