On-Orbit Calibration Method for Rotation Axis Misalignment in Rotating Mirror-Based Wide-Field Space Cameras

Abstract

Rotating mirror-based space cameras are susceptible to mirror misalignment due to the severe vibrations experienced during rocket launch and the harsh, variable conditions of the space environment, which can result in deviations of the camera’s line of sight. To mitigate this risk, this study proposes a simulation-based on-orbit calibration method for quantifying rotating mirror misalignment using a system of pointing vector equations. The method employs star coordinates as a reference to establish the reference pointing vector for stars, while simultaneously developing a model of the rotating mirror imaging system. By incorporating a misalignment matrix, the actual pointing vector of star points is derived. Subsequently, the reference star pointing vector and the actual star point pointing vector are combined to formulate a system of pointing vector equations. Solving these equations enables precise measurement of the rotating mirror’s rotational misalignment without requiring additional spaceborne equipment. Through simulations, the three-axis misalignment of the rotating mirror is deduced from imaging pixel coordinates, given the known right ascension and declination of reference star points. The influence and patterns of three-axis misalignment on pointing accuracy are analyzed separately. Although validation based on real measurement data remains to be carried out in future work, this simulation-based method provides a theoretical foundation for the calibration of internal orientation elements of space cameras equipped with moving components.

1. Introduction

With the ongoing advancement of space optical remote sensing imaging technology, increasingly stringent demands have been placed on satellite attitude positioning accuracy, optical imaging quality, and optical axis pointing precision [1,2,3,4]. However, the intense vibrations during rocket launch and the complex, fluctuating conditions of the space environment can induce misalignment in the moving components of a camera, leading to deviations in the camera’s line of sight [5,6]. Such deviations significantly impair imaging quality and target positioning accuracy. This inherent uncertainty restricts the operational reliability of space cameras in orbit and introduces a degree of unpredictability into subsequent data processing.

Optical axis calibration is a critical technology for ensuring the pointing accuracy of space optical systems, directly impacting the in-orbit reliability of space cameras. In early research, Kim et al. developed a joint calibration system utilizing a total station, calibration targets, and prism markers, achieving single-pixel-level pointing accuracy for a rover mast camera [7]. This work laid the groundwork for visual tracking of dynamic targets in planetary exploration missions. Chai et al. proposed a spatial transformation model for a two-axis turntable camera system to calibrate axis angle errors, elucidating the spatial registration mechanism between mechanical and optical systems [8]. In the realm of dynamic calibration, Yang et al. introduced a visual–inertial online self-calibration framework, underscoring the necessity of continuous calibration to maintain accuracy in dynamic environments [9]. From a control systems perspective, Danh constructed a dynamic model for a two-axis gimbal based on line-of-sight (LOS) stability theory, demonstrating superior accuracy and robustness compared to conventional control approaches [10]. Chustz et al. provided a robust benchmark for ground validation with their off-road visual–inertial dataset, maintaining orientation errors within ±1° over extended periods [11]. Wang Yu enhanced three-axis accuracy by 10–20% using an unscented Kalman filter algorithm, integrating star camera and gyroscope data for high-precision attitude determination, outperforming the extended Kalman filter method [12].

Traditional optical axis calibration methods, which rely on ground-based equipment such as total stations [13,14], find it difficult to accommodate parameter drifts induced by launch shocks and in-orbit temperature variations. Consequently, star-based calibration, leveraging the radiation stability (annual variation rate < 0.1%) and all-sky distribution of stars, has emerged as the predominant approach for on-orbit calibration. Star pattern matching aims to enhance the accuracy, efficiency, and robustness of celestial navigation systems. Feature-based matching algorithms form the core of many star identification strategies. Liu et al. proposed an improved triangle algorithm tailored for astronomical cameras, focusing on guide star selection and catalog construction to facilitate precise matching [15]. Liang et al. employed the ICP (Iterative Closest Point) algorithm combined with local normal vector-based feature point extraction to achieve low-error star identification [16]. Wei Fei et al. introduced a star pattern matching algorithm based on image distance and pose, addressing the limitations of classical quadrilateral matching methods and achieving a success rate exceeding 97% [17]. This approach improves star registration accuracy in complex scenarios. Studies indicate that optimizing star feature extraction algorithms can achieve sub-arcsecond calibration accuracy; however, the dynamic misalignment calibration of rotating mirror cameras continues to encounter technical challenges.

