INTRODUCTION

The attitude of objects in the geographical frame is commonly determined or monitored by gyroscopic aids measuring the astronomical azimuth such as gyrocompasses and gyrotheodolites [1–5]. They are periodically calibrated to reduce the systematic components of azimuth error [6, 7]. Calibration is done using a stationary astronomical azimuth determination system, which fastly and accurately determines the astronomical azimuth of reference direction by stellar observations [8–9]. However, its capabilities are limited by the operating conditions: it needs a specially equipped building on a massive vibroisolated foundation.

In field conditions, gyroscopic devices are calibrated using the reference directions specified by the landmarks (autocollimating reflectors or sighting targets) [2]. The azimuth of reference directions is determined by classical instruments: DKM-3A astronomical theodolite by Wild, Switzerland, and AU-1 automated astronomical system by the Central Research Institute of Geodesy, Airborne Survey and Cartography, Russia [2, 3, 10]. Azimuth determination is not automated in these devices, it requires long observations and highly qualified personnel.

High-precision total stations with stellar observation functions are also used in astrogeodetic surveys [11–13].

Satellite geodesy is also utilized in field determination of astronomical azimuth of reference direction. However, this method requires additional high-precision geodetic data (components of the deflection of the vertical). The accuracy of azimuth determination by this method depends on the distance between the observation points [14–16].

Currently, Concern CSRI Elektropribor develops an automated astronomical system (AAS) for precise and fast azimuth determination in field conditions. It is going to substitute the DKM-3A and AU-01 systems.

The AAS is designed as a classical astronomical theodolite comprising three main parts (Fig. 1):

Fig. 1.
figure 1

AAS design and main components.

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• sighting device (SD) consisting of the objective and TV-camera;

• vertical drive (VD) rotating the sighting axis in the vertical plane;

• azimuth drive (AD) rotating the sighting axis in the horizontal plane.

The AAS also comprises a two-axis inclination sensor (IS) installed on the AD rotating part and a leveling system.

The AAS determines the landmark azimuth as follows: it determines the azimuth of sighting axis АSA by stellar observations near the meridian plane and then turns it to the landmark, with the horizontal rotation angle of sighting axis γ calculated by AD angle sensor data. Thus the astronomical azimuth of a landmark is determined by the formula:

$$A_{L}^{{}} = A_{{SA}}^{{}} + {{\gamma }}{\text{.}}$$
(1)

Azimuth can be accurately determined only with proper mutual orientation of AD and VD rotation axes, IS sensitivity axes, and SD sighting axis. If the axes are improperly oriented due to technological errors in AAS manufacture and assembly, the horizontal error γ is determined with errors. Therefore, it is needed to analyze how AAS instrumental errors caused by technological defects in manufacture and assembly affect the azimuth determination accuracy.

LANDMARK AZIMUTH DETERMINATION ALGORITHM

The landmark azimuth determination algorithm can be conventionally divided into three major operations:

(a) determining the astronomical azimuth of the sighting axis during observation of near-meridian stars at the fixed altitude h0;

(b) determining the horizontal angle γ when proceeding from the stellar observation to landmark observation;

(c) reversing the sighting axis: rotating it successively about AD rotation axis and then about VD rotation axis so that the sighting axis points the same side as before the rotations.

The sighting axis is reversed to reduce the systematic error components caused by IS bias and AAS technological errors in manufacture and assembly.

Operations а) and b) are performed before and after the reversal of the sighting axis to determine its azimuths \(A_{{{\text{SA}}}}^{{\text{I}}}\) and \(A_{{{\text{SA}}}}^{{{\text{II}}}}\), and horizontal angles γI and γII (subscript “I” denotes prereversal position of the sighting axis, and “II”, postreversal position).

The horizontal angle γ is calculated based on the changes in AD angle sensor data ΔφAD when proceeding from stellar to landmark observations.

The resultant astronomical azimuth of the land-mark with account for the axis reversal is determined as

$$A_{{\text{L}}}^{{}} = \frac{{A_{{\text{L}}}^{{\text{I}}} + A_{{\text{L}}}^{{{\text{II}}}}}}{2},$$
(2)

where \(A_{{\text{L}}}^{{\text{I}}}{\text{, }}A_{{\text{L}}}^{{{\text{II}}}}\) are the landmark azimuth values before and after the reversal of the sighting axis, calculated by (1).

