Artificial satellites, which provide information about Earth or space regardless of weather or time, have been launched by many countries. Consequently space debris has increased rapidly, causing problems in space operations. To accurately observe these space objects and predict their movement paths, satellite-laser-ranging (SLR) systems, which measure the distance to a satellite using a laser, recently have been used in various fields such as artificial satellite operation, geophysics, space geodetics, and space surveillance [1, 2]. The National Aeronautics and Space Administration first used an SLR system to determine the orbit of the Beacon Explorer-B satellite [3]. In addition, laser-ranging systems have been developed by many countries [3]. For example, the Matera laser-ranging observatory (MLRO) located in Matera, Italy is one of the most advanced satellite and lunar laser-ranging facilities in the world [4]. Metrology and Optics (MeO) in France was designed in the framework of a laser-ranging system only for the moon [5]. These systems are based on an optical telescope with a diameter of about 1.5 m, a commonly used size. This SLR system consists of a laser, an optical-telescope assembly (OTA), a tracking mount, a transmitting and receiving optoelectronic unit, and an operation and control system [6]. To track space objects more precisely by transmitting and receiving laser light, a fast rotating driving speed is required for the transmitter and receiver on earth, including the optical system and other components. To this end, the development of a robust OTA structure is essential.

A large OTA generally has two aspherical mirrors that apply a Ritchey-Chretien optical system to correct aberrations and provide remarkable optical performance [7]. The performance of a large OTA is also determined by the size of the primary mirror (M1). As the size of the OTA increases, the performance of structures is more dependent on optical-axis misalignment due to gravity. Here the role of the OTA structure and mechanism is to maintain the collimation (i.e. alignment in all directions), even if the axis direction changes as it rotates along the object. To maintain optical alignment, it is crucial to minimize the deflection and tilt of the mirror and frame of the OTA, so that the gravitational deflections of the upper and lower frames of the OTA are balanced [8, 9]. Therefore, precise optomechanical design of the OTA structure is required to minimize optical-axis misalignment.

The design of the OTA structure generally entails a passive approach to decide on the shape and material of the structure. However, as the OTA becomes bulky and the demand for mass reduction increases, an active approach to adjusting the positions of optical objects in real time is required, to compensate for the effects of gravity [8]. The active approach also has limitations, such as the sensitivity of the sensor, the range of compensation due to optical and mechanical limitations, and errors in the control system. In other words, the passive approach is a crucial design method for gravity and thermal loads. The active approach is the last resort that could quickly compensate for precise adjustments to the displacement of the optical object. Therefore, it is recommended to distribute the structural design using these two methods in a complementary manner.

Despite the lack of a definitive OTA design in the literature, several studies on the analysis of gravity and wind deflection according to the truss structure shape of the OTA [10], as well as the design and examination of the optomechanical structure of the OTA, have been reported [11–14]. In addition, many researchers have studied the design and analysis of large OTAs [15–17], gravitational-deflection analysis depending on the geometry of the spider vane to which a secondary mirror (M2) assembly is mounted [18], and the overall design process for optomechanical structures [19, 20]. The authors of the aforementioned studies explained the design method for the final shape of the OTA, but it is necessary to focus on the performance according to the design parameters of each component, due to various design requirements and complexity.

The goal of this study is to establish an OTA-structure design procedure to minimize optical misalignment in the preliminary design stage of the optical telescope, based on a design parametric study. Based on this, we then provide a comprehensive and detailed framework for developing OTA structures, and obtain a database for basic design proposals for the OTA.

The remainder of this paper is organized as follows: Section 2 summarizes the OTA geometry. Section 3 describes a parametric study of the design parameters, followed by a description of the final design and work done to design the structure to minimize optical misalignment. Finally, we summarize the conclusions of this paper.