In the context of rotation axis error analysis, research on solar reflector systems identifies installation measurement errors and alignment inaccuracies as primary sources of deviation. Zheng et al. examined solar tracking errors in Fresnel reflectors resulting from mirror rotation axis misalignment [18]. Similarly, Yan et al. proposed precise surface installation and alignment techniques, including mathematical models for three-rotation and two-rotation alignment methods, to minimize installation errors affecting reflector axis accuracy [19]. Cao et al. reported the first light of a solar telescope, implicitly highlighting the critical role of precise mirror alignment and installation in achieving optimal optical performance [20]. Chen et al. developed a six-degree-of-freedom optoelectronic measurement system for high-resolution assessment of translation and rotation errors [21]. Lou et al. designed a five-degree-of-freedom embedded monitoring system employing multispectral self-collimation technology (sampling rate 1 kHz), enabling high-precision real-time feedback on turntable radial runout and providing data support for dynamic error compensation [22].

The evolution of star calibration technology reflects a trend toward interdisciplinary integration. Edmonds pioneered a star–solar photoelectric equivalence model, constraining systematic errors to within a 5% baseline [23]. Kelly et al. constructed an all-sky star feature database containing 120 million star parameters, integrating adaptive optics modeling to reduce star point positioning errors to 0.1 pixels [24]. Lu’s team proposed a comprehensive Kalman calibration framework [25], reducing installation error coupling by 63% through state-space modeling. Xie et al. developed an on-orbit calibration system for the APS star tracker, employing a non-uniform B-spline lens distortion compensation model to enhance calibration parameter stability by 40% [26]. In navigation systems, Lv et al. introduced an inertial/star refraction integrated calibration method, compressing positioning errors to the 300 m-level (3σ) by establishing a 14-dimensional state observation equation [27].

Through the above research, the visual axis pointing calibration methods mainly include ground calibration and in-orbit calibration, and the in-orbit calibration accuracy mostly depends on the pointing accuracy of the star sensitizer. Considering that the camera contains a movable part—the rotating mirror—and due to the problem that the rotating axis may be out of adjustment due to the thermal effect, it is urgently needed to put forward a new dynamic calibration method to ensure that a space camera can maintain both pointing accuracy and imaging quality reliability during the subsequent imaging process. A new dynamic calibration method is urgently needed to ensure that the space camera can maintain both pointing accuracy and imaging quality reliability during the subsequent imaging process.

Thus, this study addresses the on-orbit calibration challenge for rotating mirror-based wide-field space cameras by proposing an innovative dynamic calibration method based on pointing vector equations. The approach first establishes a satellite attitude reference coordinate system, constructing the theoretical pointing vector via coordinate transformations. It then develops an optical transmission model of the rotating mirror system, incorporating a misalignment matrix with six degrees of freedom, to derive the analytical relationship between actual star point coordinates and misalignment parameters. Finally, an overdetermined system of equations is formulated and solved using the Levenberg–Marquardt optimization algorithm to determine the misalignment parameters. This study is based on simulation validation rather than experimental verification with actual on-orbit data. Nonetheless, the proposed method demonstrates the capability to accurately estimate the rotational misalignment of the scanning mirror without relying on additional spaceborne instrumentation, offering a novel technical pathway for autonomous on-orbit calibration of high-resolution Earth observation satellites and providing a valuable reference for researchers and engineers in related fields.

