Measuring Antenna Elevation Mechanism Pointing Errors with Multiencoder Information Sources

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Introduction
For large aperture radio telescopes, accurate pointing is a fundamental requirement for the high-precision measurement of astrophysical targets. At present, the pointing accuracies achieved by large aperture radio telescopes in various countries are at the arcsecond level. Te 110 m radio telescope at Xinjiang [1] under construction has a design requirement of 5″ (blind pointing) for pointing accuracy, the SHAO 65 m radio telescope [2] has a pointing accuracy better than 3″ (after observing a local calibrator), and the 100 m radio telescope at Efelsberg [3] has a pointing accuracy of 10″ (blind pointing), and the Green Bank Telescope [4,5] has a pointing accuracy of 9″ (blind pointing). When higher pointing accuracy is required for radio telescopes to achieve their science goals, it also means that more error factors afecting the pointing accuracy need to be considered, including displacement of the secondary refector, azimuth track fatness, gravity loading, wind loading, and nonuniform temperature felds.
At present, the research on error factors that afect the accuracy of pointing is mainly classifed into two categories. One is the widely used software correction method, which incorporates the major error factors that afect the accuracy of pointing into the antenna pointing correction model to complete the error analysis and correction of the antenna. Kong et al. proposed a model for the pointing calibration error of the radio telescope, considering the azimuth nonlinear error, which improved the pointing accuracy by 43.6% compared to the linear model [6]. Huang et al. proposed a semiparametric regression model to improve the pointing accuracy of the telescope, and the parameters of the kinematic model were estimated by ftting the statistical model to eliminate geometric and nonlinear errors, which signifcantly improved the pointing accuracy of the telescope after calibration [7]. Te other category starts from each subfactor, afecting the pointing error, and only considers the efects of one or two factors on the antenna pointing error. Chen et al. introduced the axial displacement of the secondary refector of the telescope to the radio pointing model for SHARC II. Te results show that the reconstructed radio pointing model can improve the accuracy of estimating the pointing error [8]. Gawronski Gawronski et al. show that the pointing error in the elevation direction is reduced from 16.2″ to 5″, and the pointing error in the azimuth direction is reduced from 52.2″ to 11.2″ [9,10]. Bufa et al. and Nikolic et al. analyzed the efect of gravitational deformation of the telescope's main refecting surface on the pointing error for the SRT and the GBT, comparing the FEA gravitational deformation results with photogrammetry and updating the FEA model [11,12]. Te efect of wind loading on antenna pointing error was investigated by Hashimoto et al., who measured the dynamic characteristics of the Nobeyama 45 m radio telescope using a piezoelectric accelerometer under wind loading and determined the response frequency and vibration mode of the main refecting surface [13]. Cho et al. calculated the wind loading data from the Gemini South telescope and analyzed that the average pressure pattern on the Gemini mirror is primarily caused by the interaction of the airfow with the telescope structure [14]. Ukita used the Nobeyama 45 m telescope as a research object and analyzed thermal efects on the antenna pointing error and pointed out that the diference in pointing accuracy between night and day was still 2″ after correction [15]. Fan et al., after analyzing the insolation temperature feld of the Five-Hundred-Meter Aperture Spherical Telescope (FAST) in China, pointed out that the deformation due to thermal efects reached 2 mm, considering the breeze environment, accounting for 40% of the total accuracy requirement [16].
For the method of using software to analyze and correct antenna pointing error, due to the wide variety of error factors and the coupling problem between the error factors, it will be difcult to design an efective correction model. Regarding the infuence of only one or two factors on the accuracy of antenna pointing, few scholars have carried out relevant research specifcally on the impact of antenna pointing accuracy caused by error in the elevation mechanism. As one of the two main antenna positioning mechanisms, the impact of the error in the elevation mechanism on the accuracy of the antenna pointing cannot be ignored. In a study of the shaft angle measurement error of the 50 m radio telescope at the MiYun Station of the National Astronomical Observatory of China, Kong et al. showed that there is a very strong correlation between the shaft angle measurement error of the antenna elevation shaft and the insolation temperature gradient, and the correlation coefcient reaches 76%∼99% [17]. Zheng et al. pointed out in the analysis of the pointing error of the radio telescope that in terms of the shafting error, there are driving system errors and shaft angle transmission errors [18].
Te elevation mechanism of the radio telescope is mainly composed of transmission components and support structures. Among them, the key transmission components include the elevation gear, the driving gear, and the motor. Te key support structure includes the elevation shaft and the support bearings located at both ends of the elevation shaft. Similar to the mechanism errors in the feld of electromechanical transmission, the errors caused by the backlash error when the elevation gear meshes with the driving gear and the errors caused by the centerline deviation of the bearings at both ends of the elevation shaft are the main sources of errors in the elevation mechanism. Fernandez-Del-Rincon et al. combined the dynamic characteristics of rolling bearings and gears to propose an improved model, which states that the motion of both gears and bearings involves nonlinearity and will provide diferent resonant frequencies depending on the transmitted load [19]. Liu et al. investigated the dynamic response of the gear pair system and the interaction relationship between the bearing clearance and the backlash, and the results showed that the bearing clearance is adaptive and that gear separation and gear back meshing are suppressed as the transmitted torque increases [20]. Due to the low speed, heavy load, and highprecision operating conditions of the antenna elevation mechanism, the error correction methods in the feld of electromechanical transmission are not fully applicable to the antenna elevation mechanism.
Tere is little existing research on the efect of the elevation mechanism on telescope pointing accuracy. Terefore, aiming at the factors afecting the pointing errors of radio telescopes, this study proposes a method for measuring the error of the elevation mechanism based on multiencoder fusion, builds a scaled-down experimental platform for the elevation mechanism, and describes the measurement principle and measurement process of the elevation mechanism error in detail. Multiencoder information sources allow us to more rigorously measure the errors of the antenna elevation mechanism under diferent driving modes, and they also accurately measure the transmission errors caused by the backlash of the elevation mechanism and the centerline deviation of the bearings. Terefore, multiencoder readings and analyses will be useful in improving the correction of pointing errors in the elevation mechanism for large radio telescopes.

