Athermalization Design for the On-Orbit Geometric Calibration System of Space Cameras

Abstract

The on-orbit geometric calibration accuracy of high-resolution space cameras directly affects the application value of Earth observation data. Conventional on-orbit geometric calibration methods primarily rely on ground calibration fields, making it difficult to simultaneously achieve high precision and real-time monitoring. To address this limitation, we, in collaboration with Tsinghua University, propose a high-precision, real-time, on-orbit geometric calibration system based on active optical monitoring. The proposed system employs reference lasers to integrate the space camera and the star tracker into a unified optical system, enabling real-time monitoring and correction of the camera’s exterior orientation parameters. However, during on-orbit operation, the space camera is subjected to a complex thermal environment, which induces thermal deformation of optical elements and their supporting structures, thereby degrading the measurement accuracy of the geometric calibration system. To address this issue, this article analyzes the impact of temperature fluctuations on the focal plane, the reference laser unit, and the laser relay folding unit and proposes athermalization design optimization schemes. Through the implementation of a thermal-compensated design for the collimation optical system, the pointing stability and divergence angle control of the reference laser are effectively enhanced. To address the thermal sensitivity of the laser relay folding unit, a right-angle cone mirror scheme is proposed, and its structural materials are optimized through thermo–mechanical–optical coupling analysis. Finite element analysis is conducted to evaluate the thermal stability of the on-orbit geometric calibration system, and the impact of temperature variations on measurement accuracy is quantified using an optical error assessment method. The results show that, under temperature fluctuations of 5 °C for the focal plane and the reference laser unit, 1 °C for the laser relay folding unit, and 2 °C for the star tracker, the maximum deviation of the system’s measurement reference does not exceed 0.57″ (3σ). This enables long-term, stable, high-precision monitoring of exterior orientation parameter variations and improves image positioning accuracy.

1. Introduction

With the continuous advancement of optical remote sensing technology, high-resolution space cameras have become essential tools for Earth observation, as their imaging quality and positioning accuracy directly determine the application value of the acquired data [1,2,3]. As the core payloads of optical remote sensing satellites, space cameras are required not only to provide high-resolution imagery that reveals fine details of ground targets but also to achieve higher positioning accuracy to ensure the reliability of geographic information acquisition. However, during on-orbit operation, space cameras are affected by various factors, including attitude control errors, environmental disturbances, and low geometric calibration accuracy, all of which can degrade image positioning accuracy [4,5]. To further improve image positioning accuracy, geometric parameter calibration and optimization are typically performed multiple times during both ground-based testing and on-orbit operation. The geometric parameters mainly include the interior orientation parameters (principal point coordinates and focal length) and the exterior orientation parameters (the position and attitude of the camera in space).

On-orbit geometric calibration techniques primarily rely on ground calibration fields and have evolved through various improvements, including the use of sparse ground control points and stellar observations for calibrating space cameras on orbit, in order to meet different observational requirements and environmental constraints [6,7,8]. Even after the initial calibration, space cameras still require frequent calibration of their geometric parameters, particularly the exterior orientation parameters, to account for changes over time. With technological advancements, image positioning accuracy without ground control points has improved from hundreds of meters to the meter level [9,10]. However, the current calibration method using a ground calibration field is limited by geography, cycle, and timeliness, which cannot meet the needs of high precision and real-time monitoring at the same time. To address this limitation, based on the theory of on-orbit calibration using stellar observations, we, in collaboration with Tsinghua University, have innovatively proposed a design scheme for a high-precision, real-time, on-orbit geometric calibration system based on active optical monitoring. This scheme introduces a reference laser source to construct a serial optical system linking the space camera and the star tracker, thereby establishing the geometric parameter transformation relationship between the camera and the star tracker [11,12].

During on-orbit operation, satellites are subjected to complex thermal environments, including solar radiation, Earth-reflected radiation, and self-generated heat, resulting in nonlinear and dynamic temperature variations. Temperature fluctuations and non-uniform distributions within the optical system may induce thermal deformations in the supporting structures and optical elements, causing translations, tilts, or surface distortions of the optical components, which can compromise the measurement stability of the on-orbit geometric calibration system [13,14]. To address these thermal effects, various thermal control strategies have been proposed to enhance the thermal stability of space cameras. For example, Chenhui Xia employed indirect radiation thermal control combined with differential temperature regulation, effectively mitigating the impact of temperature fluctuations and external thermal environments on the camera structure. Elhefnawy A. investigated satellite thermal control systems to determine safe temperature margins and allowable temperature variations for internal components. Sundu H. conducted a comprehensive analysis of thermal design and control for low Earth orbit (LEO) observation satellites, focusing on the impact of extreme thermal conditions and optimizing thermal control strategies through numerical simulations [15,16,17].

Although various thermal control measures have been implemented for space cameras, the designed on-orbit geometric calibration system remains inevitably affected by dynamic temperature variations, which can affect the measurement reference frame and degrade the accuracy of exterior orientation parameter measurements. Moreover, detailed analyses of the thermal stability of this system have not yet been conducted to ensure its operational reliability. Therefore, performing thermal stability analysis and implementing athermalization design for the designed on-orbit geometric calibration system is crucial to guarantee high-precision measurement of exterior orientation parameters.