2.1. Structural Configuration

In general, truss structures have a simple connecting and manufacturing process, and cost and effort are also reduced. These advantages make truss frames attractive for designing OTA structures with diverse constraints (e.g. temperature, transport, assembly, and cost). Generally, the concept of the Serrurier truss shape using a simple A-type truss, first proposed by Serrurier in 1938, has been widely adopted [8]. In this design, the upper and lower struts resist tension and compression respectively, thereby maintaining the optical elements parallel to each other. We consider this concept for an OTA structure that can minimize displacement even if the axis direction changes as it rotates along the object. Because precise collimation is very difficult in this passive approach, a hexapod that controls M2 along six axes is adopted in the OTA structure in this study. The actuators of the hexapod finely tune the translation and tip-tilt of M2 according to the rotational position of the OTA, to maintain the collimation. As shown in Fig. 1(a), the OTA structure consists of frames (top, center, and bottom), truss struts (upper and lower), a spider vane supporting M2 in the top frame, and a mount supporting M1 in the bottom frame. Here, the masses of the 1.5-m M1 assembly and the 0.34-m M2 assembly with the hexapod are about 1.6 tons and 20 kg respectively. To make the optical path and assemble the tracking mount, the rotating shaft part of the center frame is also hollowed to a size of 340 mm. As shown in Fig. 1(b), the light falls on a concave M1, then is reflected towards a convex M2. Finally, it is reflected at 45 degrees from the flat tertiary mirror (M3) and exits to the coude path, which is an optical-design configuration that redirects the light path in a telescope. The design of the OTA structure is thus changed to approximate the design requirements based on the Serrurier concept.

Figure 1. Configuration of (a) the optical-telescope assembly, and (b) the layout of the optical design.

2.2. Finite-element Model

A finite-element model for each part of the concept design of the OTA is generated as follows in this paper. For simplification of the model, a structural model is constructed using a two-dimensional (2D) shell element (frames and spider vane) and a 1D beam element (struts). Also, the M1 and M2 assemblies are simplified using a dummy mirror assembly. Because of Earth’s gravity and the bending of the structure, it is unavoidable that a telescope has a misalignment between M1 and M2, which is an important issue for optical performance. In addition, it is well known that the most significant gravity deflection occurs when the OTA structure with elevation and azimuthal rotation based on the tracking mount is in a horizontal position [8, 9]. Therefore, a fixed boundary condition is applied to the plane of the center frame’s interface between the OTA structure and the tracking mount of the OTA, and a self-gravity load (1 g) in the −y direction is also applied for the load condition, as shown in Fig. 2. Here U_y and U_z indicate the deflection of the frame in the y- and z-directions respectively. Superscripts denote the upper and lower parts of the frame. To perform the self-gravity simulation, the material properties of steel, aluminum, and glass ceramic are used, as listed in Table 1. The simulation is performed using Ansys 2021 R2 commercial software (Ansys Inc., PA, USA), and the optical performance is evaluated. Here a computer equipped with an Intel (R) Core (TM) i7-12700KF CPU @3.60 GHz and 128 GB of RAM is used.

TABLE 1. Material properties of constituent materials.

Material	Density ρ [kg/m3]	Young’s Modulus E [GPa]	Poisson’s Ratio v	Coefficient of Thermal Expansion α [1/℃]
Steel	7,850	210.0	0.30	1.2 × 10−5
Aluminum	2,700	68.9	0.33	2.3 × 10−5
Glass Ceramic (Astrosial, M1)	2,460	92.0	0.28	1.5 × 10−7
Glass Ceramic (Zerodur, M2 and M3)	2,530	90.3	0.24	2.0 × 10−8

Figure 2. Boundary conditions for the optical-telescope assembly.

In this study, the design of the OTA structure is focused on the mechanical aspects. To this end, a parametric study is conducted on the design variables of various mechanical components to satisfy the design requirements.

