| Citation: |
Li, H. B., Peng, R. F., Shu, C. G., et al. 2024. Analysis of optical axis deviation caused by structural stiffness in equatorial telescopes. Astronomical Techniques and Instruments, 1(4): 203−210. https://doi.org/10.61977/ati2024029.
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The impact of structural stiffness on optical axis deviation poses a significant challenge in the design of equatorial telescope structures. A comprehensive analysis during the design process can reduce the reliance of a telescope on advanced control technologies, thereby improving its economic feasibility. Although full-system finite element analyses are reliable, they are encumbered by significant time requirements and limitations in covering all possible telescope orientations. Therefore, we propose an efficient and comprehensive analytical method to evaluate the optical axis deviation of equatorial telescopes across a full range of angles. To address the challenge of ensuring that the analysis covers all possible positions of an equatorial telescope, based on a model from SiTian project, we analyze the optical axis deviations caused by the fork arm at 25 different angles and then use fitting methods to obtain results for all angles. Based on the analysis results of the optical axis deviation caused by the stiffness of the optical tube in the horizontal position, we derive the results for the tube at any position using geometric relationships. Finally, we calculate the coupling factors and combine these impacts. Furthermore, we identify six discrete feature points to reflect possible telescope orientations and conduct comprehensive finite element analyses. The results are in alignment with those acquired through a comprehensive computational approach.
Telescope mounting structures predominantly use equatorial and altazimuth configurations. Equatorial telescopes are based on the equatorial coordinate system, so the adjustment process is synchronized with the rotation of the Earth, facilitating direct celestial tracking via right ascension axis rotation. However, the cantilever beam inherent in these telescopes leads to notable structural deformation. Conversely, altazimuth telescopes, despite their simpler design and mechanical superiority, contend with a zenith blind spot and challenges related to tracking errors and field rotation[1, 2]. Given the inherent advantage of equatorial telescopes in coordinate system alignment, addressing the deformation issue will significantly augment their utility in tracking celestial bodies.
Pointing accuracy is a crucial metric for assessing the performance of telescopes. The higher the pointing accuracy, the more rapidly and precisely a telescope can lock onto targets, enhancing observational efficiency; this can be affected by various errors arising during the manufacturing, assembly, installation, and operational phases of a telescope[3]. These sources of error include deformation due to weight, assembly errors, machining errors, encoder errors, thermal distortion, and dynamic errors[4]. These errors can be addressed through subsequent measurement techniques and automation control technologies. While high-performance telescope control systems can compensate for some design deficiencies, an over-reliance on them may significantly increase overall costs. Therefore, comprehensive analysis during the design process is particularly important to not only evaluate the quality of the structural design but also to provide reference data for subsequent control[5].
Full-system Finite Element Analysis (FEA) is an existing method for evaluating the structural performance of telescopes[6]. However, this approach is constrained to evaluating a singular position per analysis cycle, demanding substantial computational effort and time. To rapidly obtain the optical axis deviations caused by the structural stiffness of the telescope at any combination of right ascension and declination angles, an efficient and accurate method must be found to replace full-system FEA. In response to the significant optical axis deviation caused by structural deformation in equatorial telescope mounts, we propose a comprehensive method that combines FEA with theoretical calculations to quickly obtain optical axis deviations due to structural stiffness at any angle of rotation of the right ascension and declination axes.
SiTian is a ground-based all-sky optical monitoring project developed by the Chinese Academy of Sciences. The concept is an integrated network of dozens of 1-meter-class telescopes deployed partly in China and partly at various other sites around the world[7]. Fig. 1A shows the zero position, and Fig. 1B shows the declination axis positioned at −25° for a SiTian equatorial telescope. From the zero position, the right ascension axis spans a range of ±180°, while the declination axis covers −25° to +205°. Throughout the attitude adjustment procedure, axial pointing deviation shows variability in correspondence with the rotation of the right ascension and declination axes. This is attributable to differential structural stiffness in different orientations.