2. Imaging Mode of Rotating Mirror Wide-Field Space Camera

To achieve large-area imaging within key areas of interest, remote sensing satellites can utilize high-speed attitude maneuvering capabilities to rapidly adjust the pointing direction of the sensor, enabling multi-mode imaging of targets. However, current agile satellites have limited attitude maneuvering precision and stability, making it challenging to meet the attitude requirements of certain imaging sensors that demand high agility. Compared to agile satellite maneuvers, dual-mirror scanning offers only one-dimensional rotation, which is less flexible but provides better stability, higher positional accuracy, and a simpler control system. The dual-mirror scanning optical system, as shown in Figure 1, when combined with high-resolution optical camera imaging, can expand the imaging range in a specified direction.

While enabling rapid observation of unknown targets over large areas, it is also necessary to enhance temporal resolution to ensure frequent imaging of extensive key areas. The multi-strip vertical orbit splicing imaging mode, as illustrated in Figure 2, achieves large-area coverage in both the along-track and cross-track dimensions by adjusting the scanning rate of the dual mirror. During the next orbit, multi-strip vertical splicing imaging is performed again, allowing multiple scanning detections of the key area. Within a short overhead time, the imaging swath can be 5–10 times that of traditional push-broom imaging, significantly improving imaging efficiency.

The mentioned rotation axis of the scanning mirror, whose axis misalignment induces deviation of the line-of-sight (LOS) from its designed orientation, alters the push-broom scanning direction and generating image motion. This effect degrades dynamic imaging quality by introducing blur and spatial distortion. Consequently, maintaining accurate and stable LOS alignment is critical for preserving high-fidelity imaging performance in push-broom systems.

3. On-Orbit Calibration Method for Rotating Mirror Misalignment Based on Pointing Vector Equations

Based on an in-depth study of the imaging mode of rotating mirror-based wide-field space cameras, this paper proposes an on-orbit calibration method for quantifying rotating mirror misalignment using a system of pointing vector equations. This method first establishes an ideal model of the rotating mirror imaging system. Subsequently, a misalignment matrix is introduced to form a model of the misaligned rotating mirror imaging system, deriving the actual pointing vector of star points. Concurrently, based on actual star maps, a star catalog, and a high-precision centroid extraction algorithm, reference stars meeting brightness and initial position requirements are selected. Through coordinate transformations, the reference star pointing vector is established. Finally, by combining the reference star pointing vector and the actual star point pointing vector, a system of pointing vector equations for rotating mirror misalignment is formulated. Solving this system yields the numerical solution for the rotating mirror misalignment. The overall process is illustrated in the flowchart below (Figure 3).

3.1. Actual Star Point Pointing Vector Model

The precise and stable operation of the rotating mirror directly determines the line-of-sight pointing accuracy of the rotating mirror-based space camera. Mirror misalignment can be decomposed into two components: offset and tilt. According to applied optics principles, the offset does not affect the pointing accuracy of the line of sight as the rotating mirror is a sufficiently large planar reflector. Therefore, this study focuses on the tilt misalignment of the rotating mirror, as illustrated below (Figure 4).

To facilitate whole-satellite attitude maneuvers for correcting rotating mirror misalignment, the tilt misalignment can be decomposed into Euler angle deviations (${\delta }_x$, ${\delta }_y$, ${\delta }_z$) along three coordinate axes. Consequently, the calibration of line-of-sight deviation is transformed into the precise measurement of Euler angle deviations along these three axes.

In an ideal state, the coordinate systems and vector relationships of the rotating mirror wide-field imaging system are as shown below, involving the following three primary coordinate systems:

Camera coordinate system (C): The origin coincides with the principal point of the optical lens, with the Z-axis pointing toward the Earth’s center, the X-axis aligned with the satellite’s flight direction, and the Y-axis perpendicular to the X-Z plane, forming a right-handed coordinate system. This system can be considered coincident with the satellite’s body coordinate system.

Focal plane coordinate system (p): Obtained by translating the camera coordinate system (C) along the optical axis by the focal length $f^{\prime }$.

Rotating mirror coordinate system (RM): Move the camera coordinate system C horizontally against the optical axis by the rotating mirror installation distance $L_{RM}$ to obtain the rotating mirror coordinate system.