2. Layout and Design Principles of the On-Orbit Geometric Calibration System

The overall design principle of the on-orbit geometric calibration system is illustrated in Figure 1. The core concept of the system is to introduce a reference laser to form a series connection optical system that couples the space camera with the star tracker, thereby establishing a geometric parameter transformation relationship between them. The reference laser is emitted from the focal plane of the space camera, transmitted through the camera’s optical system and a series of mirrors, and directed to the star tracker. A dichroic mirror enables the star tracker to simultaneously receive both starlight and the laser signal, allowing joint imaging and calculation. A narrow band-pass filter is employed to attenuate non-target spectral bands, effectively reducing the impact of background noise on the measurement accuracy of the star tracker [18].

The divergence angle of the lasers should be kept constant, and the wavelength should be chosen close to the camera’s imaging band to ensure effective detection by the star tracker. In this design, an 850 nm near-infrared laser is used as the reference light source. The camera optical system parameters should be determined according to the actual design requirements; in this study, a multispectral space camera with a focal length of 14 m and a spatial resolution better than 0.5 m is employed. The passband of the narrow band-pass filter must match the laser design wavelength, with a central wavelength of 850 ± 5 nm. The dichroic mirror should ensure that it does not affect the stargazing band of the star tracker and can effectively reflect the laser to the star tracker.

The star tracker calibrates its own geometric parameters using starlight and determines its attitude information. By continuously monitoring the variation of the laser image point vector in real time, the transformation matrix from the star tracker coordinate system to the space camera coordinate system is computed, enabling real-time calibration of the camera’s exterior orientation parameters. The design principle of the star tracker unit is illustrated in Figure 2.

The on-orbit geometric calibration system was applied to the design and operation of actual space cameras, with specific functional divisions and detailed designs implemented. The overall system layout is shown in Figure 3 and consists of four main functional units: the space camera optical system, the focal plane and reference laser unit, the laser relay folding unit, and the star tracker unit. The space camera optical system is determined based on the actual space camera used and is responsible for capturing ground imagery as well as providing the imaging focal plane that serves as the mounting platform for the reference laser. The reference laser is positioned at the camera’s focal plane and is integrated with the detector. The laser relay folding unit guides the reference laser to the star tracker unit through a series of optical elements. The star tracker unit calculates the laser imaging information and establishes a reference frame based on attitude information obtained from starlight, thereby enabling real-time monitoring and calibration of changes in the camera’s exterior orientation parameters.

The core function of the focal plane and reference laser unit is to generate and transmit a stable reference laser signal, achieving precise coupling between the reference laser, the space camera, and the star tracker. This supports real-time monitoring of changes in the camera’s exterior orientation parameters. The reference laser unit consists of a highly stable laser source, collimator, and power supply module. Near-infrared laser is selected as the reference light source, which lies within the detector’s high-sensitivity response range and effectively reduces the complexity of the star tracker’s optical design, thereby ensuring the rationality and reliability of the system design.

The collimator design, shown in Figure 4, consists of three spherical lenses. Its primary function is to adjust the laser beam into a collimated beam with a fixed divergence angle, effectively suppressing stray laser light from incident on the focal plane structure surface and preventing potential stray light interference. The collimator strictly controls the laser’s divergence angle, ensuring beam quality and directional stability, thereby providing high-quality input for the subsequent laser relay folding unit.

During the calibration process, the positional variations of the laser image point along the X and Y axes must be monitored to determine the dynamic changes in the space camera’s exterior orientation parameters. Using only two lasers may cause directional bias in data acquisition. To enhance measurement accuracy, four lasers are arranged in a rectangular layout at the four corners of the camera’s focal plane, as illustrated in Figure 5.

In the on-orbit geometric calibration system, the primary function of the laser relay folding unit is to stably transmit the reference laser from the output position of the space camera to the star tracker detector. The laser relay folding unit consists of two plane mirrors coated with high-reflectivity infrared films and a rigid support structure, as shown in Figure 6. The mirrors employ high-reflectivity coatings to minimize energy loss during laser transmission, while wavelength-selective filtering suppresses stray light, enhancing the laser signal quality and preventing interference with the star tracker’s normal observations.

In this layout, the laser relay folding unit is installed before the camera’s primary mirror. The reference laser exits the camera optical system as a collimated beam and is received by plane mirrors, ensuring efficient transmission to the star tracker detector after two reflections, while maintaining a high degree of parallelism between the incident and outgoing beams.

The system design of the star tracker unit is shown in Figure 7. To enable simultaneous acquisition of laser and starlight, this design integrates a narrow band-pass filter and dichroic mirror to separate and direct the laser and starlight signals. The narrow band-pass filter ensures effective selection of the laser signal, preventing stray light interference from degrading the reference laser imaging quality. The dichroic mirror allows visible starlight to pass directly into the star tracker optical system, while the near-infrared laser is directed via a reflective path. This configuration ensures that the star tracker can simultaneously receive and process both signals, enabling combined observation of the starlight and laser.

3. Influence of Temperature on the On-Orbit Geometric Calibration System

3.1. Influence of Temperature on the Focal Plane and Reference Laser Unit

The reference laser unit is integrated into the focal plane electric control box of the space camera. During camera operation, the high power consumption of the focal plane electronic components causes a rapid temperature rise inside the electrical box. When the camera stops operating, the components no longer generate heat, making the electrical box susceptible to rapid cooling due to low temperature at radiative surfaces. To maintain thermal stability of the focal plane system, passive thermal control combining a heat sink, heat pipe, and radiant cold plate is typically adopted. This is supplemented by active thermal control to compensate for and regulate temperature fluctuations within the electrical box. A multilayer heat insulation assembly is applied to further enhance temperature stability. Passive thermal control design of the focal plane electric control box is illustrated in Figure 8.