Optical axial deviations attributable to structural stiffness are caused by deformations within the base, fork arm, and the optical tube itself. To analyze this, we begin by examining the optical axis deviation caused by the stiffness of the base before investigating the optical axis deviation caused by the directional stiffness of the fork arm. Next, we analyze the optical axis deviation caused by the stiffness of the telescope tube. Then, we combine the amounts of optical axis deviation caused by the stiffness of the telescope tube, fork arm, and base to evaluate the fluctuation in pointing. Finally, based on the regional distribution of combinations of right ascension and declination angles, we set six discrete feature points for full-system FEA, compare the results with the comprehensive analysis method, and demonstrate the reliability of the comprehensive analysis method.
Fig. 2 shows a schematic of the geometric relationships in an equatorial telescope. Here, θ=51.4∘, B is the geometric center of the fork arm base, AB represents the right ascension axis, and the line perpendicular to the plane of the paper at point D represents the initial position of the declination axis. The initial position of the geometric center of the charge-coupled device (CCD) receiving surface is C0. As the telescope orientation changes, C0 is constantly moving on a sphere with a center D and a radius R. After rotating by an angle β (where −180° ≤ β ≤ 180°) around the right ascension axis, C0 moves to C1. Then, by rotating an angle γ (where −25° ≤ γ ≤ 205°) around the declination axis, C1 moves to C2. Point E is at the surface of the sphere directly below point D, in the direction of gravity. Subsequent calculations will be based on the following coordinate system: D as the origin, \boldsymbol {D{{\boldsymbol{C}}_0}} as the positive direction of the Y-axis, \boldsymbol {AD} as the positive direction of the Z-axis, and take the direction perpendicular to the plane ADE, with the inward direction as the positive X-axis.
The coordinates of three points to be used for subsequent calculations are D(0,0,0) , E(0, - R\sin \theta , - R\cos \theta ) , and {C_2}(R\cos \gamma \sin \beta ,R\cos \gamma \cos \beta ,R\sin \gamma ) .
Theoretically, the right ascension axis spans ±180° and the declination axis extends from −25° to +205° from the zero position. However, in practice, the telescope can only operate when the optical axis is oriented toward an upward angle. Consequently, delineating the effective travel range of the telescope is essential. In Fig. 2, \boldsymbol {ED} is perpendicular to the horizontal plane, and the telescope is in the effective travel range when the angle ω between \boldsymbol {D{{\boldsymbol{C}}_2}} and the horizontal plane is positive, where
| \boldsymbol {D{{\boldsymbol{C}}_2}} = (R\cos \gamma \sin \beta ,\;R\cos \gamma \cos \beta ,\;R\sin \gamma ) , | (1) |
and
| \boldsymbol {ED} = (0,\;R\sin \theta ,\;R\cos \theta ) . | (2) |
The sine of angle ω, defined between \boldsymbol {D{{\boldsymbol{C}}_2}} and the horizontal plane, is calculated as
| \sin \omega = \cos \langle \boldsymbol {D{{\boldsymbol{C}}_2}} ,\boldsymbol {ED} \rangle = \sin \theta \cos \gamma \cos \beta + \sin \gamma \cos \theta . | (3) |
With respect to this, Fig. 3 shows the correlation between the optical axis angle relative to the horizontal plane and the rotational angles of the right ascension and declination axes. The segment exceeding 0° in the figure signifies the effective travel range of the telescope.