In Figure 5, the red ray represents the ray tracing path of a certain star point $P^{″}$ at an instantaneous moment. When the star point light shines on the rotating mirror at point $P^{\prime }$, it undergoes planar reflection. The normal vector of the reflection surface is ${\overrightarrow{n}}_{RM}$, and finally, the image is formed at point $P$ of the online array detector through the lens. Considering that the rotating mirror rotates at a speed of $\dot{\alpha }$, point $P$ will slowly move in the positive direction of $Z_p$ until it moves out of the row range of this linear array detector, completing the imaging of that row and starting the shooting of the next row of images.

In the ideal rotating mirror imaging model, it is necessary to determine the mapping relationship between the rotating mirror angle $\alpha$ and the number of image rows $N$ first. Since the imaging feature of rotating mirror scanning is to output images line by line, the row images are stitched in sequence to form a two-dimensional image. Therefore, it is only necessary to study the rotation angle change amount $\mathsf{\Delta }\alpha$ corresponding to the single-line imaging process, as well as the rotation angle change amount ${\alpha }_{\mathrm{p}}$ at the sub-pixel position of the star point centroid. Finally, based on the initial rotating mirror angle ${\alpha }_0$, by repeatedly accumulating $\left(N-1\right)\mathsf{\Delta }\alpha$ and 1 ${\alpha }_{\mathrm{p}}$, the relationship between the measured value of the rotating mirror encoder ${\alpha }_t$ and the number of image rows can be obtained. Among them, ${\alpha }_0$ is the value of the rotating mirror encoder at the beginning of the scanning imaging, which is a known quantity. The relationship between the physical quantity of the image plane and $\mathsf{\Delta }\alpha$ and ${\alpha }_{\mathrm{p}}$ will be established below (Figure 6).

As shown above, the star point corresponding to the initial rotation position is located at the center of the linear array detector. When the rotating mirror rotates by an angle $\mathsf{\Delta }\alpha$, the reflected light undergoes a shift, which can be equated to the incidence of light from a non-zero field of view. Due to the characteristics of a telephoto optical system, incident light within the same field of view is parallel and converges at the same point on the focal plane. Thus, the angular relationship is as follows:

$$\angle O_p{O}_{c}P=\angle O_c{O}_{RM}{P}_c={\omega }_p\left(p_z\right)$$

(1)

where ${\omega }_p\left(p_z\right)$ is the field-of-view angle corresponding to the offset of the star point, and $p_z$ is the movement amount of the star point in the column direction. Therefore, the relationship between the two can be expressed as follows:

$${\omega }_p\left(p_z\right)=\arctan \left({\displaystyle \frac{p_z}{f^{\prime }}}\right)$$

(2)

The change in the exit angle of the plane mirror is twice that of the rotating mirror. When $p_z$ is the pixel size $l_{pixel}$, $\mathsf{\Delta }\alpha$ can be calculated as follows:

$$\mathsf{\Delta }\alpha ={\displaystyle \frac{1}{2}}{\omega }_p\left(l_{pixel}\right)$$

(3)

When $p_z$ is the subpixel centroid offset $z_p$, ${\alpha }_p$ can be calculated as follows:

$${\alpha }_p={\displaystyle \frac{1}{2}}{\omega }_p\left(z_p\right)$$

(4)

Therefore, the rotation angle ${\alpha }_t$ corresponding to the $N$th row of the image is as follows:

$${\alpha }_t={\alpha }_0+{\displaystyle \frac{N-1}{2}}·\arctan \left({\displaystyle \frac{l_{pixel}}{f^{\prime }}}\right)+\arctan \left({\displaystyle \frac{z_p}{f^{\prime }}}\right)$$

(5)

Let the direction vector of $P\to P^{\prime }$ in the camera coordinate system C be as follows:

$${\overrightarrow{\nu }}_{in}=\left(x_c,y_c,z_c\right)$$

(6)

