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The pointing error caused by the structural deformation of large reflector antennas has become the most challenging problem of antenna servo control. Especially with the increase of the antenna diameter and the working frequency, environmental loads not only make the structure deformation more obvious, but also make the pointing accuracy influenced by deformation more sensitive. In order to solve this problem, accurately estimating the pointing error caused by the structural deformation is the key. Based on the dynamic model of antenna structure and the analyzing model of pointing error, using the displacement information of sampling points on reflector, this paper proposes a correction method to achieve the purpose of accurately estimating the pointing error caused by the structural deformation. Using a 7.3-m Ka band antenna, the results show that the antenna maximum pointing error in theoretical model calculation is 0.0041° at 10m/s wind speed condition; however, the corrected pointing error would be about 0.0054° with considering the modeling error. After compensating the controller, the pointing error could be reduced to only 0.0008° and the performance of antenna pointing was improved.

The pointing accuracy is one of the most important technical performance indicators of radio telescope which is the main instrument to receive and transmit electromagnetic waves. Qitai 110m radio telescope (QTT), which will be built in Xinjiang, China, has pointing accuracy required to reach 0.0004°. However, due to the environmental load, the pointing error would be different as follows. It is about 0.011° caused by the deformation from the gravity distribution under different conditions. It is about 0.004° caused by the deformation from temperature gradient. And, it is up to 0.024° caused by the deformation from random wind disturbance [

The study of pointing errors caused by structural deformation under different conditions such as temperature and gravity distribution is relatively mature, because the change of the loads is slow. The laser measurement techniques were used to obtain the root mean square error (RMSE) of the antenna reflector deformation at different pitch angles and estimate the effect of gravity on the deformation of the structure [

Because of slow change and high repeatability of temperature and gravity effects, it is possible to set up the error table offline whether using finite element software analysis or actual sensors measurement. Thereby, it could be overcome by look-up table method in the process of antenna servo control. However, as a random transient load which is more difficult to analyze and inhibit, the wind disturbance has also received widespread attention.

The servo error data produced by the wind disturbance of specific antenna location was collected, and the equivalent torque coefficient of wind disturbance at equivalent disturbance torque was derived, which played a guiding role in the wind disturbance simulation in other antenna design [

In recent years, although the pointing error of the structural deformation caused by wind disturbance has gradually received attention. The analysis method could be either obtaining structural deformation by the finite element software simulation to estimate the pointing error, however, it cannot be directly applied to the controller implementation, or establishing pointing error analysis model based on the modal superposition method and the approximate optical method, but this method depends on the accuracy of the model which is hard to guarantee for the load model error from the complicate wind pressure distribution on the antenna reflecting surface caused by the turbulent characteristics of the wind field and the dynamic model error. It often brings a burden to the design of the controller.

This paper presents a correction method of estimating the pointing error caused by flexible deformation of antennas. The flow chart of this algorithm is shown as Figure

The flow chart of the correction method.

To analyze the pointing error caused by the environmental load, the first step is to obtain the deformation of the antenna structure under the environmental load. As shown in Figure

The structure of dual reflector antenna.

The dynamic model of antenna structure under generalized coordinates can be expressed as follows [

To express the structure in modal coordinates, the modal displacement is introduced, which satisfies the following equations:

The modal mass matrix and modal stiffness matrix satisfy the following equation, where

The modal damping ratio matrix,

To dual reflector antenna, the factors influencing the performances of antenna pointing include the deformation of the main reflector, the lateral displacement (perpendicular to the focal axis) of the feed, the lateral displacement, and the rotation of the subreflector, as shown in Figure

The factors influencing the performances of antenna pointing.

When analyzing the influence of the deformed main reflector on pointing error, the best fitting paraboloid should be obtained based on the displacement of each node of the reflector. There are six key parameters to confirm the best fitting paraboloid. They are the displacement of the vertex of the fitted parabolic reflector,

The relationship among these parameters and the displacement of each node could be expressed as follows [

where

As shown in Figure

The pointing error caused by reflector deformation.

The distance between B′ and the focal axis of the subreflector is

where

During the estimation of the pointing error

From (

where

where

where

where

To simplify the pointing error expression, omitting the effect of the high-order term as (

where

The H_{2} norm of each transfer function could be derived, respectively:

where

When the node displacements,

Taking the antenna with a diameter of 7.3 meters as an example, the antenna and its finite element model are shown in Figure

The real and finite element model of 7.3m antenna.

In order to verify the accuracy of the finite element model, a frequency test experiment and a load deformation experiment are made to the antenna.

A load deformation experiment is applied by the 100Kg impulse in lateral loading and vertical loading, respectively. As shown in Figure

The comparison of static maximum deformation and the pointing error.