The astronomical azimuth of the sighting axis is determined as follows:

(1) the star images are recorded in the photodetector plane, and the arrays of coordinates of star image energy centers are generated \(\{ {{x}_{1}},{{y}_{1}}...{{x}_{{{{N}_{{{\text{PD}}}}}}}},{{y}_{{{{N}_{{{\text{PD}}}}}}}}\} \), where NPD is the number of star images detected in the photodetector plane;

(2) initial equatorial coordinates α0, δ0 corresponding to the equatorial coordinates of the center of the working area in star catalogue are calculated using (3) for the stars north of zenith [10]

$${{{{\alpha }}}_{0}} = {{{{\theta }}}_{{{\text{GST}}}}} + {{{{\lambda }}}_{{\text{Z}}}} + 180^\circ {\text{; }}{{{{\delta }}}_{0}} = 90^\circ - {{{{\varphi }}}_{{\text{Z}}}} + {{h}_{0}},$$
(3)

and using (4) for the stars south of zenith

$${{{{\alpha }}}_{0}} = {{{{\theta }}}_{{{\text{GST}}}}} + {{{{\lambda }}}_{{\text{Z}}}}{\text{; }}{{{{\delta }}}_{0}} = {{h}_{0}} + {{{{\varphi }}}_{{\text{Z}}}} - 90^\circ ,$$
(4)

where φZ, λZ are the astronomical coordinates of the location (initial data) that were earlier determined by the AAS by observing near-zenith stars similarly to the automated zenith telescope [17, 18], θGST is the Greenwich true siderial time [19, 20];

(3) the detected stars are identified by the coordinates of energy centers of their images in photodetector plane and star catalogue data, and the array of star pairs is generated, where the coordinates of star images \(\{ {{x}_{1}},{{y}_{1}}...{{x}_{{{{N}_{{{\text{SP}}}}}}}},{{y}_{{{{N}_{{{\text{SP}}}}}}}}\} \) in the photodetector plane are matched with their equatorial coordinates \(\{ {{\alpha }_{1}},{{\delta }_{1}}\,...\,{{\alpha }_{{{{N}_{{{\text{SP}}}}}}}},{{\delta }_{{{{N}_{{{\text{SP}}}}}}}}\} \) from the star catalogue and the calculated standard coordinates \(\{ {{\xi }_{1}},{{\eta }_{1}}\,...\,{{\xi }_{{{{N}_{{{\text{SP}}}}}}}},{{\eta }_{{{{N}_{{{\text{SP}}}}}}}}\} \), where NSP is the number of identified stars [21]. Notably, the star equatorial coordinates from the star catalogue are determined with account for the star drift, astronomical refraction, parallax, aberration, and changes in the position of the celestial axis (precession and nutation) [19];

(4) parameters for transforming the coordinates from the photodetector plane to the standard coordinates (further, transformation parameters) are determined using the star pairs’ array [22, 23];

(5) the coordinates of energy centers of star images with account for their declinations \(\{ {{\delta }_{1}}\,...\,{{\delta }_{{{{N}_{{{\text{SP}}}}}}}}\} \) are corrected for the shift during exposition caused by the Earth’s daily rotation, and the transformation parameters are determined anew;

(6) using the transformation parameters, the standard coordinates of the point on the celestial sphere corresponding to the photodetector center, and then its equatorial coordinates αSA, δSA are determined [17];

(7) apparent azimuth АSA and altitude hSA of the sighting axis are determined using the following equations [20]:

$$\begin{gathered} {{A}_{{{\text{SA}}}}} = {\text{arctan}}\left( {\frac{{\sin ({{s}_{{}}} - {{{{\alpha }}}_{{{\text{SA}}}}})}}{{\sin ({{{{\varphi }}}_{{\text{Z}}}})\cos ({{s}_{{}}} - {{\alpha }_{{{\text{SA}}}}}) - \cos ({{{{\varphi }}}_{{\text{Z}}}}){\text{tan(}}{{{{\delta }}}_{{{\text{SA}}}}}{\text{)}}}}} \right) \\ {{h}_{{{\text{SA}}}}} = \arccos \left( {\frac{{\cos ({{{{\delta }}}_{{{\text{SA}}}}})\sin ({{s}_{{}}} - {{{{\alpha }}}_{{{\text{SA}}}}})}}{{\sin ({{A}_{{{\text{SA}}}}})}}} \right), \\ \end{gathered} $$
(5)

where \(s = {{{{\theta }}}_{{{\text{GST}}}}} + 1.00274T - 0.00274{{{{\lambda }}}_{Z}}\) is the local sidereal time of the moment when the star images were recorded [24]; T is the local time, h.