Deformation of the base can lead to minor displacements in the upper structure, thereby causing deviation in the optical axis which is independent of the rotation angles of the right ascension and declination axes. We use FEA to calculate this deviation as shown in Fig. 4, in which the hidden upper structure of the fork arm is equivalently considered as a point mass of
To calculate the optical axis deviation at various angles, we attach a probe of negligible mass to the right ascension axis, with length L1 = 2 000 mm. Results (shown in Fig. 5) indicate that the total displacements at the upper and lower endpoints of the probe are Dmax =
| {\varepsilon _1} = 3600 \times \arcsin \frac{{{D_{\max }} + {D_{\min }}}}{{{L_1}}} = 3.001'' . | (4) |
Upon rotation of the right ascension axis, the fork arm experiences varying degrees of deformation attributable to the differential stiffness across directions. Consequently, it becomes imperative to assess the optical axis deviation induced by the fork arm's stiffness. As shown in Fig. 6A, the upper structure of the fork arm can be approximated as a point mass of
Considering real conditions, we apply fixed constraints to the connection point between the fork arm and the right ascension axis. To calculate the optical axis deviation under multiple directions of gravity, for ease of adjusting the coordinate system, we apply equivalent gravity using "acceleration". Fig. 7 shows the results of the FEA, indicating that the displacement of the upper and lower endpoints of the probe are D2max =
| {\varepsilon _2} = 3600 \times \arcsin \frac{{{D_{2\max }} + {D_{2\min }}}}{{{L_2}}} = 21.493'' . | (5) |
The analysis and calculations above only yield the optical axis deviation caused by the stiffness of the fork arm when the right ascension axis is rotated by 0°. To obtain the optical axis deviation caused by the stiffness of the fork arm at any angle of rotation of the right ascension axis, we select 25 characteristic points and analyze the deviation corresponding to each characteristic point by changing the direction of gravity. The relationship between the optical axis deviation caused by the fork arm and the rotation angle of the right ascension axis is shown in Fig. 8.
Because the telescope tube distorts under the influence of gravity, it is necessary to analyze the optical axis deviation caused by the optical tube's stiffness. The telescope employs a Schmidt catadioptric optical system, and its optical path is shown in Fig. 9[8].
The Schmidt catadioptric optical system consists of a Schmidt plate, a primary mirror, and a corrector lens assembly. The Schmidt plate is an aspheric surface of higher order with no focal power, the primary mirror is a spherical reflector, and the corrector lens assembly comprises three lenses. The optical axis of the system is determined by the tilt angle of the primary mirror, for the following reasons:
(1) When there is a relative positional change in the optical components of the system, the aperture stop remains on the primary mirror; therefore, the optical axis is determined by the line connecting the vertex of the spherical reflector to its center of curvature.
(2) Optical analysis suggests that the Schmidt plate and filters approximate flat glass plates. Their positional errors have a negligible effect on the optical axis. The effect of the lens assembly tilting by less than ±15" relative to the optical axis can be disregarded.
To evaluate the amount of positional deviation of the primary mirror and lens assembly, we perform an FEA on the optical tube, as shown in Fig. 10.
Fig. 11A shows the axial displacement of the lens assembly. The diameter of the lens assembly, L, equals 400 mm. The upper displacement direction is in the positive Z-axis direction with a displacement value of du, equals
| \varphi_1=3600\times\arcsin\frac{d_u+d_l}{L}=1.404''. | (6) |
The result \varphi_1 fall within the permissible optical error range and are consequently not considered in subsequent calculations, which focus only on the tilt angle of the primary mirror. During the rotation of the right ascension and declination axes, the magnitude of the radial component of gravity acting on the primary mirror is constantly changing, so the tilt angle of the primary mirror also varies accordingly. Given the minor deformation of the optical tube, changes in stiffness during the deformation process are negligible. According to linear elasticity theory, the displacement and tilt angle of the primary mirror are linearly related to the applied radial force[9]. Based on this principle, we need only measure the tilt angle of the primary mirror when gravity is radial, and data under other conditions can be calculated using the linear relationship.
