Effect of the rail unevenness on the pointing accuracy of large radio telescope

Acta Astronautica 132 (2017) 13–18

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Effect of the rail unevenness on the pointing accuracy of large radio telescope

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⁎

Na Li , Jiang Wu, Bao-Yan Duan, Cong-Si Wang School of Mechanical Electronic Engineering, Xidian University, Shannxi 710071, China

A R T I C L E I N F O

A BS T RAC T

Keywords: Rail unevenness Pointing accuracy Large radio telescope Coarse-fine mixed Modeling

Considering the stringent requirement of the pointing accuracy up to 2.5″ of the World largest full steerable telescope, this paper presents a coarse-fine mixed model to describe the azimuth rail unevenness. First, the coarse-fine mixed model is proposed. In the model, the trigonometric function is utilized to describe the error with long wavelength whilst the fractal function is used for the short wavelength errors, separately. Then the mathematic model of the pointing accuracy is developed mathematically. Finally, the coarse-fine model and point accuracy model are applied to Green Bank Telescope with valuable result. This paved the way for predicting point error of Qi Tai Telescope.

1. Introduction Over the last two decades, many countries have been racing to construct the large radio telescopes, such as the Green Bank Telescope (GBT) in America [1], the Effelsberg Telescope in Germany [2] and the Sardinia Radio Telescope in Italy [3]. From 2014, China has been planning to build the Qi Tai telescope (QTT), which will be a fully steerable telescope with the world's largest aperture (110 m). QTT is designed to work from 150 MHz to 115 GHz, which results in a repeatable pointing accuracy requirement of 1.19 s. The whole telescope structure is about 6000 t in weight and as high as a 30-floor building [4]. Therefore, such a strict pointing accuracy requirement imposes considerable difficulties on the design of QTT [5]. For the factors affecting the pointing accuracy, the telescope itself is absolutely the most important one, such as the inertia and elastic deformation of the mount, the reflector and the rail. Currently, high-precision ratio telescopes typically adopt the completely-welded rails. Compared to non-welded rails, it avoids the large deformations at rail junctions, and thereby prolongs the telescopes service life [6]. However, during the manufacturing and welding processes, it is inevitable that errors such as surface roughness and rail stress deformation are produced [7]. These errors are collectively referred to as rail unevenness. It can directly lead to the azimuth frame errors in azimuth and pitching axis, and then affect the pointing accuracy of telescope [8]. As early as 2000, Gawronski began to notice the effect of rail unevenness on radio telescope. By the aid of an inclinometer, he converted the measured unevenness data into errors of the telescope's azimuth and pitch angles in accordance with their geometric relation⁎

ship [9]. Additionally, Pisanu considered the combined effects of the deformation of the azimuth frame on pointing accuracy, which was induced by rail unevenness and temperature drift [10]. Kong performed the testing experiment on the rail unevenness and pointing accuracy and analyzed the correlation between them [11]. Besides, some researchers indirectly introduced the non-linear rail errors into the construction of a telescope's pointing model [12]. Although many fruitful results have been achieved [13], the description of rail unevenness is too ambiguous, leading to immense fitting errors and thereby severely affecting a telescope's ultimate pointing accuracy. An accurate model of rail unevenness, which is required in studies involving the effects of rail unevenness on pointing accuracy, has not yet been constructed. And so a model for antenna pointing errors, in which rail unevenness is taken into account, has still not to been established. The qualitative relationship between rail unevenness and the pointing accuracy of the antenna have not been found fundamentally. Aimed at solving the aforementioned problems, this article describes the rail unevenness using the Fourier series and the fractal function, and innovatively proposed the coarse-fine mixed description model. Finally, an influence relationship model of the rail unevenness on the pointing accuracy was established based on the mixed model. 2. Coarse-fine mixed modeling of the rail unevenness Errors that lead to rail unevenness primarily originate from two aspects – the surface roughness produced in the machining process of a single rail and the deformations produced during the welding process

Corresponding author. E-mail addresses: [email protected] (N. Li), [email protected] (J. Wu), [email protected] (B.-Y. Duan), [email protected] (C.-S. Wang).

http://dx.doi.org/10.1016/j.actaastro.2016.12.005 Received 12 July 2016; Accepted 1 December 2016 Available online 05 December 2016 0094-5765/ © 2016 IAA. Published by Elsevier Ltd. All rights reserved.