(8) corrections to the astronomical azimuth for the tilt relative to the horizon plane ΔAtilt in transverse direction (about the sighting axis) and for the shift xP, yP of the instantaneous pole with respect to the mean pole ΔAMP are determined. Correction ΔAtilt is calculated as

$$\Delta {{A}_{{{\text{tilt}}}}} = - {{\psi }_{{{\text{I}}{{{\text{S}}}_{{\text{Y}}}}}}}\tan ({{h}_{{{\text{SA}}}}}),$$
(6)

and correction ΔAMP is determined as [1]

$$\Delta {{A}_{{{\text{MP}}}}} = - ({{x}_{{\text{P}}}}\sin {{{{\lambda }}}_{{\text{Z}}}} + {{y}_{{\text{P}}}}\cos {{{{\lambda }}}_{{\text{Z}}}})\sec {{{{\varphi }}}_{{\text{Z}}}};$$
(7)

(9) the resultant azimuth of the sighting axis using stellar observations is calculated as

$$A_{{SA}}^{{\text{C}}} = A_{{{\text{SA}}}}^{{}} + \Delta {{A}_{{{\text{tilt}}}}} + \Delta {{A}_{{{\text{MP}}}}}.$$
(8)

COMPONENTS OF AAS INSTRUMENTAL ERROR

Mutual orientation of the AAS body axes and their position in the horizontal frame ONWZ is shown in Fig. 2. In the ideal case (Fig. 2а) with no technological manufacturing errors, the IS sensitivity axes (vectors ОХ and ОY) lie within the horizon plane NOW. Vector ОА coincides with the AD rotation axis, and vector O'V, with the VD rotation axis. Vector О'S corresponds to the position of the SD sighting axis when observing the landmark in the horizontal plane, then O'S is codirected with ОХ, and O'V is codirected with OY. Vector O'S' corresponds to the position of the sighting axis when observing the stars at the altitude hSA, then orientation of the sighting axis in the horizontal frame ONWZ (astronomical azimuth АSA, altitude hSA) is determined by astronomical observations.

Fig. 2.
figure 2

AAS body axes in horizontal plane. (а) body axes in ideal position; (b) body axes under technological manufacturing errors.

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AD and VD angle sensors measure the rotation of the sighting axis O’S with respect to their zero positions (φAD and θVD, respectively). AAS tilt with respect to the horizon plane is measured by IS in longitudinal and transverse directions (\({{{{\theta }}}_{{{\text{I}}{{{\text{S}}}_{{\text{X}}}}}}}\) and \({{\psi }_{{{\text{I}}{{{\text{S}}}_{{\text{Y}}}}}}}\) respectively).

Figure 2b shows the position of the AAS body axes under technological manufacturing and assembly errors, where ΔφSX is the the horizontal angle between the projection of the sighting axis in the horizontal plane and IS X axis; ΔφXY is the horizontal angle characterizing the nonorthogonality of the IS axes; ΔθXA is the vertical angle characterizing the nonorthogonality of IS X axis and AD rotation axis; ΔψYA is the vertical angle characterizing the nonorthogonality of IS Y axis and AD rotation axis; ΔφSV is the horizontal angle characterizing the nonorthogonality of the sighting axis and VD rotation axis; ΔψVA is the vertical angle characterizing the nonorthogonality of AD and VD rotation axes.

Along with technological manufacturing and assembly errors, the landmark azimuth error contains other components that can be expressed by RMS deviations for IS \({{\sigma }_{{{\text{I}}{{{\text{S}}}_{{\text{X}}}}}}}{\text{,}}\,\,\,\,{{\sigma }_{{{\text{I}}{{{\text{S}}}_{{\text{Y}}}}}}}\), VD angle sensor σVD and AD angle sensor σAD.

It is required to study how each of these error components influences the landmark azimuth accuracy.

SIGHTING AXIS ROTATIONS IN THE HORIZONTAL FRAME

In order to study how the instrumental error affects the accuracy of landmark astronomical azimuth, it is needed to mathematically describe the motion of the sighting axis in horizontal frame ONWZ when turning about AD and VD rotation axes.

Rotation about AD rotation axis by angle φАD in horizontal frame ONWZ is described by the following matrix [25]:

$${{C}_{{{{\mu {\rm A}}}}}}({{{{\varphi }}}_{{{\text{AD}}}}}) = \left[ {\begin{array}{*{20}{c}} {\cos ({{{{\varphi }}}_{{{\text{AD}}}}}) + {{B}_{{\text{A}}}}x_{{\text{A}}}^{2}}&{{{B}_{{\text{A}}}}{{x}_{{\text{A}}}}{{y}_{{\text{A}}}} - \sin ({{{{\varphi }}}_{{{\text{AD}}}}}){{z}_{{\text{A}}}}}&{{{B}_{{\text{A}}}}{{x}_{{\text{A}}}}{{z}_{{\text{A}}}} + \sin ({{{{\varphi }}}_{{{\text{AD}}}}}){{y}_{{\text{A}}}}} \\ {{{B}_{{\text{A}}}}{{y}_{{\text{A}}}}{{x}_{{\text{A}}}} + \sin ({{{{\varphi }}}_{{{\text{AD}}}}}){{z}_{{\text{A}}}}}&{\cos ({{{{\varphi }}}_{{{\text{AD}}}}}) + {{B}_{{\text{A}}}}y_{{\text{A}}}^{2}}&{{{B}_{{\text{A}}}}{{y}_{{\text{A}}}}{{z}_{{\text{A}}}} - \sin ({{{{\varphi }}}_{{{\text{AD}}}}}){{x}_{{\text{A}}}}} \\ {{{B}_{{\text{A}}}}{{z}_{{\text{A}}}}{{x}_{{\text{A}}}} - \sin ({{{{\varphi }}}_{{{\text{AD}}}}}){{y}_{{\text{A}}}}}&{{{B}_{{\text{A}}}}{{z}_{{\text{A}}}}{{y}_{{\text{A}}}} + \sin ({{{{\varphi }}}_{{{\text{AD}}}}}){{x}_{{\text{A}}}}}&{\cos ({{{{\varphi }}}_{{{\text{AD}}}}}) + {{B}_{{\text{A}}}}z_{{\text{A}}}^{2}} \end{array}} \right],$$
(9)