3.1. Tube Truss

To investigate the effect on the deflection of the OTA structures according to the truss shape, truss angle, and strut thickness, we first conduct a parametric study for the truss shape. When the top, center, and bottom frame are designed, the A-type truss and the inverted A-truss shapes [7] of the Serrurier truss [8] are generally used, as shown in Fig. 3. To compare the two truss structural concepts, the truss shape is set as a design variable. The gravity deflections in the y direction and tilts of the upper and lower frames according to the self-gravity analyses are shown in Table 2. Here |U_z^upper − U_z^lower| means the tilt of the frame. It is confirmed that the Serrurier truss shape provides better performance in terms of deflection and tilt than the inverted Serrurier shape. It is concluded that the centers of gravity of both ends of the Serrurier truss are located in the inner part of the frame, but the inverted Serrurier shape is out of balance due to the induced moment, resulting in high deflection.

TABLE 2. Deflection and tilt results according to tube-truss shape.

Case		Uy [mm]	|Uzupper − Uzlower| [mm]
Top Frame	Serrurier	−8.73 × 10−1	8.417 × 10−3
	Inverted Serrurier	-1.47	7.088 × 10−1
Bottom Frame	Serrurier	-2.25 × 10−1	1.223 × 10−1
	Inverted Serrurier	−3.88 × 10−1	3.744 × 10−1

Figure 3. Structural configuration for tube truss: (a) Serrurier, (b) inverted Serrurier.

To compare structural performance according to the angle of a pair of truss struts, the strut angle is set as a design variable. The gravity deflection of the frame according to the self-gravity analysis is shown in Table 3. It is confirmed that the truss affords good performance in terms of deflection when the angle increases. As the angle increases, the pivot-point error becomes smaller and the moment is not induced, and hence this effect is remarkable.

TABLE 3. Deflection and tilt results according to strut angle.

Case	Strut Angle [degrees]	Uy [mm]	|Uzupper − Uzlower| [mm]
Upper Struts (Top Frame)	10	-1.29 × 101	1.838 × 10−1
	25	-2.32	5.265 × 10−2
	42	−8.73 × 10−1	8.417 × 10−3
Lower Struts (Bottom Frame)	42	-2.59	2.864 × 10−1
	84	−5.22 × 10−1	1.754 × 10−1
	110	-2.25 × 10−1	1.223 × 10−1

To compare structural performance according to the thickness of the truss struts (with the outer diameter fixed), the thickness is set as a design variable. The outer and inner radii sizes are set within the range applicable to the OTA frame, in terms of manufacturing. The gravity deflection of the frame according to the self-gravity analysis is shown in Table 4. In the case of the upper strut, it is confirmed that the y- and z-direction deflections decrease as the thickness and mass decrease. On the contrary, the deflection in the z direction increases as the thickness decreases in the case of the lower strut. In general, the mass of M1 is greater than that of M2 in the large OTA. As the thickness of the lower strut decreases, the stiffness of the lower strut part also decreases, resulting in the effect of the moment. To prove this, we compare this to the Serrurier-truss model reinforced with supports in the lower strut, as shown in Fig. 4. The deflection in the z direction is reduced, because the Serrurier truss with supports has high stiffness to support the heavy mass of M1.

TABLE 4. Deflection and tilt results according to strut thickness.

Case	Strut Thickness (Router/Rinner)a) [mm]	Uy [mm]	|Uzupper − Uzlower| [mm]
Top Struts (Top Frame)	65/62	−7.29 × 10−1	7.419 × 10−3
	65/58	−8.04 × 10−1	8.358 × 10−3
	65/55	−8.73 × 10−1	8.417 × 10−3
Bottom Struts (Bottom Frame)	60/55	-2.18 × 10−1	1.350 × 10−1
	60/50	-2.22 × 10−1	1.259 × 10−1
	60/45	-2.25 × 10−1	1.223 × 10−1
	60/45 (w/support)	-2.20 × 10−1	4.329 × 10−2

Figure 4. Structural-configuration concepts for bottom struts: (a) Serrurier without support, (b) Serrurier with support.