Although active and passive thermal control designs can significantly reduce temperature fluctuations within the focal plane electronic control box, studies by Escobar E. and Yang L. indicate that under extreme high- and low-temperature conditions, the temperature inside the box can still vary by approximately 5 °C [19,20]. Such significant temperature fluctuations can severely affect the mounting structure and collimated optical system of the reference laser, leading to changes in the laser divergence angle and reduced pointing stability, ultimately compromising the measurement reference stability and accuracy of the on-orbit geometric calibration system.

The impact of temperature fluctuations on optical components is primarily reflected in thermal expansion effects. During thermal expansion, the thickness, curvature, surface shape, and spacing between optical components change, directly affecting the laser divergence angle. The variation in laser divergence angle can be quantified using the thermal expansion coefficient, expressed by the following formula:

$$\Delta {\theta }_l\approx -{\theta }_l{\displaystyle \frac{1}{f_l}}\left({\displaystyle \frac{\mathrm{d}{f}_l}{\mathrm{d}T}}\right)\Delta T$$

(1)

where $\Delta {\theta }_l$ represents the change in laser divergence angle, $f_l$ is the actual focal length of the collimator, ${\displaystyle \frac{\mathrm{d}{f}_l}{\mathrm{d}T}}$ accounts for the combined effects of thermal expansion of the lens curvature radius and the temperature coefficient of refractive index; and $\Delta T$ denotes the temperature variation range.

For the collimator of the reference laser unit, neglecting other factors and considering only pointing position changes caused by temperature variations, it can be expressed as:

$$\Delta x_l\approx {\displaystyle \frac{\mathsf{\partial }{x}_l}{\mathsf{\partial }{L}_0}}\alpha L_0\Delta T$$

(2)

In the formula, $\Delta x_l$ represents the change in pointing position; $\alpha$ is the material’s coefficient of thermal expansion; and $L_0$ denotes the distance between the laser and the collimating lens. It can be seen that variations in laser pointing and divergence angle are affected by temperature changes. Significant changes in pointing position and divergence angle directly impact the measurement accuracy of the on-orbit geometric calibration system. Therefore, thermal effects must be further considered during the design of the collimation optical system, and athermalization optimization should be performed.

3.2. Influence of Temperature on the Laser Relay Folding Unit

Thermal stability of the laser relay folding unit is critical to the measurement accuracy of the on-orbit geometric calibration system. The unit’s initial design consists of two orthogonally arranged 45° flat mirrors mounted on a rigid support structure. The mirrors are designed with a circular shape to better accommodate surface figure changes caused by thermal expansion. However, in actual operating environments, temperature variations still significantly affect the structure and optical performance of the laser relay folding unit.

To clarify the specific effects of temperature variations on the laser relay folding unit, finite element simulation analysis was conducted to study the displacement changes of the mirrors under different thermal conditions. The finite element model of the laser relay folding unit is shown in Figure 9. Using HyperMesh 10.0, the two mirrors and their supporting structures were meshed to establish the finite element analysis model. The supporting structures and mirrors were divided into hexahedral elements, comprising a total of 12,035 nodes and 8530 elements. Material properties were assigned to the supporting structures and mirrors, made of aluminum alloy and glass–ceramic, respectively.

Table 1. Statistical results of exterior orientation parameter changes during focal plane movement along the X/Y axis.

Material	Elastic Modulus (Gpa)	Poisson’s Ratio	Coefficient of Thermal Expansion (10−6/K)	Density (g/cm3)
Aluminum alloy	71.7	0.33	23.6	2.81
System length/mm	96	0.24	0.1	2.53

lists the relevant parameters of aluminum alloy and glass–ceramic [21,22].

Thermal stress and deformation analyses were conducted on the support structure and mirror under applied thermal load conditions. During the simulation, the ambient temperature was increased from 20 °C to 21 °C. The thermal deformation response of the support structure and mirror was evaluated using OptiStruct. Figure 10. presents the thermal deformation cloud map caused by the temperature change. The maximum mirror displacements along the X, Y, and Z axes were 0.64 μm, 2.78 μm, and 0.97 μm, respectively.

Nodal displacement data of the planar mirror were extracted and processed using a custom-developed program to calculate its axial rotation and translational shifts. The results indicate that, under a 1 °C temperature change, the maximum rotational variation of the mirror about the X axis is 1.37″, and about the Y axis, it is 0.76″. The corresponding surface displacement of the mirror is shown in Figure 11.

The experiment verified the rotational displacement law of the laser relay folding unit under temperature variation. Unlike the wide temperature fluctuations caused by the high power consumption of the focal plane electric control box, the laser relay folding unit experiences relatively small thermal fluctuations, which can typically be controlled within ±1 °C using active and passive thermal control techniques. However, even such minor temperature variations can still affect the measurement accuracy of the system to some extent. Therefore, further optimization of the laser relay folding unit is still required.

4. Optimized Thermal Stability Design of the On-Orbit Geometric Calibration System

4.1. Heat Elimination Design of the Reference Laser Unit Collimation Optical System

Due to the mismatch in coefficient of thermal expansion between optical glass and structural materials, their coefficient of thermal expansion differs under varying temperature conditions, leading to thermal stresses in both axial and radial directions [23]. These thermal stresses cause changes in the collimation optical system of the reference laser unit, resulting in deviations in laser pointing and divergence angle, which in turn affect the measurement accuracy of the on-orbit geometric calibration system. To meet the high-precision measurement requirements, the collimation optical system must be optimized to accommodate potential temperature fluctuations during high and low temperature operating conditions of the focal plane electric control box in the space camera, and an optical system with heat elimination capabilities must be developed.