In the image plane coordinate system p, if the position of the star point in the row is $\left(x_p,0,z_p\right)$ then the vector ${\overrightarrow{\nu }}_{in}$ satisfies the following system of equations:

$$\left\{\begin{array}{c}{x}_p·y_c-f^{\prime }·x_c=0\\ f^{\prime }·z_c-z_p·y_c=0\\ {x_c}^2+{y_c}^2+{z_c}^2=1\end{array}\right.$$

(7)

Solving this system of equations will obtain two unit vectors, which are reversed. Take the set of solutions where $y_c$ is negative. The physical meaning of the first equation in the system of equations is the plane passing through the $Z_c$-axis and point P $\left(x_p,f^{\prime },z_p\right)$ in the camera coordinate system C. The physical meaning of the second equation is the plane passing through the $X_c$-axis and point P $\left(x_p,f^{\prime },z_p\right)$ in the camera coordinate system C.

Suppose the zero position of the rotating mirror angle encoder is perpendicular to the $Z_{RM}$-axis. Therefore, the zero normal vector ${\overrightarrow{n}}_0=(0,0,1)$ at this time in the X-Y-Z rotation sequence, and so the offset rotation matrix is:

$$R_{err}=R_z\left({\delta }_z\right)·R_y\left({\delta }_y\right)·R_x\left({\delta }_x\right)$$

(8)

where $R_z\left({\delta }_z\right)$, $R_y\left({\delta }_y\right)$, and $R_x\left({\delta }_x\right)$ are rotation primitive matrices, which can be expressed by Formulas (9)–(11) as follows:

$$R_z({\delta }_z)=\left[\begin{array}{ccc}\cos {\delta }_z& -\sin {\delta }_z& 0\\ \sin {\delta }_z& \cos {\delta }_z& 0\\ 0& 0& 1\end{array}\right]$$

(9)

$$R_y({\delta }_y)=\left[\begin{array}{ccc}\cos {\delta }_y& 0& \sin {\delta }_y\\ 0& 1& 0\\ -\sin {\delta }_y& 0& \cos {\delta }_y\end{array}\right]$$

(10)

$$R_x({\delta }_x)=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos {\delta }_x& -\sin {\delta }_x\\ 0& \sin {\delta }_x& \cos {\delta }_x\end{array}\right]$$

(11)

The rotation matrix of the rotating mirror can be expressed as follows:

$$R_x({\alpha }_t)=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos {\alpha }_t& -\sin {\alpha }_t\\ 0& \sin {\alpha }_t& \cos {\alpha }_t\end{array}\right]$$

(12)

Therefore, the normal vector of the misaligned mirror surface is as follows:

$${\overrightarrow{n}}_{RM}=R_{err}·R_x({\alpha }_t)·{\overrightarrow{n}}_0=\left[\begin{array}{c}\cos {\delta }_z·\sin {\delta }_y·\cos \left({\delta }_x+{\alpha }_t\right)+\sin {\delta }_z·\sin \left({\delta }_x+{\alpha }_t\right)\\ \sin {\delta }_z·\sin {\delta }_y·\cos \left({\delta }_x+{\alpha }_t\right)-\cos {\delta }_z·\sin \left({\delta }_x+{\alpha }_t\right)\\ \cos {\delta }_y·\cos \left({\delta }_x+{\alpha }_t\right)\end{array}\right]=\left[\begin{array}{c}{n}_{xRM}\\ n_{yRM}\\ n_{zRM}\end{array}\right]$$

(13)

After knowing the incident direction vector and the mirror normal vector, the exit vector after mirror reflection is as follows:

$${\overrightarrow{\nu }}_{out}={\overrightarrow{\nu }}_{in}-2·\left({\overrightarrow{\nu }}_{in}·{\overrightarrow{n}}_{RM}\right)·{\overrightarrow{n}}_{RM}$$

(14)

Substituting the calculation results of Equations (7) and (13) into Equation (14), the actual star point pointing vector in the camera coordinate system can be obtained as follows:

$${\overrightarrow{\nu }}_{out}=\left[\begin{array}{c}{x^{\prime }}_c\\ {y^{\prime }}_c\\ {z^{\prime }}_c\end{array}\right]=\left[\begin{array}{c}{x}_c-2·C·n_{xRM}\\ y_c-2·C·n_{yRM}\\ z_c-2·C·n_{zRM}\end{array}\right]$$

(15)

Among that $C=x_c·n_{xRM}+y_c·n_{yRM}+z_c·n_{zRM}$.