Fig. 11B shows the axial displacement of the primary reflector. The diameter of the primary mirror, Lʹ, equals
| \varphi_2=3\;600\times\arcsin\frac{d_u'+d_l'}{L'}=2.62''. | (7) |
After quantifying the tilt angle of the primary mirror under radial gravity, we can calculate the relationship between the radial displacement and tilt angle of primary mirror and the rotation angles of the right ascension and declination axes based on geometric relationships, according to linear elasticity theory. As shown in Fig. 2, the direction of gravity is \boldsymbol{DE} , and the axis of the optical tube is \boldsymbol{DC_{\mathrm{2}}} , where
| \boldsymbol{DE}=(0,\ -R\sin\theta,\ -R\cos\theta), | (8) |
and
| \boldsymbol{D\boldsymbol{C}_{\mathrm{2}}}=(R\cos\gamma\sin\beta,\ R\cos\gamma\cos\beta,\ R\sin\gamma). | (9) |
If α is the acute angle between \boldsymbol{DE} and \boldsymbol{DC_{\mathrm{2}}} , then
| \cos\alpha=\left|\cos\theta\sin\gamma+\sin\theta\cos\gamma\cos\beta\right|. | (10) |
So, the tilt angle of the primary mirror \varepsilon_3 is given by
| \mathit{\mathit{\mathit{\mathrm{\mathit{\varepsilon}}}}\mathrm{_3}\mathit{\mathit{\mathit{ }}}}=\varphi_2\sin\alpha. | (11) |
Combining previous results, we calculate the relationship between the optical axis deviation caused by the stiffness of the optical tube (ε3) and the rotation angles of the right ascension and declination axes, as shown in Fig. 12. The red-to-blue color in this figure corresponds to a decreasing deviation of the optical axis caused by the optical tube.
Previous sections have calculated the optical axis deviations caused by base stiffness (ε1), fork arm stiffness (ε2), and optical tube stiffness (ε3) separately, and these deviations can all be combined. In Fig. 2, the optical axis rotation plane caused by the fork arm stiffness and base deformation is plane C0DB, and the optical axis rotation plane caused by optical tube deformation is plane C2DE. We need to merge the optical axis deviations along these two different planes into a total optical axis deviation.
The normal vector of plane C0DB is
| \boldsymbol{n_{\mathrm{2}}}=(1,\ 0,\ 0), | (12) |
and the normal vector of plane C2DE is
| \begin{split} \boldsymbol{n_{\mathrm{3}}}= & \boldsymbol{C\mathrm{_2}D}\times \boldsymbol{DE}=R^2(\mathrm{cos}\gamma\mathrm{cos}\beta\mathrm{cos}\theta-\mathrm{sin}\gamma\mathrm{sin}\theta, \\ &-\mathrm{cos}\gamma\mathrm{sin}\beta\mathrm{cos}\theta,\mathrm{\ cos}\gamma\mathrm{sin}\beta\mathrm{sin}\theta). \end{split} | (13) |
Therefore, the angle between the two planes is
| \begin{split} \cos \phi =& \cos \left\langle { {{{\boldsymbol{n}}_2}} , {{{\boldsymbol{n}}_3}} } \right\rangle =\\& \frac{{\cos \beta \cos \theta \cos \gamma - \sin \gamma \sin \theta }}{{\left| {\cos \gamma } \right|\sqrt {{{(\cos \beta \cos \theta - \tan \gamma \sin \theta )}^2} + {{(\sin \beta )}^2}} }} . \end{split} | (14) |
According to Rodrigues' rotation formula[10], the total optical axis deviation δ is given by
| \cos \delta = \cos {\varepsilon _3}\cos ({\varepsilon _2} + {\varepsilon _1}) + \sin {\varepsilon _3}\sin ({\varepsilon _2} + {\varepsilon _1})\cos \phi . | (15) |
Using equation (15), we calculate the extent of optical axis deviation, caused by structural stiffness, over various rotational angles of the right ascension and declination axes, as illustrated in Fig. 13. The red-to-blue color in this figure corresponds to a decreasing deviation of the optical axis caused by the overall structural stiffness.
Fig. 13 is created using numerous discrete points. Let n be the total number of points, with each point numbered from 1 to n. Xi represents the optical axis deviation of the ith point, which is determined by β and γ, and is reflected by the colors shown in Fig. 13. \overline X denotes the average of all X. XRMS represents the root mean square error of the optical axis deviations, attributable to structural stiffness, which is given by
| {X}_{{\mathrm{RMS}}}=\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^{n}{\left({X}_{i}-\overline{X}\right)}^{2}}}=1.410''. | (16) |
To verify the reliability of the results analyzed by the comprehensive calculation method, we conduct an FEA of the entire system and compare the results of both analyses. Given the significant computational demand inherent in executing precise FEA on the entire assembly, we only conduct FEA in six different positions.