3.2. Spider Vane

A spider vane is a structure that supports the M2 assembly in the top frame. The spider vane should be thin, to minimize light interference. In addition, it should hold the M2 assembly stably for fine tuning in the tip-tilt direction in an active optics system. First, the shape of the spider vane is divided into four types, in terms of mass reduction and fine tuning in the tip-tilt direction. To compare structural performance according to the shape of the spider vane and the presence of a counterweight, as shown in Fig. 5, these parameters are set as design variables. Here the thickness of the spider vane’s rib is fixed at 6 mm. The gravity deflection of the frame according to the self-gravity analysis is shown in Table 5. From the results, it is confirmed that the fourth shape [Fig. 5(d)] with the counterweight has low deflection. The deflection of M2 is compensated by a counterweight on the opposite side of the system because the system’s center of gravity is in the symmetry plane of the spider vane. In general, the spider vane has low stiffness in the axial direction (piston mode). This is improved by altering the X shape to a cruciform shape arranged in the vertical direction of the top frame, as shown in Figs. 6(a) and 6(b).

TABLE 5. Deflection and tilt results according to spider-vane shape and counterweight.

Counter Weight	Spider-vane Shape		Uy [mm]	|Uzupper − Uzlower| [mm]
Without Counterweight	Model A	X Shape	-2.48 × 10−3	3.28 × 10−3
	Model B		-2.65 × 10−3	1.29 × 10−3
	Model C		-2.52 × 10−3	1.45 × 10−3
	Model D		-2.23 × 10−3	1.14 × 10−3
	Model D	Cruciform Shape	-1.63 × 10−3	7.72 × 10−4
With Counterweight	Model A	X Shape	-2.36 × 10−3	3.75 × 10−3
	Model B		-2.62 × 10−3	1.56 × 10−4
	Model C		-2.39 × 10−3	1.94 × 10−4
	Model D		-2.07 × 10−3	1.15 × 10−4
	Model D	Cruciform Shape	-1.53 × 10−3	1.09 × 10−4

Figure 5. Structural-configuration concepts for the spider vane: (a) Model A, (b) model B, (c) model C, (d) model D, and (e) with counterweight.

Figure 6. The layout for the spider vane: (a) X shape, and (b) cruciform shape.

4.1. Design Requirements

First, the system used in this study is based on an optical telescope with a diameter of about 1.5 m, a frequently used size [3, 4]. The specifications of the optical elements and the compensation range of the hexapod are set in terms of optical and mechanical performance. To meet the motor specifications for a tracking mount rotating along the target, the total weight and maximum moment of inertia are also designed for less than 4,000 kg and 4,500 kg∙m² respectively, and the center of gravity is designed to be within 1 mm from the rotational center. The size and height of the OTA structure are also limited, to meet the space limitation of the dome size. A set of design requirements is established from the author’s knowledge and experience with optical telescopes, as listed in Table 6.

TABLE 6. Structural-design requirements for the optical-telescope assembly.

Parameter	Requirement
Diameter of M1 [mm]	1,520
Distance from M1 to M2 [mm]	1,800
Travel Range of Hexapod	(X, Y, Z) mm = (±22.5, ±22.5, ±12.5) (θX, θY, θZ) = (±7.5°, ±7.5°, ±12.5°)
Mass [kg]	≤4,000
Moment of Inertia [kg∙m2]	≤4,500
Center of Gravity [mm]	≤1.0 (from rotation center)
Size (W × D × H) [m]	2.0 × 2.0 × 4.0