Analyzing Errors and Building an Experiment Platform
According to the actual working conditions and the working principle of the antenna elevation mechanism, the error of the antenna elevation mechanism is analyzed, and the error measurement principle is briefy described. Based on this principle, a scaled-down experiment platform of the antenna elevation mechanism is built, and data acquisition with multiencoders is performed.

Te Principle of Error Measurement.
When the antenna performs the elevation action, there is a rotational error between the actual elevation angle of the elevation gear and its theoretical elevation angle. Te error is a comprehensive error, related to both the backlashes between the elevation gear and the driving gear and the centerline deviation of the bearings at both ends of the elevation shaft. If the elevation shaft is driven by a single motor to perform the antenna elevation action, a transmission error will occur due to the presence of backlash. When the elevation shaft is driven by two motors, each performing the elevation action of the antenna, it serves to eliminate the backlash by mechanical means. If the bearings at both ends of the elevation shaft are installed or run for a long time, there will be a transmission error due to the centerline deviation of the bearings. In order to accurately measure the error caused by the backlashes between the elevation gear and the driving gear, as well as the error caused by the centerline deviation of the bearings at both ends of the elevation shaft, in this study, we utilized a multiencoder information source to measure the real-time position of three key parts of the elevation mechanism: the servo motor, the elevation gear, and the elevation shaft. Te measurement principle is shown in Figure 1.
In Figure 1, the elevation shaft passes through the elevation gear and its front and rear ends are mounted, respectively, on the position adjusting device and the bracket through bearings. Te position adjusting device has three degrees of freedom, which can adjust the rotation angle θ along the Z axis, the displacement z 1 along the Z axis, and the displacement y 1 along the Y axis of the front end of the elevation shaft. Te reduction ratio between the elevation gear and the driving gear is 198:38. Te elevation angle of the antenna ranges from 5°to 90°, and the rotation angle of the driving gear ranges from 0°to 443°. When the elevation mechanism performs the elevation action, the encoders located at the ends of the left-and right-hand servo motors output the pulse signal B and the pulse signal A, respectively, to the motion controller. Te encoder at the front of the elevation shaft measures the angle of rotation in this position and outputs the pulse signal C to the motion controller. Te readhead located below the magnetic scale outputs the pulse signal D to the motion controller. By comparing pulse signal A with pulse signal D, the error caused by the backlashes between the driving gear and the elevation gear can be measured. By comparing pulse signal C with pulse signal D, the error caused by the centerline deviation of the bearings at both ends of the elevation shaft can be measured.

