Track-Level Compensation

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The track-level compensation technique is an open-loop control of the RF beam position. The beam position is calculated with respect to the antenna azimuth and elevation position from the look-up table, and the antenna position is corrected to reduce the pointing error.

The whole antenna structure rotates in azimuth on a circular azimuth track. It is manufactured with the level of precision of ±0.5 mm for the 34-m DSN antennas. For more discussion of the azimuth track design, see [1]. The pointing accuracy of the antennas and radiotelescopes is impacted by the unevenness of the antenna azimuth track. The track unevenness causes repeated antenna tilts, hence repeatable pointing errors. In this chapter, track-level errors are described along with their compensation using the look-up table. Also, the creation of the table is described, which includes the collection and processing of the inclinometer data, and determining azimuth axis tilt. Next, the antenna pointing errors are derived from the table. Finally, pointing improvement is discussed using the table.

1 Description of the Track-Level Problem

The track is shown in Fig. 13.1. Manufacturing and installation tolerances, as well as the foundation compliance and the gaps between the segments of the track, are the sources of the pointing errors that reach over 20 mdeg peak-to-peak magnitude.

Fig. 13.1

The 34-m antenna: (a) alidade and azimuth track, and (b) azimuth track

This chapter discusses the improvement of the pointing accuracy of the antennas by implementing the track-level-compensation look-up table; see [2, 3]. The table consists of three axis rotations of the antenna lower structure (alidade) as a function of the azimuth position. The development of the table is presented, based on the measurements of the inclinometer tilts, and describe the processing the measurement data. Also the determination of the elevation and cross-elevation errors of the antenna is presented as a function of the alidade rotations. The pointing accuracy of the antenna with and without a table was measured using various RF beam pointing techniques. It was shown that the pointing error decreased when the table was used, from 7.5 mdeg to 1.2 mdeg in elevation, and from 20.4 mdeg to 2.2 mdeg in cross-elevation.

Track-level compensation (TLC) look-up tables were created for the DSN antennas in Goldstone, California, in Canberra, Australia, and in Madrid, Spain. However, the James Clerk Maxwell Telescope^{1 , 2} used inclinometers to perform track profile measurements to overcome possible systematic errors, but the results have not been published. The track-level unevenness compensation is planned for the Sardinia Radio Telescope [5]. The Green Bank Telescope memo³ reports on the pointing errors due to the azimuth track-level unevenness. GBT says⁴ that “in the antenna engineering and operations area work on the Green Bank Telescope azimuth track was seen as the most important.” Inclinometers were used also for the thermal deformation of the IRAM telescope [4].

2 Collection and Processing of the Inclinometer Data

The TLC system hardware consists of four inclinometers, the interface assembly, and a PC computer. Four digital inclinometers (model D711 of Applied Geomechanics) are mounted on the antenna. The inclinometers are located on the alidade, as shown in Fig. 13.2. Each inclinometer measures tilt in two axis, denoted x and y. The manufacturer describes the inclinometer rotation as tilts. Note that the x-axis tilt is equivalent to the y-axis rotation, and vice versa, as shown in Fig. 13.3.

Fig. 13.2

The location of the inclinometers at the alidade and X, Y, and Z rotations of the alidade

Fig. 13.3

x-axis tilt is a rotation with respect to y-axis

First, the track profile was measured. A shim of 2.5 mm was placed on the track, and the antenna wheel positioned on the shim. The inclinometer tilts were measured to determine the relationship between the inclinometer tilt and the azimuth track unevenness. The shim caused 14.8 mdeg x-tilt of the inclinometer No. 3. Based on this scaling and the continuous records of the lower inclinometer x-tilt measurements during the antenna constant velocity slewing the azimuth track profile was determined, and is shown in Fig. 13.4. It is seen from this plot that the maximum, peak-to-peak, track profile variation is 1.2 mm, slightly higher than the specification (1 mm).

Fig. 13.4

Azimuth track profile of the 34-m antenna

Next, the inclinometer data were collected to determine the alidade rotations. The antenna moves at constant azimuth axis velocity of 0.05 deg/s. Due to the environmental disturbances the inclinometer data are extremely noisy. Take for example the x-axis measurement of the inclinometer 1 shown in Fig. 13.5. The unfiltered data are represented by the gray line. Using a zero-phase filter to prevent filtering delay, the data is smoothed, as represented by the black line.