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and post-processing. These two different kinds of surface error have different characteristics [14,15]. The surface roughness is randomness, high-frequency, low-amplitude, and small scale, while the deformation is systematic, low frequency, and large amplitude. Based on these distribution features, the large-scale deformation of rail unevenness was fitted using the Fourier series [16]. Next, the WeierstrassMandelbrot (W-M) fractal function was used to describe the fitting residual error, and then, fitting function of the small-scale roughness was conducted [17]. Finally, the coarse-fine mixed model of rail unevenness was established, as shown below in Eq. (1). k

F (x ) = f1 ( x ) + ∑i =1 f2i (Ai , Di , L i, x 0i , y0i) m

=(a 0 + ∑n =1 [an cos(nω0 x ) + bn sin(nω0 x )]) k

+ (∑i =1 f2i (Ai , Di , L i, x 0i , y0i)) M

f2i (Ai , Di , L i, x 0i , y0i) = Ai (Di −1) ∑ j =i n1i

1 γ (2− Di ) j

cos(2πγ j (x + x 0i )) + y0i

n1i = lg(1/ L i )/ lg (γ ) Mi = lg(Nγ n1i /2)/ lg (γ )

(1)

where, f1 (x ) denotes the Fourier series; a 0 , a1, …, an , b1, …, bn denotes the undetermined coefficient of f1 (x ); m denotes the expansion order of f1 (x ); ω0 denotes the fundamental frequency of f1 (x ); S denotes the rail length described by F (x ) and is generally set as one third of the rail's overall length; k denotes the number of rail segments when the W-M fractal function is used for fitting rail unevenness; l denotes the fitting length of f2i (k = S / l ) and is set as a round number; f2i (Ai , Di , L i, x 0i , y0i) denotes the i-th segment of the W-M fractal function f2 (x ); Ai denotes the amplitude height coefficient of f2 (x ), reflects the value of f2i , and determines the specific size of f2i ; L i denotes the sampling length of f2 (x ); Di denotes the fractal dimension of f2 (x ); y0i and x 0i denote the longitudinal and lateral displacements of f2 (x ), respectively. As shown in Eq. (1), the Fourier series f1 (x ) and the W-M fractal function f2 (x ) codetermine the fitting precision of rail unevenness. Moreover, higher expansion order of the Fourier series (m ), will result in higher the fitting precision is. Similarly, shorter the sampling length of the fractal function (L ), will result in the higher the fitting precision is. In this work, the sampling length of the Weierstrass-Mandelbrot (W-M) fractal function (L ) equals the fitting length of the small-scale function, and so determining the values of m and L is the key to determining the unevenness model [18]. The determination methods for these two key parameters are described in detail below [19]. The optimization model based on the W-M fractal function can be described as:

Fig. 1. Displays the detailed process for determining the parameters.

3. Effects of rail unevenness on the QTT's pointing accuracy 3.1. Effect of the rail unevenness on the azimuth frame error

N

F ( y) =

[ ∑ ( f 2i (xj , y) − y (xj ))2 ]/ N

M

f2 (x ) = A(D −1)

∑ n = n1

1 cos(2πγ n (x + Δx )) + Δy γ (2− D) n

Find

y = ( Ai , Di , L i, x 0i , y0i )T

Min .

F ( y) =

S. T .

To describe telescope's various errors and investigate their effects on the pointing accuracy, four coordinate systems were constructed, as shown in Fig. 2, the detailed descriptions of these coordinate systems are provided below: OXYZ - Geodetic coordinate system, in which the origin is located at the center of the azimuth orbit, the Z-axis is perpendicular to the ground, and the negative direction of the Y-axis points to true north. Oa Xa Ya Za - Coordinate system rigidly connected to the azimuth axis, in which the origin is located at the center of the azimuth orbit and the Za -axis coincides with the azimuth axis. Overall, the coordinate follows the rotation and deflection of the azimuth axis. When the telescope has no errors along the azimuth axis and the azimuth angle equals zero (i.e., A = 0°), Oa Xa Ya Za is identical to OXYZ . Oe Xe Ye Ze -Coordinate system rigidly connected with the pitch axis in which the origin is located at the middle of the pitch axis and the Xe -axis coincides with the pitch axis. Overall, the coordinate follows the rotation and deflection of the pitch axis. When the telescope has no axis errors and the pitch angle and azimuth angle equal 90° and 0°, respectively, Oe Xe Ye Ze only differs from OXYZ along the direction of the Z-axis with a difference of the height of the azimuth axis h .