Error Analysis of Encoders.
Te measurement terminals used in this study include the encoder that comes with the servo motor, the encoder connected to the end of the elevation shaft, and the magnetic encoder system. Te precise installation of these measurement terminals is the basis for ensuring the accuracy of the measurement results. Terefore, it is necessary to analyze the encoder's own errors and reduce their interference with the measurement results.
Te eccentricity error of the magnetic scale in the magnetic encoder system will seriously afect its accuracy. Tis eccentricity error is caused when the geometric center of the magnetic scale does not coincide with the actual center of rotation during installation, as shown in Figure 2.
Te installation radius of the magnetic scale is R, O is the center of the magnetic scale, O′ is the center of the magnetic scale after eccentricity, and the eccentricity is e. Assuming that the actual rotation angle and the theoretical rotation angle of the magnetic scale are β 1 and β, respectively, equations (1) and (2) can be obtained according to the geometric relationship in Figure 2.
When Δβ is sufciently small, tan α ≈ α. Since R is much larger than ecos β, equation (2) is simplifed to obtain the following equation: Based on the working principle of an encoder, we know that encoders continuously process the signal sampled by the front end through the microprocessor. However, the signal is not synchronized in time from the time it is sampled to the time it enters the microprocessor for position angle calculation. Tis dynamic angle error due to system delay is easy to ignore among many error factors. Assuming that the time delay from when the signal is sampled to when the microcontroller starts the corresponding solution is Δt, and the rotational angular velocity of the elevation shaft is ω, the dynamic delay error can be expressed by the following equation within the delay time Δt.
(4) Figure 3 shows the relationship between the dynamic delay error and the rotational angular velocity for diferent delay times [21]. It can be seen that the dynamic delay error is proportional to the rotational angular velocity. In highspeed rotation, the error caused by the dynamic delay angle has a great impact on the accuracy of the encoder.
On the basis of the analysis of the encoder's own errors and the parameters of the experimental platform, the maximum value of the eccentricity error of the magnetic scale installation and the dynamic delay error of the encoder are calculated. In the experimental platform, the maximum speed of the elevation shaft is about 1°/s, and the installation radius of the magnetic scale is 71.5 mm. If the delay time is 0.6 × 10 −3 s, then the encoder dynamic delay error can be calculated as 2.06″ according to equation (4) and Figure 3. If the eccentricity between the geometric center and the center of rotation of the magnetic scale is 10 μm, the maximum Advances in Astronomy value of the eccentricity error of the magnetic scale installation is 0.5″ according to equation (3). It can be seen that the errors caused by the encoder itself are on the order of angular seconds or even subangular seconds.

Building Experiment Platform.
After analyzing the measurement method and the principle of antenna elevation mechanism error, it is also necessary to design the scaleddown experiment platform for error measurement according to its operating environment and actual working conditions. In actual antenna working conditions, the elevation gear is driven by the dual-motor, and the error caused by the backlash is generally removed by both mechanical and electrical backlash elimination methods. After long-term operation and wear of the elevation gear, the error caused by the backlash cannot be eliminated by electrical means, so it is necessary to measure the law of change of the mechanical backlash error. Furthermore, the measurement of the angle of elevation during antenna operation can only be performed by the encoder at the end of the elevation shaft, and the measured value of the encoder has been mixed with various error factors after passing through multiple drive chains. Terefore, it is necessary to measure and compare the errors of each transmission chain of the elevation shaft and to determine their law of change by multiencoder information sources.
Based on the analysis presented above, this study builds the scaled-down experimental platform for measuring antenna elevation mechanism error, as shown in Figure 4.