4.2. Optical-telescope Mechanical Structure

In the detailed design step, the design is improved to be more accurate or extended to additional operating conditions. In this section, the design is subdivided based on the previous results of the parametric study for each structural part of the OTA. The main consideration for the design of the OTA structure is to meet the design requirements for size and mass while minimizing the gravity deflection when the OTA is horizontal. In this study, the structure of the OTA is designed to support M1, M2, and M3 to withstand the weight load as much as possible, as shown in Table 7 and Fig. 7. To have the necessary stiffness in the design requirements and the operating environment, the shape and thickness of each part are set as shown in Table 8. It should be noted that the dimensions of all of the parts are set within the ranges applicable to OTA structures in terms of manufacturing. For reduction of mass and cost, the existing top and bottom frames with a square shape are adjusted to a dodecagon shape by cutting off the rest of the frame, except for the strut-support parts. A welded structure is also employed for ease of assembly; However, a ball head instead of welding is used in the upper strut, since the mass of M2 is less than that of M1. The advantage is that a moment is not induced when a force is applied to the ball head, because the ball moving freely minimizes the wear of the contact surface. In addition, the hollow strut is used to support the weight transmitted to the OTA. The spider vane is connected to the top frame in a cruciform shape to effectively support the M2 assembly. The large OTA structure generally is out of balance, and therefore the balance (center of gravity) of the OTA structure is finely adjusted by moving the dummy mass in the upper strut in this study. A mirror cover is also used, to prevent dust from settling onto the surface of the mirror while the OTA is not in operation.

TABLE 7. Specifications for the optical-telescope assembly.

Parameter	Value
Size (W × D × H) [m]	2.0 × 2.0 × 3.348
Mass [kg]	3,921
Moment of Inertia [kg∙m2]	4,210
Center of Gravity [mm]	(-1.0 × 10−1, 3.0 × 10−2, -2.5 × 10−1)

TABLE 8. Specifications for each part of the optical-telescope assembly.

Part	Size [mm]	Remark
Spider	6.0t	Cruciform Shape
Top Frame	150 × 150 × 200 × 4.5t	Dodecagon Shape
Top Struts	Φ139.7 × 4.0t	Serrurier
Center Frame	150 × 150 × 500 × 6.0t	Square Shape
Bottom Frame	150 × 150 × 150 × 4.5t	Dodecagon Shape
Bottom Struts	Φ101.6 × 4.0t	Serrurier

Figure 7. Detailed design of the optical-telescope assembly.

Subsequently, the finite-element model for the OTA structure is generated in detail, with about 1.64 × 10⁶ nodes and 9.0 × 10⁵ solid elements, as shown in Fig. 8. First, the six rigid body modes are confirmed in modal analysis without boundary condition for mathematical verification. From the modal-analysis results, the mode shapes are broadly classified into four local mode shapes (spider vane, cover, M1 mount, and M3 assembly) and global mode shapes for the OTA, as shown in Fig. 9. The first natural frequency of the OTA is designed to be greater than 95 Hz, and the natural frequency of the first global mode is confirmed as 98 Hz. In addition, the margin of safety for all parts of the OTA is 0 or more. Therefore, it is concluded that the OTA structure satisfies the stiffness and strength requirements.

Figure 8. Finite-element model of the optical-telescope assembly.

Figure 9. Main mode shapes and natural frequencies of the optical-telescope assembly (OTA): (a) Local mode–spider (23.8 Hz), (b) local mode–cover (23.9 Hz), (c) local mode–M1 mount (38.9 Hz), (d) local mode–M3 assembly (131.8 Hz), and (e) global mode–OTA (98.9 Hz).

In general, four external environmental factors (gravity, temperature, wind, and slew acceleration) are considered to evaluate the stability of the optical-telescope structure. The effects of wind and slew acceleration are not considered in this study, because the OTA operates inside the dome and the slew acceleration is very small, in the case of the optical telescope developed in this study.

First, a self-gravity load (1 g) in the −y and −z directions is also applied for the load condition, the simulation results are analyzed in terms of the optical performance, and optical deflection occurs, as shown in Table 9 and Fig. 10. As a result, we conclude that a horizontal position is the worst case, from the perspective of deflection along the y axis. In addition, this fact supports that the most significant gravity deflection occurs when the OTA structure is horizontal [8, 9]. Subsequently, we verify the stability of the optical-telescope structure through a thermal analysis by applying the temperature deviation inside the dome, which changes according to the external environment. This step is critical because the reliability of the thermal-analysis results can be obtained when the temperature conditions are similar to the actual operating conditions. Here the optical telescope is operated within a temperature range of -20 to 40 ℃, depending on the seasonal temperature. Therefore, the maximum thermal conditions (-20 and 40 ℃) are applied for the load condition, and the simulation results are analyzed for optical performance. Frame and mirror deflection occur as shown in Table 10 and Fig. 11. It is confirmed that the optical deflection is within the range of compensation via hexapods.