The reference laser unit outputs a near-infrared fiber laser source with a fixed divergence angle. The space camera used in this study has a f-number of 11.67, corresponding to an optical system convergence angle of 4.89°. To prevent excessive laser divergence, causing the beam to illuminate the sidewalls of the electrical box and generate stray light, the laser divergence angle is precisely controlled within the range of 4.89° ± 0.01°. This requirement demands not only high accuracy of the laser divergence angle under temperature variations but also ensures that the laser pointing direction does not exhibit significant deviation.

The laser output follows a Gaussian distribution. During free-space propagation, the beam radius $\omega (z)$ changes with propagation distance z due to diffraction effects, causing the beam to gradually expand. For a Gaussian beam, this variation can be expressed as [24]:

$$\omega \left(z\right)={\omega }_0\sqrt{1+{\left({\displaystyle \frac{z}{z_R}}\right)}^2}$$

(3)

Here, ${\omega }_0$ represents the beam waist radius, and $z_R$ is the Rayleigh length, which denotes the distance over which the beam expands from the waist to $\sqrt{2}{\omega }_0$ during propagation, specifically expressed as:

$$z_R={\displaystyle \frac{\pi {\omega }_0^2}{\lambda }}$$

(4)

In the equation, $\lambda$ represents the laser wavelength.

Transformation of the Gaussian beam through the optical system follows the system’s paraxial ray transfer matrix. Based on this, the beam waist radius ${\omega }_f$ after passing through a single thin lens can be derived as:

$${\omega }_f={\displaystyle \frac{\lambda f_l}{\pi {\omega }_0}}$$

(5)

The far-field divergence angle ${\theta }_f$ after transformation by the thin lens is as follows:

$${\theta }_f={\displaystyle \frac{{\omega }_0}{f_l}}$$

(6)

For Gaussian beams with finite size, the divergence angle cannot be compressed infinitely after passing through an optical system. However, when the beam waist is located at the back focal plane of the lens, the divergence angle can reach the desired target, with the condition:

$$z_0=f_l$$

(7)

Thus, divergence angle design value is derived as:

$${\theta }_d={\displaystyle \frac{\lambda }{\pi {\omega }_0}}$$

(8)

Collimation system magnification can be expressed by the following formula:

$$M={\displaystyle \frac{D}{2{\omega }_0}}$$

(9)

Here, D is the effective aperture of the lens, and M is the collimation magnification.

According to the above formula, when the lens focal length is fixed and the waist of the incident Gaussian beam is located at the lens’s back focal plane, different output beam divergence angles can be achieved. The initial optical structure of the reference laser unit collimator is shown in Figure 12. Considering that the system must be installed inside the focal plane electric control box, size constraints are fully taken into account during design. A three-piece spherical lens configuration is adopted, with detailed parameters listed in

Table 2. Material parameters of the laser relay folding unit.

Surface Number	Radius (mm)	Thickness (mm)	Materials	Diameter (mm)
0	/	5	/	/
1	31.641	2	ZF6	6
2	17.794	5.2	/	/
3	−13.772	3	H-ZF3	8
4	−8.319	3	/	/
5	−74.016	3	H-ZF1A	10
6	−24.266	/	/	/

[25].

To quantitatively analyze the effects of temperature fluctuations on laser pointing and divergence angle, simulations were performed using Zemax Professional and HyperWorks 10.0 to evaluate the pointing positions and divergence angle variations of the reference laser unit over a temperature range of 20–25 °C. A finite element model of the laser collimation system was established for thermal stability analysis, with the system structure meshed accordingly. The finite element model of the system is shown in Figure 13, comprising 16,302 nodes and 11,416 elements, all hexahedral. Material properties were assigned to both the optical and mechanical components of the system. The mechanical structure was made of Invar steel with an extremely low coefficient of thermal expansion, and the lens material parameters are listed in

Table 3. Material properties of the collimation system.

Materials	Elastic Modulus (Gpa)	Poisson’s Ratio	Coefficient of Thermal Expansion (10−6/K)	Density (g/cm3)
ZF6	54.7	0.216	8	4.78
H-ZF3	88.3	0.254	9	3.06
H-ZF1A	82.6	0.237	8.5	2.79

.

A uniform temperature field was applied by setting temperature boundary conditions to perform thermal deformation analysis. The displacement cloud diagram of the collimation system at 25 °C is shown in Figure 14. The maximum displacements along the X, Y, and Z axes were 0.258 μm, 0.485 μm, and 0.1 μm, respectively. Sigfit 2010R2Dwas used to fit the surface change data of the collimation optical system, and the eccentricity and tilt parameters of each lens element after being subjected to thermal stress were imported into Zemax and converted into the pointing offset and offset angle of the laser along the X and Y axes after passing through each lens. The statistical results are shown in

Table 4. Statistics of laser pointing offset from individual collimation components.

Component Number	Direction	Offset (μm)	Offset Angle (″)
M1	X	0.008	0.0172
	Y	1.014	2.062
M2	X	0.031	0.062
	Y	0.974	2.008
M3	X	0.013	0.0262
	Y	0.045	0.0996
System changes	X	0.051	0.105
	Y	0.021	4.16

.