where \({{B}_{{\text{A}}}} = 1 - \cos ({{{{\varphi }}}_{{{\text{AD}}}}})\); xA, yA, zA are the Cartesian coordinates of vector OA (AD rotation axis) in ONWZ frame.

Coordinates xA, yA, zA are calculated as follows:

$$\left[ \begin{gathered} {{x}_{{\text{A}}}} \hfill \\ {{y}_{{\text{A}}}} \hfill \\ {{z}_{{\text{A}}}} \hfill \\ \end{gathered} \right] = C_{{{{\Delta }}{{{{\theta }}}_{{{\text{XA}}}}}}}^{*}C_{{{{{{\theta }}}_{{{\text{I}}{{{\text{S}}}_{{\text{X}}}}}}}}}^{*}C_{{{{\Delta }}{{{{\psi }}}_{{{\text{YA}}}}}}}^{*}{\text{C}}_{{{{{{\psi }}}_{{{\text{I}}{{{\text{S}}}_{{\text{Y}}}}}}}}}^{*}\left[ \begin{gathered} {{x}_{{{\text{A0}}}}} \hfill \\ {{y}_{{{\text{A0}}}}} \hfill \\ {{z}_{{{\text{A0}}}}} \hfill \\ \end{gathered} \right],$$
(10)

where xA0 = 0, yA0 = 0, zA0 = 1 are the coordinates of ideal vector ОА; \(C_{{{{{{\theta }}}_{{{\text{I}}{{{\text{S}}}_{{\text{X}}}}}}}}}^{*}\), \(C_{{{{\psi }_{{{\text{I}}{{{\text{S}}}_{{\text{Y}}}}}}}}}^{*}\) are the rotation matrices for AAS tilt with respect to horizon plane \({{{{\theta }}}_{{{\text{I}}{{{\text{S}}}_{{\text{X}}}}}}}\), \({{\psi }_{{{\text{I}}{{{\text{S}}}_{{\text{Y}}}}}}}\):

$$C_{{{{{{\theta }}}_{{{\text{I}}{{{\text{S}}}_{{\text{X}}}}}}}}}^{*} = {{C}_{{{{{{\theta }}}_{{{\text{I}}{{{\text{S}}}_{{\text{X}}}}}}}}}}{{C}_{{\Delta {{{{\varphi }}}_{{{\text{SX}}}}}}}},\,\,\,C_{{{{{{\psi }}}_{{{\text{I}}{{{\text{S}}}_{{\text{Y}}}}}}}}}^{*} = {{C}_{{{{{{\psi }}}_{{{\text{I}}{{{\text{S}}}_{{\text{Y}}}}}}}}}}{{C}_{{\Delta {{{{\varphi }}}_{{{\text{XY}}}}}}}},$$
(11)