Advances in Astronomy
Te operating principle of the experimental platform in Figure 4 is the same as the error measurement principle described in Subsection 2.1.

Experiments and Results
According to the analysis of the antenna elevation mechanism error in the previous section, the real-time positions of diferent parts of the elevation mechanism are measured and recorded by using a multiencoder to collect pulse signals from multiple positions simultaneously. Measurements and recordings are divided into four cases: elevation action driven by a single motor, elevation action driven by two motors, elevation action under diferent centerline deviations of the bearings, and elevation action under diferent backlashes.

Te Elevation Action Driven by the Single Motor.
When the elevation gear is driven by a single motor, the elevation shaft rotates from 5°to 90°at a speed of 1°/s. Referring to Figure 1, it can be seen that signal D-signal A represents the backlash error between the elevation gear and the driving gear, and signal D-signal C represents the error caused by the centerline deviation of the bearings at both ends of the elevation shaft. As can be seen in Figure 5 that when the elevation shaft rotates from 5°to 90°, that is, the antenna is pitched up, the transmission error caused by the backlash error and the centerline deviation of the bearings continues to increase, and the backlash error increases faster. When the elevation shaft rotates to 90°, the backlash error reaches 0.4°.
In order to further observe the law of change of the two errors in the complete elevation action of the antenna, we then rotated the elevation shaft from 5°to 90°and then from 90°to 5°at a speed of 1°/s. Te rotation of the elevation shaft is suspended at 90°for 1 second to eliminate interference from rotational inertia. Te error results are shown in Figure 6. It can be seen from the fgure that the transmission error caused by the backlash error and the centerline deviation of the bearings decreases with the pitched down action of the antenna. Both errors are close to zero when the elevation shaft rotates back to 5°.
Combining Figures 5 and 6, it can be seen that when the elevation shaft is driven by a single motor, the transmission error caused by the backlash error and the centerline deviation of the bearings will gradually increase when the antenna is elevated during a complete elevation cycle. Te maximum is reached when the antenna is rotated to 90°, which are 0.4°and 0.063°, respectively. When the antenna is pitched down, the two kinds of error will gradually decrease; the errors approach zero when the antenna is rotated to 5°. Tis error law of change is a nonlinear elastic error.

Te Elevation Action Driven by the Dual-Motor.
To compare the error law of change when driven by the single motor and the dual motor, as well as the efect of the dual motor on the elimination of mechanical backlash, a motor is Advances in Astronomy 5 installed on each side of the elevation gear, respectively, as shown in Figure 7. Te right motor drives the elevation gear from 5°to 90°at a speed of 1°/s, and the left motor drives the elevation gear from 90°to 5°at a speed of 1°/s. Tat is, the right motor drives the antenna pitched up, and the left motor drives the antenna pitched down. Tis process can ensure that the elevation gear always engages with the driving gear. When the dual-motor drives the elevation gear, the error results are shown in Figure 8. It can be seen from the fgure that the transmission error caused by the backlash error and the centerline deviation of the bearings gradually increases as the antenna's elevation increases and reaches the maximum value of 0.128°and 0.062°, respectively, when the antenna rotates to 90°. Te two kinds of error gradually decrease as the elevation gear rotates back. Te change trend and law of the two types of error are similar to those of the elevation gear driven by a single motor, and they are both nonlinear elastic errors.
Comparing Figures 6 and 8, it can be seen that when the antenna performs the elevation action, the error caused by the backlash is signifcantly reduced by adopting the dualmotor driving method. It shows that the dual-motor driving method has a good mechanical backlash elimination efect. However, it can also be seen that the dual-motor driving method does not have a signifcant efect on the error caused by the centerline deviation of the bearings.