(2)

j =1

0

(3)

N

[ ∑ j =1 ( f 2i (xj , y) − y (xj ))2 ]/ N < x 0i < x max L min < L i < L max ymin < y0i < ymax Amin < A2i < Amax Dmin < Di < Dmax

(4)

The upper and lower limits of the variables were determined based on their physical meanings. The physical meanings and the determination methods for the related parameters in the optimization model of fractal function are described in the Fig. 1. 14

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the X-axis and Y-axis, which are denoted as Δx2 and Δy2 , respectively, and shown in Eq. (6).

h1

Yr

Zr Or Ze

Oe

⎡ Z (A + π ) − Z (A − 4 Δx1 = ⎢ ⎢ 2ρ ⎣ ⎡ Z (A + Δx2 = ⎢ ⎢ ⎣

Ye Xr Xe

⎡ Z (A − Δy1 = ⎢ ⎢ ⎣

Za O(Oa)

Ya Xa

Y

− Z (A− 4 ) ⎤ ⎥ cos(π /4) ⎥ 2ρ ⎦ π

(5)

− Z (A + 4 ) ⎤ ⎥ cos(π /4), ⎥ 2ρ ⎦ π

− Z (A− 4 ) ⎤ ⎥ cos(π /4) ⎥ 2ρ ⎦

3π ) 4

π

(6)

ϕtx = Δx1 + Δx2 , ϕty = Δy1 + Δy2

A

(7)

In addition, the rail unevenness can also lead to the stress deformation of the azimuth frame and thereby, the torsion of the azimuth frame, which can make the pitch axis rotate around the axis of Za , as shown in Fig. 4. The displacements of the upper end-points on the pitch axis induced by Wheel 1, Wheel 2, Wheel 3, and Wheel 4 are denoted as δ12 and δ34 , respectively, they can be written as the following:

Fig. 2. Illustration of the telescope's coordinate system.

Or Xr Yr Z r -Coordinate system fixedly connected with the reflector. When the pitch angle equals 90° (i.e., E = 90°), Or Xr Yr Z r differs from Oe Xe Ye Ze along the direction of the Z-axis with a difference of h1. As shown in Fig. 3, Oa Xa Ya Za is the coordinate rigidly connected with the azimuth axis, and the origin is located at the center of azimuth orbit. The black line shows the positions of the four wheels of the azimuth coordinate system. The red line shows the errors resulting in the incline and distortion of the azimuth housing. The four wheels, π denoted as Wheel 1, Wheel 2, Wheel 3, and Wheel 4, are Z (A + 4 ), 3π

cos(π /4),

Under the combined action of four points, according to the righthand rule, the pitch axis is rotated by φty and φty along the axis of Xa and Ya , respectively, as in the following:

X

π

⎥ ⎦

3π ) 4

3π ) 4

⎡ Z (A + Δy2 = ⎢ ⎢ ⎣

Z

h

3π ⎤ ) 4 ⎥

⎡ −Z (A + π ) + Z (A − π ) ⎤ 4 4 ⎥ δ12 = h ⎢ , ⎢⎣ ⎥⎦ 2r cos(π /4)

⎡ Z (A − 3π ) − Z (A + 4 δ34 = h ⎢ ⎢ 2r cos(π /4) ⎣

3π ⎤ ) 4 ⎥

⎥ ⎦ (8)

Under the combined action of four points, the pitch axis is rotated by ϕtz along the axis of Za . ϕtz is the rotation of the azimuth coordinate system around the Z-axis induced by rail unevenness, it can be written as the following:

3π

Z (A− 4 ), Z (A − 4 ) and Z (A + 4 )in height, respectively. A very small height difference between Wheel 1 and Wheel 3 can produce rotations of the azimuth axis along the directions of the X-axis and Y-axis, which are denoted as Δx1 and Δy1, respectively, and shown in Eq. (5). Similarly, a very small height difference between Wheel 2 and Wheel 4 can also produce rotations of the azimuth axis along the directions of

ϕtz =

δ12 − δ34 , l

l=

2r

(9)

in where, h denotes the height of the pitch axis far from the ground, l denotes the length of the pitch axis, and r denotes radius of the rail. Therefore, the rotation value can be obtained [20]: π

ϕtx ≈

π

Z (A + 4 ) − Z (A− 4 ) − Z (A −

Fig. 3. Inclined deformations on the azimuth frame caused by rail unevenness. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

+ Z (A +

3π ) 4

(10)

2 2r π

ϕty ≈

3π ) 4

π

−Z (A + 4 ) − Z (A− 4 ) + Z (A −

3π ) 4

+ Z (A +

3π ) 4

2 2r

Fig. 4. Distortion of azimuth frame induced by rail unevenness.