As shown in Table 4, the decentering and tilt of lens surfaces M1 and M2 have a significant impact on the laser spot center drift, with the collimation system exhibiting a maximum laser pointing deviation of 4.16″ in the Y direction. Meanwhile, at 25 °C, the system’s divergence angle changes by 0.26°, which may affect the measurement accuracy of the on-orbit geometric calibration system. These results indicate the need for further system optimization based on the simulation analysis.

The temperature-dependent variation in the absolute refractive index of the optical material is given by Equation (10), which is obtained by SCHOTT Company through the fitting of a large number of experimental data, and specifically expressed as [26]:

$$\Delta n_{abs}\left(\lambda ,T\right)={\displaystyle \frac{n^2\left(\lambda ,T_0\right)-1}{2n\left(\lambda ,T_0\right)}}\cdot \left(D_0\cdot \Delta T+D_1\cdot \Delta T^2+D_2\cdot \Delta T^3+{\displaystyle \frac{E_0\cdot \Delta T+E_1\cdot \Delta T^2}{{\lambda }^2-{\lambda }_{TK}^2}}\right)$$

(10)

In this equation, $\Delta n_{abs}\left(\lambda ,T\right)$ represents the absolute refractive index change of optical material when the temperature changes by $\Delta T$. $n\left(\lambda ,T_0\right)$ denotes the refractive index at the working wavelength $\lambda$ under the reference temperature $T_0$(20 °C). $D_0$, $D_1$ and $D_2$ are temperature-dependent coefficients that characterize the linear and nonlinear variations in refractive index relative to the deviation of temperature baseline. $E_0$ and $E_1$ depend on both temperature and wavelength, indicating that chromatic dispersion is also influenced by temperature. ${\lambda }_{TK}$ is the characteristic wavelength parameter used in the material dispersion equation.

Based on Equation (10), the specific variation in refractive index with temperature can be determined. In the Multi-Configuration Editor, optical system structures corresponding to 20 °C and 25 °C were established, and optimization was performed for the lens curvature radii, refractive indices, and overall system design parameters. After implementing athermalization, the final design parameters of the collimation optical system are summarized in

Table 5. Optimized optical design parameters of the collimation system.

Surface Number	Radius (mm)	Thickness (mm)	Material	Diameter (mm)
0	/	5	/	/
1	17.95	2	H-QF1	6
2	12	5.2	/	/
3	−17.94	3	H-ZF5	8
4	−9.34	3	/	/
5	−74.016	3	H-ZF1A	10
6	−24.266	/	/	/

.

The optimized results indicate that the curvature radii and material parameters of lenses M1 and M2 exhibit notable changes.

Table 6. Statistical analysis of laser pointing and divergence angle variation after optimization.

Direction	Offset (μm)	Offset Angle (″)	Divergence Angle Variation (″)
X	−0.01 (Before optimization 0.051)	−0.02 (Before optimization 0.105)	2.0 (Before optimization 0.26°)
Y	0.08 (Before optimization 0.021)	0.16 (Before optimization 4.16)	2.3 (Before optimization 0.26°)

summarizes the variations in laser pointing and divergence angle at 25 °C. The maximum laser pointing deviation is 0.08 μm with a pointing angle shift of 0.16″, and the divergence angle variation is only 2.3″. Through athermalization design optimization, the collimation optical system maintains excellent pointing stability and minimal divergence variation within a 5 °C temperature range, providing a reliable reference for high-precision measurements in the on-orbit geometric calibration system.

4.2. Design of the Optical System of the Perturbation-Resistant Laser Relay Folding Unit

To address the issues of temperature sensitivity and insufficient thermal stability in the initial design of the laser relay folding unit, which utilized dual orthogonal planar mirrors, this article presents an optimized optical system design for the unit and proposes two improved solutions: a reflective right-angle prism and a right-angle cone mirror.

The reflective right-angle prism is formed by machining a cylinder along its axis to create three mutually perpendicular isosceles right triangles. These three triangles serve as orthogonal reflective surfaces, while the cylindrical base acts as the incident and exit surfaces for the light beam, as shown in Figure 15.

An important feature of the reflective right-angle prism is its unique retroreflective capability. The geometric principle of light propagation through the prism is illustrated in Figure 16. When a light beam enters the prism perpendicularly through its base, it undergoes three successive reflections on the three mutually perpendicular reflective surfaces before exiting again from the base. The incident ray A₁ and the exiting ray A₄ are parallel but propagate in opposite directions.

However, due to manufacturing limitations, it is difficult to achieve a complete vertical relationship between the three reflective surfaces in the actual processing process. These slight angular deviations cause minor changes in the propagation direction of the reflected beam, preventing it from remaining perfectly parallel to the incident ray, thereby affecting the system’s optical performance [27,28]. This angle error is a constant systematic error, which can be eliminated by theoretical calculation and practical calibration.