Because the FEA method for each position of the entire system is similar, we only describe the analysis process for one position here, for brevity. Specifically, the verification is undertaken with the right ascension axis at −45° and the declination axis at 60°. In finite element model processing, structures that are inconvenient for modeling, such as counterweights, CCD, and motors, are treated with distributed mass for equivalence. To facilitate acquisition of the optical axis deviation caused by the full-system structural stiffness, we place a probe at the geometric center of the primary mirror, with the probe connected to its corresponding end face via a "fixed joint". The details for the FEA are shown in Fig. 14. We extract the tilt angle data for the primary mirror. Then, using the same method, we conduct six FEA analyses of the entire system and summarize the analysis results in Table 1.
| β=−135° γ=180° |
β=−90° γ=120° |
β=−45° γ=60° |
β=45° γ=60° |
β=90° γ=120° |
β=135° γ=180° |
|
| Results obtained from overall FEA/(") | 23.989 | 24.176 | 25.771 | 25.763 | 24.165 | 23.976 |
| Results obtained from the combined calculation method/(") |
23.015 | 24.479 | 25.299 | 25.299 | 24.479 | 23.015 |
| Relative error/(%) | 4.060 | 1.253 | 1.832 | 1.801 | 1.299 | 4.345 |
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Results obtained by the method proposed in this study have a relative error within 5% compared with those obtained by the whole machine FEA method. The possible reasons for the error are as follows:
(1) Limited number of fitting points
To obtain the relationship between the optical axis deviation caused by the fork arm and the rotation angle of the right ascension axis, we fitted only 25 measurement points. Increasing the number of fitting points can enhance computational accuracy.
(2) Simplifications in the complete finite element model
We simplify the upper structure of the fork arm to a point mass in this study. In the processing of the complete finite element model, we treat structures difficult to model, such as the counterweight, CCD, and motor, with equivalent distributed masses. Because of the position errors of the distributed masses, the center of gravity of the local structure may slightly deviate from being a point mass.
This study aims to find a method to replace full-system FEA for calculating optical axis deviations caused by structural stiffness in equatorial telescopes. This approach combines FEA with theoretical calculations and addresses the challenge of quickly obtaining optical axis deviations caused by structural stiffness at any rotation angle of the right ascension and declination axes through full-system FEA. Initially, we decompose the factors causing optical axis deviation and calculate the contribution mechanisms of each factor to optical axis deviation through geometric and mechanical relationships. Then, we combine these factors through geometric relationships and calculate the optical axis deviation caused by structural stiffness for arbitrary cases. Comparing the results obtained with this method to those from full-system FEA, with a relative error within 5%, validates the effectiveness of this method.
Overall, this study provides significant theoretical support and practical methods for the design and performance optimization of equatorial telescopes, especially in terms of enhancing pointing accuracy and observational efficiency. Future research can further explore the application of this method to different types of telescopes, as well as how to further optimize their structural design and control systems.
Shaoming Hu initiated the project. Haibo Li conceived the idea. Runfu Peng mainly wrote the manuscript and provided algorithm support. Chenggang Shu and Yong Bi supervised the project. All authors read and approved the final manuscript.
Haibo Li is the associate editors-in-chief and Yong Bi is editorial board member for Astronomical Techniques and Instruments. They were not involved in the editorial review or the decision to publish this article. The authors declare no competing interests.
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Figures(14) / Tables(1)
Supported by: Beijing Renhe Information Technology Co., Ltd. 
| β=−135° γ=180° |
β=−90° γ=120° |
β=−45° γ=60° |
β=45° γ=60° |
β=90° γ=120° |
β=135° γ=180° |
|
| Results obtained from overall FEA/(") | 23.989 | 24.176 | 25.771 | 25.763 | 24.165 | 23.976 |
| Results obtained from the combined calculation method/(") |
23.015 | 24.479 | 25.299 | 25.299 | 24.479 | 23.015 |
| Relative error/(%) | 4.060 | 1.253 | 1.832 | 1.801 | 1.299 | 4.345 |
DownLoad:
CSV