Te Elevation Action under Diferent Centerline Deviations of the Bearings.
When the antenna is installed, or after a period of operation, the centerline of the bearings at both ends of the elevation shaft will deviate. Terefore, it is necessary to measure the transmission error caused by the centerline deviation of the bearings and analyze the error law of the change at diferent degrees. Te centerline deviation of the bearings can occur randomly in the radial direction of the bearings. In this study, the centerline deviation of the bearings along the Y axis is taken as an example. Before installing the elevation gear, we used a dial indicator to measure along the axis of the elevation shaft. At the same time, we continuously adjust the Z axis of the position adjusting device so that the heights of both ends of the elevation shaft in the direction of the Z axis are consistent, as shown in Figure 9.
After adjusting the heights of both ends of the elevation shaft along the direction of the Z axis, it is also necessary to adjust its position along the direction of the Y axis. Again, we use a dial indicator to measure along the axial direction of the elevation shaft and continue to adjust the Y axis of the position adjusting device so that the position of both ends of the elevation shaft along the direction of the Y axis is consistent, as shown in Figure 10. After adjusting the positions of both ends of the elevation shaft along the directions of the Z axis and the Y axis, we use this as an initial reference. By changing the y 1 of the position adjusting device along the direction of the Y axis and the rotation angle θ around the center, the centerline deviation of the bearings increases by 0.5 mm. Te values of y 1 and θ are given in Table 1. Figure 10, since the right end of the elevation shaft is fxed, when the value of y 1 increases, the elevation shaft will be slightly deformed due to the bending moment. In this case, it is necessary to fne-tune the rotation angle θ around the center of the position adjustment device to eliminate the impact of deformation at both ends of the elevation shaft as much as possible. According to the geometric relationship in Figure 10, it can be obtained as follows:

Referring to
In equation (5), b is the center distance of the bearings at both ends of the elevation shaft.
In the experimental platform, the bearing center distance at both ends of the elevation shaft is 150 mm, so the relationship between the rotation angle θ of the center of the position adjusting device and y 1 is shown in Table 1: Te right motor was controlled to drive the elevation shaft rotated from 5°to 90°and then rotated back to 5°at a speed of 1°/s. Transmission error was measured under the four centerline deviations of the bearings of y 1 � 0 mm, y 1 � 0.5 mm, y 1 � 1.0 mm, and y 1 � 1.5 mm, respectively, and the error results are shown in Figure 11. It can be seen in Figure 11 that diferent centerline deviations of the bearings will cause diferent transmission errors, and the law of change of transmission errors still belongs to nonlinear elastic errors. Furthermore, by comparing Figures 11(a)-11(d), it can be seen that with increasing y 1 , the maximum value of transmission error caused by centerline deviation of the bearings at both ends of the elevation shaft also increases at a constant rate. In Figure 11(d), when the centerline deviation of the bearings at both ends of the elevation shaft is 1.5 mm, the maximum transmission error caused by it reaches 0.257°in one complete elevation cycle of the antenna.