Firstly, the error analysis of two orthogonal mirrors is considered. Let the two mirrors be denoted as $M_1$ and $M_2$, with their respective unit normal vectors $n_1$ and $n_2$. Ideally, the angle between the two normals is 90°, $n_1\cdot n_2=0$. If the included angle becomes $\pi /2-\Delta \alpha$ due to the error caused by processing and manufacturing, then:

$$n_1\cdot n_2=\cos \left({\displaystyle \frac{\pi }{2}}-\Delta \alpha \right)=\sin \Delta \alpha$$

(11)

Express the $n_1$ and $n_2$ in component form as follows:

$$n_{1x}{n}_{2x}+n_{1y}{n}_{2y}+n_{1z}{n}_{2z}=\sin \Delta \alpha$$

(12)

Since the normals are unit vectors, they satisfy the following conditions:

$$n_{ix}^2+n_{iy}^2+n_{iz}^2=1$$

(13)

Next, the reflection characteristics of the single-sided mirror are analyzed. Let the incident light direction vector be $I_0$, and the reflected light direction vector be $I_r$. According to the law of reflection, their relationship can be expressed as:

$$I_r=I_0-2(I_0\cdot n_r)n_r$$

(14)

where $n_r$ is the normal vector of the reflecting surface. Rewriting it into matrix form can be expressed as:

$$I_{r,i}=H_r{I}_{0,i}$$

(15)

The general form of reflection matrix $H_r$ is:

$$H_r=\left[\begin{array}{ccc}1-2n_x^2& -2n_x{n}_y& -2n_x{n}_z\\ -2n_y{n}_x& 1-2n_y^2& -2n_y{n}_z\\ -2n_z{n}_x& -2n_z{n}_y& 1-2n_z^2\end{array}\right]$$

(16)

Finally, extended to the reflective right-angle prism, under ideal conditions, the three mirror surfaces are mutually perpendicular, with their normal vectors defined as follows:

$$\left\{\begin{array}{c}{n}_1=(1,0,0)\\ n_2=(0,1,0)\\ n_3=(0,0,1)\end{array}\right.$$

(17)

When manufacturing errors are present, it is assumed that the angle between each pair of the three reflective surfaces is not 90°, with error values denoted as $\Delta {\alpha }_{12}$, $\Delta {\alpha }_{13}$, and $\Delta {\alpha }_{23}$, respectively. Assuming equal errors, i.e., $\Delta {\alpha }_{12}=\Delta {\alpha }_{13}=\Delta {\alpha }_{23}={\theta }_p$, the expressions for the normal vectors can be approximated as follows:

$$\left\{\begin{array}{c}{n}_2\approx ({\theta }_p,1,0)\hfill \\ n_3\approx ({\theta }_p,{\theta }_p,\sqrt{1-2{\theta }_p^2})\hfill \end{array}\right.$$

(18)

When the light ray is incident perpendicularly onto the base surface of a reflective right-angle prism and subsequently reflects off the three mirror surfaces in sequence, its final direction is determined by the consecutive multiplication of the three reflection matrices. Assuming the incident light direction is $I_0=(0,0,-1)$, the outgoing light direction vector after three reflections can be expressed as:

$$I_r=H_3{H}_2{H}_1{I}_0$$

(19)

Considering the angular deviations introduced by errors, each reflection causes a slight deflection in the outgoing light direction. By approximating the deflection angle, the deviation introduced by a single reflection is denoted as ${\delta }_1\approx 2{\theta }_p$. After three reflections, it can be seen from the geometric relationship that the final deflection angle is the superposition of the deviations caused by three reflections:

$$\delta =\sqrt{3}{\delta }_1\approx 3.46{\theta }_p$$

(20)

It can be seen that when the angular errors between the three surfaces are equal, the deviation angle of the outgoing light exhibits a linear relationship with the error and is amplified by a factor of 3.46 due to the triple reflections. If the angular errors differ among the three surfaces, complementary or cumulative effects may occur, and the resulting deviation must be determined through experimental measurement.

Compared to the reflective right-angle prism, which involves three reflections and may introduce larger angular errors, the right-angle cone mirror requires only two reflections, significantly reducing the angular deviation. The design of the right-angle internal conical reflector is formed by rotating the principal cross-section of an isosceles right-angle reflector around the perpendicular line passing through the vertex of the inclined surface, thereby creating a reflective structure with conical surface characteristics, as illustrated in Figure 17.

The optical axis of the right-angle cone mirror passes through the cone vertex and is perpendicular to the base surface. When light enters along the optical axis from the base, the reflected rays converge onto a cross-sectional plane that intersects the optical axis. This characteristic allows an ideal right-angle cone mirror to impart a certain focusing effect on parallel light beams, thereby reducing spatial dimensional requirements. Moreover, compared to the reflective right-angle prism, the right-angle cone mirror further reduces angular errors by minimizing the number of reflections and optimizing the structural design, thereby enhancing the stability and accuracy of the optical path. The structural design of the right-angle cone mirror within the laser relay folding unit is illustrated in Figure 18.

In the design of the laser relay folding unit, in addition to focusing on the angular errors of the two types of mirrors, comprehensive consideration must also be given to system volume, weight, manufacturing complexity, and thermal stability. These parameters directly affect the overall performance and reliability of the on-orbit geometric calibration system.

The laser relay folding unit is mounted on the sidewall of the load-bearing cylinder of the space camera. If the system volume is too large, it may obstruct the optical path required for normal imaging. Meanwhile, system mass is another critical factor; excessive weight increases the mechanical load on the load-bearing structure, potentially leading to stress-induced deformation, which can negatively impact the imaging quality of the optical system.

Table 7. Design parameters of the laser relay folding unit.

Laser Relay Folding Unit	Volume (cm3)	Material	Mass (g)
Reflective right-angle prism	9.49	SI	2212 3037 2400
		SIC
		Glass–ceramic
Right-angle cone mirror	2.74	SI	638
		SIC	877
		Glass–ceramic	693

summarizes the specific mass parameters of the two design schemes, a right-angle prism and a right-angle cone mirror, constructed from different materials under the condition of constant system volume.