where

$${{C}_{{\Delta {{{{\varphi }}}_{{{\text{SX}}}}}}}} = \left[ {\begin{array}{*{20}{c}} {\cos (\Delta {{\varphi }}_{{{\text{SX}}}}^{{}})}&{ - \sin (\Delta {{\varphi }}_{{{\text{SX}}}}^{{}})}&0 \\ {\sin (\Delta {{\varphi }}_{{{\text{SX}}}}^{{}})}&{\cos (\Delta {{\varphi }}_{{{\text{SX}}}}^{{}})}&0 \\ 0&0&1 \end{array}} \right],\,\,\,\,{{C}_{{\Delta {{{{\varphi }}}_{{{\text{XY}}}}}}}} = \left[ {\begin{array}{*{20}{c}} {\cos (\Delta {{\varphi }}_{{{\text{XY}}}}^{{}})}&{ - \sin (\Delta {{\varphi }}_{{{\text{XY}}}}^{{}})}&0 \\ {\sin (\Delta {{\varphi }}_{{{\text{XY}}}}^{{}})}&{\cos (\Delta {{\varphi }}_{{{\text{XY}}}}^{{}})}&0 \\ 0&0&1 \end{array}} \right];$$
(12)
$${{C}_{{{{{{\theta }}}_{{{\text{I}}{{{\text{S}}}_{{\text{X}}}}}}}}}} = \left[ {\begin{array}{*{20}{c}} {\cos ({{\theta }}_{{{\text{I}}{{{\text{S}}}_{{\text{X}}}}}}^{{}})}&0&{ - \sin ({{\theta }}_{{{\text{I}}{{{\text{S}}}_{{\text{X}}}}}}^{{}})} \\ 0&1&0 \\ {\sin ({{\theta }}_{{{\text{I}}{{{\text{S}}}_{{\text{X}}}}}}^{{}})}&0&{\cos ({{\theta }}_{{{\text{I}}{{{\text{S}}}_{{\text{X}}}}}}^{{}})} \end{array}} \right],\,\,\,\,{{C}_{{{{\psi }_{{{\text{I}}{{{\text{S}}}_{{\text{Y}}}}}}}}}} = \left[ {\begin{array}{*{20}{c}} 1&0&0 \\ 0&{\cos (\psi _{{{\text{I}}{{{\text{S}}}_{{\text{Y}}}}}}^{{}})}&{ - \sin (\psi _{{{\text{I}}{{{\text{S}}}_{{\text{Y}}}}}}^{{}})} \\ 0&{\sin (\psi _{{{\text{I}}{{{\text{S}}}_{{\text{Y}}}}}}^{{}})}&{\cos (\psi _{{{\text{I}}{{{\text{S}}}_{{\text{Y}}}}}}^{{}})} \end{array}} \right];$$
(13)

\(C_{{{{\Delta }}{{{{\theta }}}_{{{\text{XA}}}}}}}^{*}\), \(C_{{{{\Delta }}{{{{\psi }}}_{{{\text{YA}}}}}}}^{*}\) are the rotation matrices for the angular deflections of the AAS body axes from their ideal positions, corresponding to technological errors ΔθXA, ΔψYA:

$$C_{{\Delta {{{{\theta }}}_{{{\text{XA}}}}}}}^{*} = {{C}_{{\Delta {{{{\theta }}}_{{{\text{XA}}}}}}}}{{C}_{{\Delta {{{{\varphi }}}_{{{\text{SX}}}}}}}},\,\,\,\,C_{{{{\Delta }}{{\psi }_{{{\text{YA}}}}}}}^{*} = {{C}_{{{{\Delta }}{{\psi }_{{{\text{YA}}}}}}}}{{C}_{{\Delta {{{{\varphi }}}_{{{\text{XY}}}}}}}},$$
(14)

where

$${{C}_{{\Delta {{{{\theta }}}_{{{\text{XA}}}}}}}} = \left[ {\begin{array}{*{20}{c}} {\cos (\Delta {{\theta }}_{{{\text{XA}}}}^{{}})}&0&{ - \sin (\Delta {{\theta }}_{{{\text{XA}}}}^{{}})} \\ 0&1&0 \\ {\sin (\Delta {{\theta }}_{{{\text{XA}}}}^{{}})}&0&{\cos (\Delta {{\theta }}_{{{\text{XA}}}}^{{}})} \end{array}} \right],\begin{array}{*{20}{c}} {} \end{array}{{C}_{{\Delta {{{{\psi }}}_{{{\text{YA}}}}}}}} = \left[ {\begin{array}{*{20}{c}} 1&0&0 \\ 0&{\cos (\Delta \psi _{{{\text{YA}}}}^{{}})}&{ - \sin (\Delta \psi _{{{\text{YA}}}}^{{}})} \\ 0&{\sin (\Delta \psi _{{{\text{YA}}}}^{{}})}&{\cos (\Delta \psi _{{{\text{YA}}}}^{{}})} \end{array}} \right].$$
(15)

The similar matrix CμV(θVD) describes the rotation of vector O'V – which has coordinates xV0 = 0, yV0 = 1, zV0 = 0 in the ideal position – about the VD rotation axis by angle θVD. Apart from the mentioned angles \({{{{\theta }}}_{{{\text{I}}{{{\text{S}}}_{{\text{X}}}}}}}\), \({{{{\psi }}}_{{{\text{I}}{{{\text{S}}}_{{\text{Y}}}}}}}\), ΔθXA, ΔψYA, the orientation of VD rotation axis also depends on the rotation about AD rotation axis described by the matrix CμA(φАD), and angles ΔφSV, ΔψVA described by the matrices