Te Elevation Action under Diferent Backlashes.
In Subsection 3.1, when analyzing the elevation action driven by the single-motor, it is found that there is a certain law in the change of the backlash error when the driving gear meshes with the elevation gear. Te change of this error is further analyzed under diferent backlashes.
Te variation of backlash in the gear transmission structure is related to the negative tolerance of the tooth thickness and the center distance, and their geometric relationship is shown in Figure 12. Referring to Figure 12, it can be seen that the size of the tooth clearance is proportional to the negative tolerance of the tooth thickness and the center distance of the gear, respectively. By adjusting the displacement y 2 of the motor base along the direction of the Y axis, the center distance between the elevation gear and the driving gear can be changed, and then the backlash can be adjusted. When the driving gear meshes with the elevation gear, the minimum backlash between them can be calculated according to equation [22] as follows: In equation (6), J min is the minimum backlash, a is the center distance, m and n are the modulus of the two gears, respectively.
In the experimental platform, the elevation gear and the driving gear parameters are shown in Table 2. It can be calculated that the center distance between the two gears a � 118 mm and the minimum backlash J min ≈ 0.1 mm. Te position of y 2 at this time is marked as 0, and the backlash of the two gears is adjusted based on this position.
In terms of geometric relationship, the minimum backlash J min between the elevation gear and the driving gear is not directly equal to the displacement y 2 of the driving gear along the direction of the Y axis. As shown in Figure 13, there is the following geometric relationship between J min and y 2 : In equation (7), d is the reference diameter of the elevation gear.
According to the geometric relationship between J min and y 2 in Figure 13 and the parameters of the two gears in Table 2, the corresponding relationship between the displacement y 2 of the driving gear along the direction of the Y axis and the minimum backlash J min between the two gears can be obtained, as shown in Table 3.
Controlling the motor on the right to drive the elevation shaft from 5°to 90°and then rotating back from 90°to 5°at a speed of 1°/s, the errors of four displacements of y 2 � 0 mm, y 2 � 0.5 mm, y 2 � 1.0 mm, and y 2 � 1.5 mm were measured, respectively, and the results are shown in Figure 14.
As can be seen in Figure 14, the change in the minimum backlash J min causes the backlash error to change. By comparing Figures 14(a)-14(d), it can be seen that the backlash error when the driving gear meshes with the elevation gear increases continuously with the increase in J min .     Advances in Astronomy As shown in Figure 14(d), when y 2 is 1.5 mm, that is, J min is 1.26 mm, the backlash error between the elevation gear and the driving gear reaches the maximum value of 0.969°in a complete antenna elevation cycle.

Comparison and Analysis
In the previous section, elevation action experiments were carried out for the antenna elevation shaft with the singlemotor drive and the dual-motor drive. Under the two different drive modes, the backlash error and the transmission error caused by the centerline deviation of the bearings have diferent efects. Additionally, the elevation action under diferent centerline deviations of the bearings and diferent backlashes will also cause diferent errors. Tis section further compares and analyses these errors. Figure 15 shows the standard deviation, mean value, and maximum value of the errors caused by the backlash when the elevation shaft is driven by the single-motor and the dual-motor, respectively. In this plot, the standard deviation refects the dispersion of the error caused by the backlash over a complete elevation cycle of the antenna. As can be seen in Figure 15, when the elevation shaft driving mode changes from the single-motor drive to the dual-motor drive, the mean value of the error caused by the backlash is reduced from 0.21°to 0.06°, decreasing by 71.4%. Tis indicates that the drive of the dual-motor has a greater efect in eliminating the error caused by the backlash compared to the single-motor drive. Moreover, in terms of standard deviation, the error caused by backlash is relatively less dispersed when the elevation shaft is driven by the dual motor.

Te Error Caused by the Driving Modes.
Te standard deviation, mean value, and maximum value of the transmission error caused by the centerline deviation of the bearings are shown in Figure 16.
It can be seen in Figure 16 that the mean value of the transmission error caused by the centerline deviation of the bearings is 0.032°for both drive modes, and the standard deviation and the maximum value are also approximately equal.

Te Error Caused by the Centerline Deviation of the
Bearings. Te standard deviation, mean value, and maximum value of errors caused by the centerline of bearings at the four types of deviation values are shown in Figure 17.
It can be seen from Figure 17 that the standard deviation, mean value, and maximum value of the transmission error caused by the centerline deviation of the bearings all increase with the increase of the deviation value. When the deviation value is 0, the mean value of the transmission error caused by the centerline deviation of the bearings is 0.03°in one complete elevation cycle of the antenna. Tis average error may be caused by the slight deformation of the elevation shaft during the antenna elevation process. When the deviation value increases from 0 to 1.5 mm, the mean value of the transmission error caused by the deviation value increases from 0.03°to 0.15°, which is an increase of about 5 times.