Compared with the reflective right-angle prism, the right-angle cone mirror offers significant advantages in terms of fabrication precision and assembly process. Taking into account key factors such as system volume, mass, manufacturing complexity, and angular error, the right-angle cone mirror was ultimately selected as the core component of the laser relay folding unit. This choice ensures higher measurement accuracy and long-term operational reliability for the on-orbit geometric calibration system.

Due to the fact that temperature variations affect the structural stability and optical performance of the laser relay folding unit, its thermal stability becomes a key consideration in the design of the on-orbit geometric calibration system. A thermo–mechanics–optical coupling finite element analysis method was employed to conduct an in-depth study on the thermal deformation and stability of the right-angle cone mirror system under different material conditions, in order to evaluate its anti-interference ability to temperature change [29]. The finite element model of the right-angle cone mirror is shown in Figure 19. A uniform temperature operating condition was applied for thermal deformation simulation analysis to model the system response as the temperature increased from 20 °C to 21 °C. For different material combinations, the displacement variation of the mirror surface was analyzed, and the statistical results are shown in

Table 8. Statistics of right-angle cone mirror displacement under different materials.

Material (Structure/Mirror Body)	Maximum Displacement (μm)
Aluminum Alloy/Silicon	4.45
Aluminum Alloy/Glass–ceramic	4.37
Invar/Silicon	0.39
Invar/Glass–ceramic	0.38

. The results show that when the support structure is made of Invar and the mirror is made of glass–ceramic, the displacement variation of the mirror surface is the smallest, only 0.38 μm, indicating that this material combination provides better thermal stability under temperature variations and can effectively reduce the impact of thermally induced deformation on the measurement accuracy of the on-orbit geometric calibration system.

Subsequently, by calculating the nodal displacement parameters under different material combinations, the rotational angles around the three axes and the rigid body displacements of the right-angle cone mirror surface were obtained, with the statistical results shown in

Table 9. Statistics of parameter variations of the right-angle cone mirror surface.

Material	Mirror Surface	Rotation about the Three Axes (″)			Rigid Body Displacement (μm)
		X Axis	Y Axis	Z Axis	X Direction	Y Direction	Z Direction
Aluminum Alloy/Silicon	R1	1.25	2.87	1.02	−9.73	0.11	11.98
	R2	0.46	−0.78	0.99	3.4	−2.68	−0.39
Aluminum Alloy/Glass–ceramic	R1	1.43	3.46	1.1	−12.1	0.82	14.35
	R2	0.24	−1.74	0.97	7.12	−3.46	−3.25
Invar/Silicon	R1	0.05	−0.39	−0.01	1.65	−0.19	−1.53
	R2	0.04	0.16	0.01	−0.82	0.18	0.28
Invar/Glass–ceramic	R1	0.04	−0.29	0.01	1.28	0.14	−1.18
	R2	0.01	0.01	0.01	0.27	0.04	0.16

.

The surface parameters of the right-angle cone mirror were imported into optical design software to calculate the pointing error under different material conditions. The statistical results show that the combination of Invar and glass–ceramic yields the smallest change in beam pointing angle, only 0.34″, followed by the combination of Invar and silicon with a pointing angle change of 0.37″. Among these, selecting Invar and glass–ceramic minimizes the displacement of the cone mirror but results in a higher mass. The thermal stability of glass–ceramic combined with silicon is relatively close and can reduce the system mass to some extent. Considering both thermal stability and mass, Invar and silicon were ultimately chosen as the design materials for the laser relay folding unit.

5. Thermal Stability Simulation Analysis of the On-Orbit Geometric Calibration System

During on-orbit operation, under the combined effects of external heat flux, cold space, and internal heat sources, various units of the on-orbit geometric calibration system experience different temperature fluctuations. Due to the differences in coefficients of thermal expansion among different materials, such temperature variations induce thermal stress and elastic deformation in system components, leading to changes in the relative positions of optical elements and surface shape errors of optical components, which ultimately affect the measurement accuracy of the system. Therefore, conducting thermal stability simulation analysis is of great significance for evaluating the variation in system measurement accuracy errors.

The thermal stability simulation workflow of the on-orbit geometric calibration system is illustrated in Figure 20. First, the nodal displacements and rigid-body rotations of each optical element surface under specified thermal conditions are obtained through finite element analysis. Second, the surface displacements of each optical element are fitted and decomposed into two components: “rigid-body displacement and rotation” and “local surface deformation residuals.” Finally, the fitted rigid-body displacements and rotations, together with the surface deformation terms, are applied as corrections to the corresponding surface parameters in the optical software, after which the system’s exit pupil and chief ray are recalculated to determine pointing deviations and focal plane displacements. The analysis workflow primarily comprises three parts: structural finite element analysis to obtain nodal displacement data, fitting of surface deformation data, and optical performance evaluation. By applying different thermal conditions to the reference laser unit, laser relay folding unit, and star tracker unit, the impact of temperature variations on the performance of the on-orbit geometric calibration system can be comprehensively assessed.

In the design process of the on-orbit geometric calibration system, the model matching accuracy among optics, mechanics, and thermodynamics disciplines is a key issue. To improve accuracy and reduce coordinate transformation operations between different software analysis models, a unified absolute coordinate system will be adopted to construct models across disciplines. Due to its priority and importance, the absolute coordinate system defined in the optical design can serve as the reference for other disciplinary models. In the optical design software, the positive Z axis of the absolute coordinate system is defined as the direction of light propagation, the positive Y axis points vertically upward in the meridional plane, and the positive X axis is determined by the right-hand rule. In addition, optical design requires establishing local coordinate systems for each optical element to precisely describe the translation and rotation characteristics of their surfaces. This unified coordinate system framework can not only enhance the collaborative accuracy of multidisciplinary models but also reduce errors introduced by coordinate transformations, thereby significantly improving the overall performance and reliability of the on-orbit geometric calibration system.