$$\begin{gathered} {{C}_{{{{\Delta }}{{{{\varphi }}}_{{{\text{SV}}}}}}}} = \left[ {\begin{array}{*{20}{c}} {{{\cos(\Delta \varphi }}_{{{\text{SV}}}}^{{}}{\text{)}}}&{ - {{\sin(\Delta \varphi }}_{{{\text{SV}}}}^{{}}{\text{)}}}&{\text{0}} \\ {{{\sin(\Delta \varphi }}_{{{\text{SV}}}}^{{}}{\text{)}}}&{{{\cos(\Delta \varphi }}_{{{\text{SV}}}}^{{}}{\text{)}}}&{\text{0}} \\ {\text{0}}&{\text{0}}&{\text{1}} \end{array}} \right], \\ {{C}_{{\Delta {{\psi }_{{{\text{VA}}}}}}}} = \left[ {\begin{array}{*{20}{c}} {\text{1}}&{\text{0}}&{\text{0}} \\ {\text{0}}&{{{\cos(\Delta }}\psi _{{{\text{VA}}}}^{{}}{\text{)}}}&{ - {{\sin(\Delta }}\psi _{{{\text{VA}}}}^{{}}{\text{)}}} \\ {\text{0}}&{{{\sin(\Delta }}\psi _{{{\text{VA}}}}^{{}}{\text{)}}}&{{{\cos(\Delta }}\psi _{{{\text{VA}}}}^{{}}{\text{)}}} \end{array}} \right]. \\ \end{gathered} $$
(16)

Then coordinates xV, yA, zV are calculated as follows:

$$\begin{gathered} \left[ \begin{gathered} {{x}_{{\text{V}}}} \hfill \\ {{y}_{{\text{V}}}} \hfill \\ {{z}_{{\text{V}}}} \hfill \\ \end{gathered} \right] \\ = \,\,{{C}_{{{{\mu A}}}}}({{{{\varphi }}}_{{{\text{AD}}}}}){{C}_{{{{\Delta }}{{{{\varphi }}}_{{{\text{SV}}}}}}}}{{C}_{{{{\Delta }}{{{{\psi }}}_{{{\text{VA}}}}}}}}C_{{{{\Delta }}{{{{\theta }}}_{{{\text{XA}}}}}}}^{*}C_{{{{{{\theta }}}_{{{\text{I}}{{{\text{S}}}_{{\text{X}}}}}}}}}^{*}C_{{{{\Delta }}{{{{\psi }}}_{{{\text{YA}}}}}}}^{*}C_{{{{{{\psi }}}_{{{\text{I}}{{{\text{S}}}_{{\text{Y}}}}}}}}}^{*}\left[ \begin{gathered} {{x}_{{{\text{V0}}}}} \hfill \\ {{y}_{{{\text{V0}}}}} \hfill \\ {{z}_{{{\text{V0}}}}} \hfill \\ \end{gathered} \right]. \\ \end{gathered} $$
(17)

Therefore, rotation of the sighting axis about AD rotation axis is described by the formula

$$\left[ \begin{gathered} x_{{{\text{SA}}}}^{{\text{R}}} \hfill \\ y_{{{\text{SA}}}}^{{\text{R}}} \hfill \\ z_{{{\text{SA}}}}^{{\text{R}}} \hfill \\ \end{gathered} \right] = {{C}_{{\mu {\text{A}}}}}({{{{\varphi }}}_{{{\text{AD}}}}})\left[ \begin{gathered} {{x}_{{{\text{SA}}}}} \hfill \\ {{y}_{{{\text{SA}}}}} \hfill \\ {{z}_{{{\text{SA}}}}} \hfill \\ \end{gathered} \right],$$
(18)

and about VD rotation axis, by the formula

$$\left[ \begin{gathered} x_{{{\text{SA}}}}^{{\text{R}}} \hfill \\ y_{{{\text{SA}}}}^{{\text{R}}} \hfill \\ z_{{{\text{SA}}}}^{{\text{R}}} \hfill \\ \end{gathered} \right] = {{C}_{{\mu {\text{V}}}}}({{{{\theta }}}_{{{\text{VD}}}}})\left[ \begin{gathered} {{x}_{{{\text{SA}}}}} \hfill \\ {{y}_{{{\text{SA}}}}} \hfill \\ {{z}_{{{\text{SA}}}}} \hfill \\ \end{gathered} \right],$$
(19)

where xSA, ySA, zSA are the initial prerotation Cartesian coordinates of the sighting axis in horizontal frame; \(x_{{{\text{SA}}}}^{{\text{R}}},y_{{{\text{SA}}}}^{{\text{R}}},z_{{{\text{SA}}}}^{{\text{R}}}\) are the postrotation Cartesian coordinates of the sighting axis in the horizontal frame.