3.2. Reference Star Pointing Vector Model

After the actual star map is captured, the imaging sky area can be precisely determined through star map recognition and matching technology, and a star with appropriate brightness can be selected as a reference star (Figure 7).

During the scanning imaging of the rotating mirror wide-field camera, the satellite’s attitude is fixed, and the Euler angles $\left(\phi ,\theta ,\psi \right)$ describing the satellite’s attitude are known.

The direction of the reference star in the J2000 inertial system is represented by ${\overrightarrow{S}}_{inertial}^I$, which is as follows:

$${\overrightarrow{S}}_{inertial}^I=\left[\begin{array}{c}\cos \lambda ·\cos \eta \\ \cos \lambda ·\sin \eta \\ \sin \lambda \end{array}\right]$$

(16)

where $\eta$ represents the right ascension of the star in the Earth’s inertial system, and $\lambda$ represents the declination. The specific values can be queried through astronomical databases. Because the distance between the reference star and the Earth is so great that the diameter of the Earth can be ignored, in the same reference frame the direction vector from the center of the Earth to a certain star and the direction vector from the center of mass of a near-Earth satellite to that star can be regarded as parallel. Under the I series, it can be described as follows:

$${\overrightarrow{n}}_{BS}^I={\overrightarrow{n}}_{IS}^I$$

(17)

In the rotation sequence of the Z-X-Y coordinate system, the transformation matrix from the Earth inertial system to the satellite orbit system is as follows:

$$R_{oI}=R_y\left(-u-{\displaystyle \frac{\pi }{2}}\right)·R_x\left(i-{\displaystyle \frac{\pi }{2}}\right)·R_z\left(\mathsf{\Omega }\right)=\left[\begin{array}{ccc}-\sin u·\cos \mathsf{\Omega }-\cos u·\cos i·\sin \mathsf{\Omega }& -\sin u·\sin \mathsf{\Omega }+\cos u·\cos i·\cos \mathsf{\Omega }& \cos u·\sin i\\ -\sin i·\sin \mathsf{\Omega }& \sin i·\cos \mathsf{\Omega }& -\cos i\\ -\cos u·\cos \mathsf{\Omega }+\sin u·\cos i·\sin \mathsf{\Omega }& -\cos u·\sin \mathsf{\Omega }-\sin u·\cos i·\cos \mathsf{\Omega }& -\sin u·\sin i\end{array}\right]$$

(18)

where u represents the true perigee angle, i represents the orbital inclination angle, and Ω represents the right ascension of the ascending node.

The coordinate transformation matrix from the satellite orbit system to the satellite body system is as follows:

$$R_{Bo}=R_X\left(\phi \right)·R_Y\left(\theta \right)·R_Z\left(\psi \right)$$

(19)

where the rotation matrix of the primitive coordinate axes is as follows:

$$R_X(\phi )=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \phi & \sin \phi \\ 0& -\sin \phi & \cos \phi \end{array}\right]$$

(20)

$$R_Y(\theta )=\left[\begin{array}{ccc}\cos \theta & 0& -\sin \theta \\ 0& 1& 0\\ \sin \theta & 0& \cos \theta \end{array}\right]$$

(21)

$$R_Z(\psi )=\left[\begin{array}{ccc}\cos \psi & \sin \psi & 0\\ -\sin \psi & \cos \psi & 0\\ 0& 0& 1\end{array}\right]$$

(22)