The finite element model of the on-orbit geometric calibration system is shown in Figure 21. The overall system analysis further includes the focal plane electric control box structure, the main support structures of the camera and star tracker, and the load-bearing barrel structure that secures the laser relay folding unit, thereby providing a more comprehensive evaluation of the system’s measurement accuracy. The optical element mounting structures of each system unit are made of low coefficient of thermal expansion materials and combined with the thermal control system to effectively regulate the overall temperature variation amplitude, reducing the impact of thermal deformation on measurement accuracy. The optimized design keeps the structural deformation of the reference laser unit, laser relay folding unit, and dichroic mirror within a minimal range.

Table 10. Statistics of surface parameter variations in on-orbit calibration optics.

On-Orbit Geometric Calibration System	Mirror Surface	Rotation About the Three Axes (″)			Rigid Body Displacement (μm)
		X Axis	Y Axis	Z Axis	X Direction	Y Direction	Z Direction
Reference laser unit	M1-1	0.93	−0.04	0.01	3 × 10−3	0.22	−0.073
	M1-2	0.93	−0.03	0.01	3.14 × 10−3	0.23	−0.065
	M2-3	0.96	−0.01	0.01	8.27 × 10−4	0.23	−0.042
	M2-4	1.02	−0.05	0.02	2.8 × 10−3	0.23	−0.031
	M3-5	1.08	0.02	0.05	−6.98 × 10−3	0.28	−0.015
	M3-6	0.08	−0.01	0.05	5.14 × 10−3	0.21	−0.018
Laser relay folding unit	R1	0.05	−0.42	−0.01	1.72	−0.28	−1.64
	R2	0.04	0.35	0.01	−0.93	0.19	0.32
Star tracker unit	Dichroic mirror	0.06	0.07	0.01	0.08	0.08	0.26

summarizes the rotation around the three axes and the rigid body displacement data of the optical element surfaces in each unit. The results indicate that the optimized design effectively controls the thermal deformation of the system, keeping the displacements of the reference laser unit, laser relay folding unit, and dichroic mirror within a minimal range.

However, the structural analysis results only serve as an intermediate step in the thermal stability simulation and cannot directly reflect changes in the system’s measurement accuracy. To accurately evaluate the measurement accuracy error under varying thermal conditions, the rigid-body displacement and rotation data of each optical element surface must be imported into optical analysis software. By updating the surface parameters of the original optical model and performing optical analysis, the measurement accuracy error of the system under temperature variations can be obtained. The results indicate that when the focal plane and the reference laser unit experience a 5 °C temperature change, the laser relay folding unit undergoes a 1 °C change, and the star tracker undergoes a 2 °C change; the maximum deviation of the in-orbit geometric calibration system’s measurement reference is only 0.57″ (3σ). For a low-Earth-orbit satellite at an altitude of approximately 500 km, and neglecting orbital and attitude errors, the use of the in-orbit geometric calibration system can improve the uncontrolled geolocation accuracy of the spaceborne camera imagery to the meter level.

6. Conclusions

To address the issues of low accuracy and poor timeliness in the on-orbit geometric calibration of high-resolution space cameras, this article proposes a high-precision real-time on-orbit geometric calibration scheme based on active optical monitoring and conducts in-depth analysis and optimization of the system’s thermal stability. The influence of temperature fluctuations on the focal plane, reference laser unit, and laser relay folding unit is analyzed, and a targeted athermal design optimization strategy is proposed. By optimizing the structural and material parameters of the collimation optical system, an athermal collimation system is developed, in which the pointing deviation is controlled within 0.16″ and the divergence angle variation remains below 2.3″ under 5 °C temperature change, ensuring laser stability.

In addition, to address the thermal stability issues of the laser relay folding unit, this paper proposes replacing the conventional pair of orthogonal plane mirrors with a right-angle cone mirror, aiming to reduce the impact of thermal deformation on the system’s measurement accuracy. Through thermo–mechanical–optical coupled analysis and optimization of structural materials, a combination of Invar and silicon is ultimately selected, ensuring that the beam pointing angle variation remains below 0.34″ under a 1 °C temperature change, thereby enhancing the system’s resistance to thermal disturbances.

Based on finite element simulation analysis, this article evaluates the stability of the on-orbit geometric calibration system under different thermal conditions. The study shows that, when the temperature fluctuation of the focal plane and reference laser unit is 5 °C, that of the laser relay folding unit is 1 °C, and that of the star tracker is 2 °C; the maximum deviation of the system’s measurement baseline does not exceed 0.57″(3σ), meeting the requirements for high-precision on-orbit geometric calibration. This research provides a theoretical foundation and engineering support for the long-term stable operation of geometric calibration systems for space cameras.

In the future, we will further investigate the on-orbit experiments of the geometric calibration system, enhance system accuracy through more stable optical measurement techniques, and extend the method to various spaceborne payloads to achieve high-precision on-orbit geometric calibration adapted to different mission requirements. The research presented in this paper provides theoretical and technical support for the high-precision, real-time calibration of high-resolution remote sensing cameras and will contribute to the application and advancement of next-generation remote sensing satellites in Earth observation, deep space exploration, and other fields.