SIMULATION

The systematic components of the instrumental error ΔφSX, ΔφXY, ΔθXA, ΔψYA, ΔφSV, ΔψVA were set within the range 60–360 arcsec. The random error components were simulated by processes with Gaussian distribution, zero mean values and RMS deviations \({{{{\sigma }}}_{{{\text{I}}{{{\text{S}}}_{{\text{X}}}}}}}\) = 0.2 arcsec, \({{{{\sigma }}}_{{{\text{I}}{{{\text{S}}}_{{\text{Y}}}}}}}\)= 0.2 arcsec, σVD = 60 arcsec, and σAD= 0.2 arcsec. The astronomical observations were simulated for the star sighting at h0 = 40°, which is the optimal altitude. At higher altitude, the error in astronomical azimuth of the sighting axis grows, and at lower altitude, the effect of lateral refraction increases [10].

The simulation was performed as follows:

(1) the vectors corresponding to the body axes in ideal orientation (without instrumental errors) as shown in Fig. 2a were generated in the horizontal frame;

(2) instrumental errors ΔφSX, ΔφXY, ΔθXA, ΔψYA, ΔφSV, ΔψVA were set and the vectors corresponding to AD and VD rotation axes were generated using (9)–(17), then the tilt angles with respect to the horizon plane \({{{{\theta }}}_{{{\text{I}}{{{\text{S}}}_{{\text{X}}}}}}}\), \({{\psi }_{{{\text{I}}{{{\text{S}}}_{{\text{Y}}}}}}}\) were set as random processes with zero mean values and RMS deviations \({{{{\sigma }}}_{{{\text{I}}{{{\text{S}}}_{{\text{X}}}}}}}\),\({{{{\sigma }}}_{{{\text{I}}{{{\text{S}}}_{{\text{Y}}}}}}}\);

(3) using (19), the AAS sighting axis О'S was turned to the position for observing the near-meridian stars by rotating the vector О'S about VD rotation axis through the angle θVD = h0 + Δθ, where Δθ is the error in VD rotation angle simulated as a random process with zero mean and RMS deviation σVD. After the sighting axis rotation about VD axis, its azimuth and altitude were calculated by the formulas

$$\begin{gathered} {\text{ }}A_{{{\text{SA}}}}^{{}} = \arcsin \left( {\frac{{y_{{{\text{SA}}}}^{{\text{R}}}}}{{\sqrt {{{{(x_{{{\text{SA}}}}^{{\text{R}}})}}^{2}} + {{{(y_{{{\text{SA}}}}^{{\text{R}}})}}^{2}}} }}} \right);{\text{ }} \\ {{h}_{{{\text{SA}}}}} = \arcsin \left( {\frac{{z_{{{\text{SA}}}}^{{\text{R}}}}}{{\sqrt {{{{(x_{{{\text{SA}}}}^{{\text{R}}})}}^{2}} + {{{(y_{{{\text{SA}}}}^{{\text{R}}})}}^{2}} + {{{(z_{{{\text{SA}}}}^{{\text{R}}})}}^{2}}} }}} \right), \\ \end{gathered} $$
(20)

(4) the readings of AD angle sensor φAD and the sighting axis azimuth during stellar observations \(A_{{{\text{SA}}}}^{{\text{I}}}\)were recorded, with AD angle sensor error simulated as a random process with zero mean and RMS deviation σAD;

(5) true horizontal coordinates of the landmark (azimuth AL ∈ [0; 90]° and altitude hL = 0°) were set and the sighting axis pointing at the landmark was simulated: the sighting axis was rotated about AD and VD axes so that its azimuth and altitude calculated by formulas (20) coincide with AL and hL. The axis was rotated about AD and VD axes using the formulas (18) and (19);

(6) after pointing at the landmark the data of AD angle sensor were calculated, and the horizontal rotation angle γI was determined by the difference between the output during stellar observation and during the landmark observation. During the landmark observation, the error of AD angle sensor σAD was simulated;

(7) the landmark azimuth \(A_{{\text{L}}}^{{\text{I}}}\) was calculated using (1);

(8) reversal of the sighting axis was simulated: AAS body axes were successively rotated about AD rotation axis using the rotation matrix CμA by 180°, and then about the VD rotation axis using the matrix CμV by 180°;

(9) the sighting axis was again set to the meridian plane, the steps 3–7 were repeated, and the sighting axis azimuth during stellar observations \(A_{{{\text{SA}}}}^{{{\text{II}}}}\), horizontal angle γII and azimuth of the landmark \(A_{{\text{L}}}^{{{\text{II}}}}\) after reversal were determined;

(10) the landmark azimuth \(A_{{\text{L}}}^{{{\text{calc}}}}\) was calculated using (2), then the azimuth error was estimated by the formula

$$\Delta A_{{\text{L}}}^{{}} = {{A}_{{\text{L}}}} - A_{{\text{L}}}^{{{\text{calc}}}}.$$
(21)

As a result of simulation, the error array \(\Delta A_{{\text{L}}}^{{}}\) was formed comprising NK = 1000 elements, and its mean value \(\Delta _{{{{A}_{{\text{L}}}}}}^{{{\text{syst}}}}\) characterizing the systematic component of landmark azimuth error and RMS deviation \({{\sigma }_{{{{A}_{{\text{L}}}}}}}\) characterizing the random error component were estimated.