Te Error Caused by the Backlash.
Te standard deviation, mean value, and maximum value of errors caused by the four types of backlash when the driving gear meshes with the elevation gear are shown in Figure 18.
It can be seen in Figure 18 that the standard deviation, mean value, and maximum value of the error caused by the backlash all increase as it increases. When the minimum backlash between the elevation gear and the driving gear is 0.1 mm, the mean value of the error caused by this backlash is 0.22°. When the minimum backlash increases from 0.1 mm to 1.26 mm, the average error caused by the backlash increases by 1.9 times. Tis also shows that the antenna error during the elevation process is greatly afected by the change in backlash.
Te error variation law of the elevation mechanism caused by multiple error factors measured in the experimental platform is important for improving the pointing accuracy of the radio telescope directly or indirectly.
During the pointing of the radio telescope, the motor adjusts the elevation angle by a signal from the control room, while the actual elevation angle is fed back to the control room by an encoder at the end of the elevation shaft. According to the experimental results, diferent driving modes of the elevation mechanism afect the actual elevation angle of the encoder feedback. Terefore, at the beginning of the design of the radio telescope, the elevation shaft needs to be driven by multiple motors to reduce the deviation  Figure 13: Te geometric relationship between J min and y 2 .   between the actuation angle and the feedback angle of the elevation mechanism. In addition, a more efcient way is to compensate the actuation angle of the elevation mechanism in the control signal according to the error variation law measured in the experiment.
Te centerline deviation of the bearings at both ends of the elevation shaft is generated during the installation or operation of the radio telescope. Although the direction of the centerline deviation is random, the components of this deviation in the x, y, and z directions can be measured and calculated at the radio telescope. According to the relationship between the centerline deviation of the bearings and transmission error in the experimental results, the error caused by the centerline deviation of the bearings can be corrected in the existing error correction model.
Wear between the elevation gear and the driving gear during the long-term operation of the radio telescope can reduce the fatigue life of the gear material. It also leads to the continuous expansion of the backlash, which afects the pointing accuracy of the radio telescope. According to the variation law of backlash and error in the experimental results, the reasonable arrangement and dynamic adjustment of the observation plan of the radio telescope can help reduce the wear of the transmission gear, prolong the fatigue life of the gear material, and reduce the error of the elevation mechanism.

Conclusions
Te pointing errors of large aperture radio telescopes directly afect their observation results. Measurement and analysis of the error of its various parts are an important means of improving pointing errors. Aiming at the error factors caused by the elevation shaft of the radio telescope, this study measured and analyzed the law of change of the elevation shaft rotation error under diferent driving modes, diferent centerline deviations of the bearings, and diferent backlashes.
When the driving mode of the elevation shaft was changed from single-motor drive to dual-motor drive, the maximum value of the error caused by the backlash between the elevation gear and the driving gear was reduced from 0.4°t o 0.13°, and the mean value was reduced from 0.21°to 0.06°. Tis shows that the driving mode of the dual-motor can eliminate about 70% of the mechanism error caused by the backlash. However, the change in driving mode will hardly afect the error caused by the centerline deviation of the bearings at both ends of the elevation shaft. When the centerline of the bearings at both ends of the elevation shaft has diferent deviation values, it will afect the transmission error of the elevation shaft. Experiments show that when the deviation value increases from 0 to 1.5 mm, the mean value of the transmission error caused by the deviation value increases from 0.03°to 0.15°, which is an increase of about 5 times.
When the minimum backlash between the elevation gear and the driving gear increased from 0.1 mm to 1.26 mm, the mean value of the error caused by this backlash increased by a factor of 1.9.
From the results of measurement and analysis, factors such as the driving mode of the elevation shaft, the backlash, and the centerline deviation of the bearings will greatly afect the rotation error of the elevation shaft. Terefore, in the design of radio telescopes, the way multiple motors drive the elevation shaft should be considered. Te measurement results can also be applied to the control program and error correction model to obtain a more accurate elevation angle. It is also possible to adjust the observation time and observation plan dynamically in future observations, so as to ensure the long-term stable pointing accuracy of the radio telescope. Te research results have important reference values for the correction of the pointing errors of the existing Nan Shan 26 m radio telescope and the Qi Tai 110 m radio telescope to be built.

Data Availability
Te data that support the fndings of this study are available from the corresponding author (xuqian@xao.ac.cn) upon reasonable request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.