The pointing vector of this satellite point in satellite system B can be obtained as follows:

$${\overrightarrow{n}}_{BS}^B=R_{Bo}·R_{oI}·{\overrightarrow{S}}_{inertial}^I$$

(23)

Also, since the satellite system and the camera system are re-consolidated and fixedly connected, the reference stellar pointing vector in the camera system C is equal to that in the satellite system.

$${\overrightarrow{n}}_{BS}^C={\overrightarrow{n}}_{BS}^B$$

(24)

Solve the vector system of Equations (13) and (23) simultaneously. The number of unknowns is equal to the number of equations, and the system of equations is numerically solvable. Then the misalignment quantities ${\delta }_x$, ${\delta }_y$, and ${\delta }_z$ of the rotating lens can be obtained.

4. Simulation Analysis

4.1. Actual Star Vector Calculation

At present, linear array push-scan cameras are commonly used in optical satellites. In order to better analyze the imaging characteristics of the vertical orbit search camera and conduct a quantitative analysis of the optical axis pointing error in this mode, the simulation parameters of the camera are as follows (

Table 1. Typical camera parameters.

Parameter	Value
Orbital Altitude	500 km
Focal Length	500 mm
Wavelength	450 nm–800 nm
Resolution	3.5 m
Field of View	18°
F-number	5
Pixel Size	3.5 μm
Integration Levels	1–128
Average Quantum Efficiency	65%

).

To ensure imaging quality, the scanning mirror must maintain high stability during high-speed rotation to avoid degradation caused by mirror jitter. The rotating mirror adopts a U-shaped symmetrical support structure, offering high rigidity and excellent balance performance. Driven by a stepper motor, the mirror achieves a scanning speed exceeding 10°/s with a stability better than 1%. Based on the actual conditions of the mirror and motor, the simulation considers a mirror angular velocity range of 1–11°/s.

To analyze the variation in reference star points during imaging, simulations were conducted to generate star map data under different misalignment error conditions, enabling high-precision calculation of installation errors. Based on star map simulation data with varying misalignment levels, centroid extraction, star map identification, and vector calculations were performed. The computed misalignment values were compared with the input true misalignment values to evaluate the model’s computational accuracy.

According to the aforementioned conclusion, the deflection of the rotating mirror can be decomposed into the angle deviations around the X, Y, and Z axes, totaling three degrees of freedom. The vector equation $v_{out}({\delta }_x,{\delta }_y,{\delta }_z)=n_C$ established by solving three misalignment angles with a single reference star point is essentially a condition for the equality of three-dimensional vectors and can be mathematically decomposed into three scalar equations. However, these three equations are not always linearly independent—in some cases they may be underdetermined or pathological. Especially when the incident and exit directions are coplanar or approximately parallel, coupling occurs between the rotation angles, making it impossible for the system of equations to uniquely determine these three angles. However, this problem can be avoided by solving with multiple star points. At this time, there is the following superimposed residual function:

$$J({\delta }_x,{\delta }_y,{\delta }_z)={{\displaystyle \sum _{i=1}^N‖v_{out}^{(i)}({\delta }_x,{\delta }_y,{\delta }_z)-n_C^{(i)}‖}}^2\hspace{1em},\hspace{1em}\hspace{1em}i=1,2,3,\cdots N$$

(25)

Therefore, using multiple star points makes the system overdetermined. Through least-squares fitting it is possible to eliminate equation degeneracy, thereby improving the stability and accuracy of the solution.

In conclusion, considering the application requirements a sky region with three simultaneously visible stars within the same field of view is suitable for on-orbit calibration. Smaller angular separations increase computational errors, so three stars with angular separations exceeding 1° were selected for calibration. The right ascension and declination of the stars are as follows (

Table 2. Star parameters.

Target Star	Right Ascension and Declination (Hour Angle)	Right Ascension and Declination (Degrees)
Star 1	(18 h 00 m 00 s, −85°)	(270°, −85°)
Star 2	(17 h 44 m 00 s, −88°)	(266°, −88°)
Star 3	(18 h 12 m 00 s, −89.5°)	(273°, −89.5°)

).