SIMULATION RESULTS

The simulation was performed in Matlab environment.

Table 1 presents the simulated total effect of the technological manufacturing and assembly errors ΔφSX, ΔφXY, ΔθXA, ΔψYA, ΔφSV, ΔψVA on the systematic component of landmark azimuth error \(\Delta _{{{{A}_{{\text{L}}}}}}^{{{\text{syst}}}}\). The maximum errors \(\Delta _{{{{A}_{{\text{L}}}}}}^{{{\text{syst}}}}\) were observed for the case when the angular distance between the landmark and the meridian plane was AL = 90°. The random components of the instrumental error were not specified.

Table 1.   Systematic components of the instrumental error and their effects
Full size table

The dependency of the landmark azimuth error on the angular distance between the landmark and the meridian plane is shown in Fig. 3.

Fig. 3.
figure 3

Landmark azimuth error vs. angular distance between the landmark and the meridian plane.

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Figure 4 presents the estimated effect produced by the individual components of the instrumental error and their combinations on the landmark azimuth accuracy as a percentage of the resultant \(\Delta _{{{{A}_{{\text{L}}}}}}^{{{\text{syst}}}}\) depending on the angular distance between the landmark and the meridian plane.

Fig. 4.
figure 4

Effect produced by the components of the instrumental error and their combinations as a percentage of the resultant error.

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Table 2 presents the results of simulating the random components of the instrumental error: \({{{{\sigma }}}_{{{\text{I}}{{{\text{S}}}_{{\text{X}}}}}}}\), \({{{{\sigma }}}_{{{\text{I}}{{{\text{S}}}_{{\text{Y}}}}}}}\), σVD and σAD.

Table 2. Random components of the instrumental error and their effects
Full size table

The following conclusions can be drawn from the simulation:

(1) the error systematic component \(\Delta _{{{{A}_{{\text{L}}}}}}^{{{\text{syst}}}}\) depends on the angular distance between the landmark and the meridian plane and reaches maximum values when the angular distance is 90° (Fig. 3), with the astronomical observations conducted both north and south of zenith, when the azimuth ASA = 180°;

(2) the main contribution to the error systematic component \(\Delta _{{{{A}_{{\text{L}}}}}}^{{{\text{syst}}}}\) is made by the components of the instrumental error ΔθXA, ΔψYA characterizing nonorthogonality of IS axes and AD rotation axis. The instrumental error components ΔφSX, ΔφXY, ΔφSV, ΔψVA produce a negligibly small effect (Fig. 4);

(3) IS error about axis X does not have a serious effect on the landmark azimuth error \({{{{\sigma }}}_{{{{A}_{{\text{L}}}}}}}\), and the effect of IS error about axis Y can be expressed as \({{{{\sigma }}}_{{{\text{I}}{{{\text{S}}}_{{\text{Y}}}}}}}\tan ({{h}_{{{\text{SA}}}}}){{\sqrt 2 } \mathord{\left/ {\vphantom {{\sqrt 2 } 2}} \right. \kern-0em} 2}\);

(4) the error of VD angle sensor σVD does not influence the landmark azimuth error \({{{{\sigma }}}_{{{{A}_{{\text{L}}}}}}}\), and AD angle sensor error σАD is completely included into \({{{{\sigma }}}_{{{{A}_{{\text{L}}}}}}}\);

(5) the resultant random error component \({{{{\sigma }}}_{{{{A}_{{\text{L}}}}}}}\)conditioned by the errors of AD and IS angle sensors can be expressed as

$${{{{\sigma }}}_{{{{A}_{{\text{L}}}}}}} = \sqrt {{{\sigma }}_{{{\text{AD}}}}^{2} + \frac{{{{\sigma }}_{{{\text{I}}{{{\text{S}}}_{{\text{Y}}}}}}^{2}{\text{tan}}{{{({{h}_{{{\text{SA}}}}})}}^{2}}}}{2}} .$$
(22)

CONCLUSIONS

The paper presents the results of simulating the effect produced by the instrumental error of the automated astronomical system on the accuracy of landmark azimuth determination. The results have demonstrated that the random component of azimuth error is mainly affected by the errors of the inclination sensor and angle sensor of the azimuth drive, and the systematic component is mostly affected by the vertical angles characterizing nonorthogonality of the inclination sensor axes and azimuth drive rotation axis. It is planned to further apply the obtained results to construct the mathematical model of the system error and form the requirements for mutual orientation of the system axes when designing the system.