Fig. 13.5

Raw inclinometer data (gray line) and the filtered data (black line)

3 Estimating Azimuth Axis Tilt

The additional processing includes the removal of the azimuth axis tilt from the data. The tilt is present in the inclinometer data as harmonic functions in x- and y-axis tilt, of period 360 deg; see Fig. 13.6a,b. Its amplitude (a) and phase ($\varphi$) need to be determined. Let $\alpha _{1x} (i)$ and $\alpha _{1y} (i)$ be the ith sample of the x- and y-tilts of the inclinometer 1, and e(i) be the ith sample of the azimuth encoder. The inclinometer harmonics caused by the azimuth axis tilt are described as

$$ \alpha _{1x} (i) = a\cos (e(i) + \varphi ),\, \,{\rm{and}}\enspace \alpha _{1y} (i) = a\sin (e(i) + \varphi )$$

((13.1))

or, in short, as

$$ \alpha _{1x} (i) = a_c c_i - a_s s_i\,\,{\rm{and}}\enspace \alpha _{1y} (i) = a_c s_i - a_s c_i$$

((13.2))

where $a_c = a\cos (\varphi )$, $a_s = a\sin (\varphi )$, $c_i = \cos (e(i))$, and $s_i = \sin (e(i))$. For n samples define the following vectors and matrices

$$ \alpha _1 = \left\{ \begin{matrix} \alpha_{1x}\\ \alpha _{1y} \end{matrix} \right\}, \ {\rm{where}}\enspace \alpha _{1x} = \left\{ \begin{matrix} {\alpha _{1x} (1)} \\ {\alpha _{1x} (2)} \\ \vdots\\ {\alpha _{1x} (n)} \\ \end{matrix} \right\} \ {\rm{and}}\enspace \alpha _{1y} = \left\{ \begin{matrix} {\alpha _{1y} (1)} \\ {\alpha _{1y} (2)} \\ \vdots\\ {\alpha _{1y} (n)} \\ \end{matrix} \right\} $$

((13.3))

$$ P = \left[\begin{matrix} c &\quad {- s} \\ s &\quad c \\ \end{matrix} \right], \ {\rm{where}}\ c = \left\{ \begin{matrix}{l} {c_1 } \\ {c_2 } \\ \vdots\\ {c_n } \\ \end{matrix} \right\} \ {\rm{and}}\enspace s = \left\{ \begin{matrix}{l} {s_1 } \\ {s_2 } \\ \vdots\\ {s_n } \\ \end{matrix} \right\} $$

((13.4))

and

$$ A = \left\{ \begin{matrix} {a_c } \\ {a_s } \\ \end{matrix} \right\} $$

((13.5))

Fig. 13.6

Removing the azimuth axis tilt from the inclinometer data (solid line, inclinometer data; dash-dot line, inclinometer tilt caused by the azimuth axis tilt; and dashed line, inclinometer data after azimuth axis tilt removal)

For the above notations equations (13.2) can be rewritten in a compact form

$$ PA = \alpha _1 $$

((13.6))

The least-square solution A of equation (13.5) is as follows

$$ A = (P^T P)^{ - 1} P^T \alpha _1 $$

((13.7))

But, from equation (13.5) it follows that

$$ a\cos (\varphi ) = a_c,\,{\rm{and}}\enspace a\sin (\varphi ) = a_s $$

((13.8))

therefore

$$a = \sqrt {a_c^2 + a_s^2 }\enspace {\rm{and}}\enspace \varphi = \tan ^{ - 1} \left( {\frac{{a_s }}{{a_c }}} \right)$$

((13.9))

Based on several sets of data one obtains from (13.9) the following amplitude and phase of the azimuth axis tilt

$$a = 4.2\ {\rm{ mdeg}}\quad{\rm{and}}\quad \varphi = 274.5\,{\rm {deg}}$$

that is, the tilt magnitude 4.2 mdeg and phase 274.5 deg. The x- and y-axis movements of the inclinometer 1 after the tilt removal is shown in Fig. 13.6a,b (dashed line).

4 Creating the TLC Table

The TLC look-up table consists of X, Y, and Z rotations of the alidade, as shown in Fig. 13.2. They are obtained from the inclinometer tilts. Namely, a rotation with respect to the antenna x-axis, denoted X, is a rotation with respect to the antenna elevation axis. It is measured as the y-tilt of the second inclinometer ($\alpha _{2y}$):

$$X = \alpha _{2y} $$

((13.10))

The Y rotation is a tilt of the elevation axis. It is an average of the x-tilts of the inclinometers 1 and 2, that is,

$$Y = 0.5(\alpha _{1x} + \alpha _{2x} )$$

((13.11))

The Z rotation of the alidade is a twist of the alidade, and is not directly measured by the inclinometers. It is determined from x-tilts of inclinometers 3 and 4, as follows. From Fig. 13.7, which represents the view from the top of the alidade, one has

$$Z = \frac{d_3 - d_4 }{L}$$

((13.12))

where d ₃ and d ₄ are horizontal displacements of the locations of inclinometers 3 and 4, and L = 12.396 m is the distance between the two inclinometers. The displacements d ₃ and d ₄ are determined from the tilts of inclinometers 3 and 4, respectively, by assuming that the horizontal displacement of the alidade side due to azimuth track unevenness is caused predominantly by the rigid-body motion of each side of the alidade. This assumption has been checked with the finite-element model of the alidade, giving a 93% accuracy in the estimation of displacements d ₃ and d ₄. It was also confirmed by the comparison of the rotations of the inclinometers located at the bottom, the middle, and the top of the alidade.