15

(11)

Acta Astronautica 132 (2017) 13–18

N. Li et al. π

ϕtz ≈

π

h (−Z (A + 4 ) − Z (A− 4 ) − Z (A −

3π ) 4

+ Z (A +

3π )) 4

among subreflector, feed source, and main reflector. At last, only considering the effect of rail unevenness on the pointing accuracy, let ϕax = ϕay = ϕaz = 0 , ϕex = ϕey = 0 , and ϕez = ϕox = ϕoy = 0 , and by substituting them into Eq. (17), the pointing accuracy can be obtained as following:

(12)

2r 2

The above expressions can be rewritten in matrix form: 1 1 ⎡ 1 ⎡ ϕtx ⎤ ⎢ 2 2 r − 2 2 r − 2 2 r ⎢ ⎥ ⎢ 1 1 1 ⎢ ϕty ⎥ = ⎢− 2 2 r − 2 2 r 2 2 r ⎢ϕ ⎥ ⎢ h h ⎣ tz ⎦ ⎢ − h − 2 ⎣ 2r 2 2r 2r 2

⎡ Z (A + π ) ⎤ 4 ⎥ π Z (A − 4 ) ⎥ ⎥ 1 ⎥ ⎢ 2 2 r ⎥ ⎢ Z (A − 3π ) ⎥ ⎥ 4 ⎥ ⎢ h ⎥ 3π ⎥ 2r 2 ⎦ ⎢ Z (A + ) ⎣ 4 ⎦

1 ⎤ ⎢ 2 2 r ⎥⎢

⎡ 1 ⎢ 2 2r − ⎡ α ⎤ ⎡ 0 tan E − 1⎤ ⎢ 1 − ⎢ β⎥ = ⎢ ⎥ − 0 ⎦ ⎢⎢ 2 2 r ⎣ ⎦ ⎣− 1 0 h ⎢⎣ − 2

(13)

2r

The perturbation of the azimuth coordinate system induced by the rail unevenness can be written as the following matrix form:

⎡ 1 ϕtz − ϕty ⎤ ⎢ ⎥ 1 ϕtx ⎥ Rt = ⎢− ϕtz ⎢ ϕ 1 ⎥⎦ ⎣ ty − ϕtx

1 2 2r 1 2 2r h 2r 2

−

1 2 2r

1 2 2r h

−

2r 2

⎡ Z (A + π ) ⎤ 4 ⎥ π Z (A − 4 ) ⎥ 1 ⎥⎢ ⎥ 2 2 r ⎥ ⎢ Z ( A − 3π ) ⎥ ⎥ 4 ⎥ ⎢ h ⎥ 3π ⎥ 2r 2 ⎦ ⎢ Z ( A + ) ⎣ 4 ⎦ 1 ⎤ ⎢ 2 2 r ⎥⎢

(18)

(14)

4. Experiments and verification 4.1. Experimental verification on the GBT

3.2. Effects of the rail unevenness on pointing accuracy As shown in Fig. 5, the GBT is chosen as the sample to validate the reliability and effectiveness of the proposed methods. GBT is the world's largest fully steerable radio telescope, its structure weighs 8500 t and is 450 feet tall. The surface area of the GBT is a 100 by 110 m active surface with 2209 actuators for the 2004 surface panels. The panels are made from aluminium to a surface accuracy of better than 0.003 in. (76 µm) RMS. Its rail diameter is 64 m. There are 48 rail segments with an overall length of 201 m and the unevenness degree (RMS) is 0.0568 mm [21]. Using the correlation coefficient test method, the non-scaling ranges of GBT is [0.116, 0.901] [22]. And then, using the wavelet transform, the fractal dimension of it is calculated, DGBT = 1.602 . Fig. 6 describes comparison of the model result and the measured data on the rail unevenness of GBT. Table 1 lists the RMS values and amplitude ranges of the expansion order (m ), the fitting length (l ), and the description error of rail unevenness. RMS0 denotes the RMS of original rail, RMSm denotes RMS of fitting error, B0 denotes the amplitude of original rail unevenness, and Bm denotes the amplitude of the fitting error. Based on the rail unevenness model, the antenna's pointing errors at any azimuth and pitch angle can be acquired by the proposed influence relation function. According to the design of the GBT antenna, h=60 and r=34 can be set. Figs. 7 and 8 show the pointing errors of the azimuth and pitch angles induced by rail unevenness when the antenna's pitch angle equals 45° (i.e., E = 45o ). The indexed values are list in the Table 2, in where, ΔPE = α 2 + β 2 .