TABLE 9. Deflection and tilt results according to the position.

Position	Part	Uy [mm]	|Uzupper − Uzlower| [mm]
0 degrees (Horizontal direction, 1 g in −y direction)	Top Frame	−8.79 × 10−2	8.16 × 10−4
	Bottom Frame	−6.21 × 10−2	2.21 × 10−2
	M1	-2.13 × 10−1	2.76 × 10−2
	M2	−8.70 × 10−2	3.5 × 10−3
	M3	-1.29 × 10−1	3.27 × 10−2
45 degrees	Top Frame	−7.02 × 10−2	1.63 × 10−3
	Bottom Frame	−5.91 × 10−2	1.35 × 10−2
	M1	-1.22 × 10−1	6.19 × 10−2
	M2	−7.14 × 10−2	1.84 × 10−3
	M3	-1.29 × 10−1	2.29 × 10−2
90 degrees (Vertical direction, 1 g in −z direction)	Top Frame	3.0 × 10−3	-1.5 × 10−3
	Bottom Frame	3.0 × 10−3	4.5 × 10−3
	M1	4.8 × 10−4	-2.10 × 10−1
	M2	−9.6 × 10−4	−5.7 × 10−2
	M3	−5.5 × 10−4	-1.30 × 10−1

TABLE 10. Deflection and tilt results according to the temperature inside the dome.

Temperature	Part	Uy [mm]	|Uzupper − Uzlower| [mm]
40 ℃	Top Frame	4.7 × 10−4	2.38 × 10−3
	Bottom Frame	4.7 × 10−4	7.20 × 10−4
	M1	−6.7 × 10−2	2.48 × 10−4
	M2	-1.9 × 10−3	8.60 × 10−4
	M3	−7.3 × 10−2	2.48 × 10−3
−20 ℃	Top Frame	2.3 × 10−4	1.90 × 10−3
	Bottom Frame	2.3 × 10−4	1.38 × 10−2
	M1	3.4 × 10−2	4.18 × 10−2
	M2	−9.6 × 10−4	1.32 × 10−3
	M3	−5.5 × 10−4	-1.30 × 10−1

Figure 10. Plots of displacement relative to the optical-telescope assembly (OTA)’s position: (a) 0°, (b) 45°, and (c) 90°.

Figure 11. Plots of displacement relative to the temperature condition: (a) 40 ℃, (b) -20 ℃.

Among the structural-shape concepts of various OTAs, the Serrurier truss structural shape was selected in this study, and the collimation of the telescope according to the design variables was extensively discussed. The results of a parametric study on the major design variables of the OTA part were obtained, and based on them the design of the OTA structure was changed to approximate the design requirements. To check the deflection due to gravity in a horizontal position, the performance and features of the structure were analyzed and described according to the design parameters. Through a series of parameter studies of the OTA-structure design, it was possible to obtain design specifications that satisfied the design requirements and produced a structure that could minimize deflection. Therefore, the design process proposed in this study could be effectively applied to the design of the OTA. It is also expected that development costs and time for design and manufacture will be reduced. It should be noted that the shape and dimensions for each component might be different with respect to the design requirements of an OTA structure. However, the OTA design results will present a similar trend if the struts have identical Serrurier structures. This approach can be extended to establish a database for the design of OTAs as they are developed, and has potential application to the manufacture of OTAs. Studies of this are in progress and will be published when completed.

The authors declare no conflicts of interest.

Data underlying the results presented in this paper are not publicly available at the time of publication.

This work was supported by the Defense Rapid Acquisition Technology Research Institute (DRATRI) - Grant funded by Defense Acquisition Program Administration (DAPA) (UC200012D).