Loading | EL angle | Simulation results | Simulation results | Test results of deformation | Test results of | Relative error |
---|---|---|---|---|---|---|

Lateral | 70 | 0.80 | 2.88 | 0.68 | 2.41 | 19.50% |

50 | 0.66 | 1.80 | 0.55 | 1.52 | 18.42% | |

30 | 0.41 | 1.13 | 0.50 | 1.36 | 16.91% | |

| ||||||

Vertical | 70 | 1.30 | 4.32 | 1.23 | 4.11 | 5.11% |

50 | 1.45 | 5.04 | 1.44 | 4.89 | 3.07% | |

30 | 1.63 | 6.12 | 1.52 | 6.30 | 4.44% |

The position and direction of the applied load.

An example of field testing.

Take the dynamic process into account, taking elevation angle of 70°, for example, API test results and simulation results are compared as shown in Figure

The comparison of the dynamic oscillatory of the sampling point.

The dynamic oscillatory in time domain

The error between test and simulation

It can be seen from the comparison of the results that static load deformation error is less than 20% and dynamic response characteristics are basically consistent.

Frequency test experiments were performed in two directions by loading impulse loads (to generate oscillations of the antenna structure in the two directions). The oscillation directions corresponding to orientation and elevation were adopted, and accelerometer sensors were placed on the two points with the largest amplitudes. Then, by using the modal analyzer, the natural frequencies in the two directions were acquired by analyzing the data collected by the accelerometer sensors and comparing the results with those of the ANSYS model with the same oscillation modal (with the same oscillation). Table

Natural frequency measurement tests.

Test content | Load (kg) | Test result (Hz) | Simulation result (Hz) | Relative error (%) |
---|---|---|---|---|

Rotation mode in EL | 100 | 6.84 | 7.625 | 11.48 |

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Rotation mode in EL | 50 | 6.84 | 7.625 | 11.48 |

| ||||

Rotation mode in AZ | 100 | 27.34 | 26.893 | 1.63 |

| ||||

Rotation mode in AZ | 50 | 27.34 | 26.893 | 1.63 |

From the results of frequency test experiment and load deformation experiment, it is considered that the finite element analysis results basically meet the accuracy requirement. The finite element model can be used as a basis for dynamic modeling and reference for real-testing structural deformation.

For pointing error simulation analysis, taking a random disturbance as an example, the average wind speed is 10m / s according to the spectrum characteristics of wind disturbance. The equivalent wind force acting laterally on the 7.3m antenna reflector is shown in Figure

The equivalent wind force acting laterally on reflector.

The maximum pointing error is about 0.0041°, which caused by the structural deformation under the wind disturbance, by applying the pointing error analysis model described in this article, see Figure

The pointing error caused by deformation of the reflector.

Because the wind disturbance modeling error is not considered in the above-mentioned pointing error estimation, the correction method described in the article will be used to correct the pointing error.

Suppose the reflector system uniformly distributes 15 deformation sampling points; see Figure

The distribution of sampling points on the reflector.

When the wind disturbance modeling error is not introduced, the node deformation is calculated from the dynamic model described in this paper; see Figure

The nodal displacement obtained from dynamic model.

The finite element software analysis results replace the actual deformation information in pointing error correction. In the process of the finite element analysis, the load modeling error is introduced, by the transient analysis of the ANSYS software, the deformation of the nodes (s1, s2, s5, s9, s10, and s14) is shown in Figure

The nodal displacement with considering the load modeling error.

The deformation information of these nodes is regarded as the measured deformation information. The correction method proposed in this paper is used to correct the pointing error shown in Figure

The comparison of pointing error before and after the correction.

According to the correction of estimating the pointing error caused by the structural deformation, a controller is designed as shown in Figure

Antenna servo system.

Under a wind with the speed of 10 m/s, Figure

The pointing performance after compensation.

The influence of the large reflector antenna flexible deformation on the pointing accuracy has become more and more significant. The traditional pointing error estimation method either depends on finite element software analysis and thus cannot be directly applied to pointing control, or depends on the modeling accuracy of the load model and the pointing error analysis model and thus cannot guarantee the accuracy of the error estimation. Against these problems, this paper proposes a correction method of pointing error caused by structural deformation based on optimization of correction weight factor of each order of pointing error. Through the experiment and simulation of the 7.3m antenna, it shows that the method can effectively correct the pointing error estimation by using the measured deformation, consequently to improve the accuracy of the pointing error after compensation with PID controller.

The authors declare that they have no conflicts of interest.

This work was supported by the National 973 Program under Grant no. 2015CB857100, the National Natural Science Foundation of China under Grant nos. 51705387, 51575419, and 51490660, the National 111 Project under Grant no. B14042, the China Postdoctoral Science Foundation under Grant no. 2017M613078, the Fundamental Research Funds for the Central Universities under Grant no. JBX170414, and the CAS “Light of West China” Program under Grant nos. 2017-XBQNXZ-B-024 and 2017-XBQNXZ-B-023.