By taking the rail unevenness into account, the transformation of coordinates can be written:

⎡ xr′⎤ ⎡ ⎤ ⎡ x ⎤ ⎢− ϕtz cos E − ϕty sin E ⎥ ⎢ ⎥ r e a ϕtx ⎢ yr′ ⎥ = Re Ra Rt Rb ⎢ y ⎥ = ⎢ ⎥ ⎣z⎦ ⎢ ⎢⎣ z ′ ⎥⎦ ⎥⎦ ⎣ 1 r ⎡ 0 ⎤ ⎡− ϕtz cos E − ϕty sin E ⎤ ⎢ ⎥ = ⎢0⎥ + ⎢ ϕtx ⎥ ⎢⎣ ⎥⎦ 1 ⎢⎣ ⎥⎦ 0

(15)

The pointing error induced by the rail unevenness is described as the following:

⎡ xr′⎤ ⎡− ϕtz cos E − ϕty sin E ⎤ ⎢ ⎥ ⎡ xr ⎤ ⎢ ⎥ ⎢ ⎥ y ′ Δt = ⎢ yr ⎥ − r = ⎢ ϕtx ⎥ ⎢ ⎥ ⎢⎣ z ′ ⎥⎦ ⎣ zr ⎦ ⎢⎣ ⎥⎦ 0 r

(16)

Then, the above-described pointing error in the coordinate system of the reflector can be converted into the azimuth and pitching deviations in a geodetic coordinate system, the errors of the azimuth and pitch angels are α = Δθr / cos(E ) and β = Δϕr , respectively. ⎡ −(ϕ + ϕ + ϕ ) − (ϕ + ϕ )tan E − ϕ cos A tan E + ϕ sin A tan E ⎤ tz ez az ty ey ay ax ⎥ ⎡ α ⎤ ⎢⎢ ⎥ ⎢ β ⎥ = ⎢ − ϕoysecE ⎥ ⎣ ⎦ ⎢−ϕ − ϕ cos A − ϕ sin A − ϕ − ϕ ⎥ ax ay ex ox ⎣ tx ⎦ ⎡ ϕtx ⎤ ⎢ ⎥ ⎢ ϕty ⎥ ⎢ϕ ⎥ ⎢ tz ⎥ ⎢ ϕax ⎥ ⎢ϕ ⎥ ay ⎡ 0 tan E −1 sin A tan E − cos A tan E −1 0 tan E −1 0 −secE ⎤ ⎢ ⎥ =⎢ ⎥ ⎢ϕ ⎥ 0 − cos A − sin A 0 −1 0 0 −1 0 ⎣−1 0 ⎦ ⎢ az ⎥ ⎢ ϕex ⎥ ⎢ϕ ⎥ ⎢ ey ⎥ ⎢ ϕez ⎥ ⎢ ⎥ ⎢ ϕox ⎥ ⎣⎢ ϕoy ⎦⎥

5. Conclusion According to the error characteristic of the rail unevenness in two scales, the Fourier series was introduced to describe the large scale error and the W-M fractal function for the small one. And the coarsefine mixed description model of the rail unevenness was developed. Based on the model, the influence relation function on pointing accuracy was established. For verification, the mixed model and the influence function were applied to GBT. Comparative analysis on the model results and the measured data indicated that the proposed methods are effective and feasible.

(17)

in where, A denotes the antenna's azimuth angle; E denotes the antenna pitch angle; α denotes the error of azimuth angle; ϕtx , ϕty , and ϕtz denote errors of the azimuth frame's coordinate system induced by rail unevenness, and can be calculated based on the mixed model; ϕax , ϕay , and ϕaz denote errors of the azimuth frame's coordinate system primarily induced by the wheel rail's overall incline and installation error of the azimuth axis; ϕex , ϕey , and ϕez denote errors of the pitch axis coordinate system primarily induced by the deformation of the azimuth frame and installation error of the azimuth axis. Finally, ϕox and ϕoy denote errors of RF axis primarily induced by the position deviation

Acknowledgment The authors acknowledge the financial support from the National Natural Science Foundation of China (Grant no. 51305322, 51405364 and 51490660) 16

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Table 1 Related parameters and indexes of three different antennas.

GBT

m

l (mm)

RMS0 (mm)

RMSm (mm)

B0 (mm)

Bm (mm)

18

60

0.0568

0.0107

± 0.2

± 0.02

Fig. 7. Pointing error induced by rail unevenness after the rail compensation.

Fig. 5. Photograph of the GBT and its rail detail.

Fig. 8. Pointing error induced by rail unevenness after the rail compensation. Table 2 Effects of rail unevenness on pointing index/azimuth angle. Index (second of arc)

α

β

ΔPE

RMS Peak-to-peak value

0.5218 3.012

0.2407 1.601

0.5746 3.547

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Fig. 6. Comparison of model results versus measured data on GBT rail unevenness.

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