Fig. 13.7

Top view on the inclinometers 3 and 4

The rigid-body angle is measured as the x-tilt of the inclinometers 3 and 4 (denoted as $\alpha _{3x}$ and $\alpha _{4x}$, respectively), therefore

$$d_3 = H\alpha _{3x}\quad{\rm{and}}\quad d_4 = H\alpha _{4x}$$

((13.13))

where H is the height at which the inclinometers are located, H = 9.292 m. Introducing (13.13) to (13.12) one obtains the Z rotation of the alidade as

$$ Z = \frac{H}{L}(\alpha _{3x} - \alpha _{4x} ) $$

((13.14))

where H is the alidade height and L is the distance between the inclinometers 1 and 2. Because for the 34-m antennas L = 12.39 m, the ratio is $H \left/\right. L = 0.75$, therefore

$$Z = 0.75(\alpha _{3x} - \alpha _{4x} )$$

((13.15))

The X, Y, and Z alidade rotations obtained from the inclinometer data, for azimuth angles varied from 0 to 360 deg, are shown in Fig. 13.8. The plots show that the X rotation (the elevation correction) is comparatively small, and that the largest is the Z rotation. But, it will be shown later that the Z rotation is compensated for by the azimuth encoder and hence it is not a part of pointing error.

Fig. 13.8

The TLC look-up table of the 34-m antenna

5 Determining Pointing Errors from the TLC Table

The antenna elevation and cross-elevation pointing errors are determined. The elevation error $\Delta _{EL}$ is simply determined as the alidade X rotation

$$\Delta _{EL} = X$$

((13.16))

The cross-elevation error, $\Delta _{XEL}$, depends on the antenna elevation position, EL, and on the alidade Y and Z rotations as illustrated in Fig. 13.9

$$\Delta _{XEL} = Z\cos (EL) - Y\sin (EL)$$

((13.17))

Fig. 13.9

The relationship between the cross-elevation error and the X and Y rotations of the alidade

However, the Z-rotation contributions are assumed zero in the TLC table because this error is measured by the azimuth encoder and therefore eliminated by the azimuth servo. The following experiment at the 34-m antenna was conducted at the Madrid Deep Space Communication Complex to verify this hypothesis. With the antenna dish positioned at EL = 30 deg a shim of 1 mm thick was placed on the azimuth track, as shown in Fig. 13.10. The antenna was then moved slowly with constant speed in azimuth over the shim. The same antenna movement was repeated when the shim was removed. The difference between azimuth encoder reading with and without the shim is plotted in Fig. 13.11. It shows the azimuth position rising sharply (section A) when the antenna is climbing the shim. But the azimuth servo compensates for the shim disturbance (section B), and the azimuth position returns to the initial position (section C). As result the antenna does not need correction in z-axis, and the Z component of the TLC table shall be zero.

Fig. 13.10

Azimuth wheel crosses 1 mm shim

Fig. 13.11

The azimuth encoder reading when crossing the shim. Section A shows a sharp rise in encoder reading at the beginning of the shim; Section B shows an azimuth servo correction to the shim disturbance; and Section C shows the stabilized azimuth position

Based on the above experiment the following equation

$$\Delta _{\mathit{XEL}} = - Y\sin (EL)$$

((13.18))

is the resulting formula for the cross-elevation error.

6 Antenna Pointing Improvement Using the TLC Table

The improvement of pointing accuracy with the look-up table was evaluated using the RF pointing data. The following RF beam measurement techniques were used: boresight, monopulse, and conscan. The data were measured with the installed TLC table (“TLC table on”) and without the TLC table (“TLC table off”). Both methods are useful in the validation of the effectiveness of the TLC table. Namely, when the table is on, the pointing errors should be significantly smaller than the errors predicted from the TLC table (or the errors obtained for the same track with the TLC table off). When the table is off, the RF pointing errors should match the errors predicted from the TLC table.

The measurements with the TLC table off were taken for the trajectory shown in Fig. 13.12. Figure 13.13a shows that the measured elevation pointing errors and the errors predicted by the look-up table coincide. The elevation error predicted from the TLC table varies by 7 mdeg, from –3 to 4 mdeg (dashed line). Figure 13.13b shows that the cross-elevation errors (predicted and measured) coincide when the antenna elevation position is below 72 deg, and that a deterministic residual is uncompensated when the antenna elevation position is above 72 deg The cross-elevation error predicted from the TLC table vary by 12 mdeg, from –5 to 7 mdeg (dashed line); they show a “deterministic” error (for AZ < 240 deg, where antenna elevation position above 72 deg).

Fig. 13.12

The 34-m antenna tracking trajectory

Fig. 13.13

The 34-m antenna pointing errors, measured (black solid line) and predicted from the TLC table (dashed line): (a) the elevation pointing error, and (b) the cross-elevation pointing error

The measurements of the radio beam position with a TLC table have a standard deviation of 0.41 mdeg (or 1.2 mdeg peak-to-peak), while the radio beam data with the TLC table shows its standard deviation of 0.72 mdeg (or 2.2 mdeg peak-to-peak), for the antenna at elevation position below 72 deg.

Table 13.1 summarizes the antenna tracking accuracy with the TLC table on and off. The elevation pointing error decreased 6-fold, and the cross-elevation pointing error decreased 10-fold.

Table 13.1 Peak-to-peak pointing errors of the 34-m antenna