The positions of the three stars are illustrated in the following Figure 8.

During star calibration, the celestial sphere is treated as a sphere with an infinite radius(R ⟶ ∞) as the projected positions of all celestial objects (regardless of distance) are determined solely by their direction, i.e., celestial coordinates (right ascension, declination, etc.), independent of distance. Thus, the satellite and Earth can be considered at the center of the celestial sphere, neglecting the Earth’s radius. This relationship is depicted in Figure 9 below:

Using the geometric imaging model as a theoretical foundation, the target centroid information measured by the camera is used to precisely solve for the target’s pointing direction in the celestial coordinate system. The satellite operates in a sun-synchronous orbit, with the flight direction along the X-axis, the optical axis along the Z-axis, and the Y-axis satisfying the right-hand rule with respect to the X-Z plane. The rotating mirror rotates about the X-axis.

Assuming no installation misalignment, the right ascension and declination coordinates of the stars are determined for the observation period (Figure 10), ensuring that the stars are precisely imaged at their theoretical positions (calibrated position 1).

In the absence of mirror misalignment, the initial normal vector of the scanning mirror, n₀, is set as (0, $1/\sqrt{2}$, $-1/\sqrt{2}$), as noted in Figure 9, corresponding to the starting position at the South Pole. Virtual star points in the negative Y-axis hemisphere within the push-broom region are selected, and their corresponding mirror deflection positions are calculated based on their declination. The rotation matrix about the X-axis is obtained, and the actual mirror normal vector at the corresponding position is computed by multiplying with n₀. For the selected virtual star points, the normal vector pointing toward the celestial sphere’s center is reflected by the mirror, yielding the normal vector of the light ray entering the optical system post-reflection. The reverse extension of this normal vector corresponds to a right ascension and declination on the celestial sphere, equivalent to the virtual image produced by the mirror. From the computed virtual object point, the right ascension and declination or the angle between the incident light ray and the optical axis are derived, allowing calculation of the ideal image height and corresponding pixel coordinates on the push-broom focal plane (Figure 11,

Table 3. Virtual coordinates and pixel positions of reference stars without misalignment.

		Star 1	Star 2	Star 3
Scanning Mirror Rotation Angle		2.5°	1°	0.25°
Celestial Coordinates of Virtual Object	RA	270°	270.139825°	269.973828°
	Dec	−10°	−3.995120°	−0.999315°
Pixel Coordinates		(0, −25,189.6)	(348.6, −9977.3)	(−65.3, −2491.9)

).

4.2. Reference Star Vector Calculation

When the scanning mirror exhibits misalignment, the imaging coordinates of reference stars on the focal plane deviate from their ideal positions. The real-time normal vector of the scanning mirror comprises the following three components: the initial normal vector ${\mathit{n}}_0$, the rotation matrix about the X-axis, and the misalignment matrix for the three angles (${\delta }_x$, ${\delta }_y$, ${\delta }_z$). The resulting mirror normal vector, incorporating misalignment, is calculated for the corresponding declination. The outgoing light ray vector and the corresponding right ascension and declination of the virtual image are computed for the light ray passing through the misaligned mirror. By reverse-tracing the light ray from the focal plane coordinates, the outgoing light ray vector from the optical system is determined, confirming that its corresponding right ascension and declination match those from forward ray tracing (

Table 4. Virtual coordinates and pixel positions of reference stars with misalignment.

		Star 1	Star 2	Star 3
Scanning Mirror Rotation Angle		2.5°	1°	0.25°
Celestial Coordinates of Virtual Object	RA	269.951194°	270.094582°	269.930266°
	Dec	−10.029991°	−4.025219°	−1.029288°
Pixel Coordinates		(−121.7, −25,266.7)	(235.8, −10,052.7)	(−173.9, −2566.6)

).

For the virtual coordinates of reference stars, reverse ray tracing is performed. The light ray, after reflecting off the mirror with unknown misalignment, yields an outgoing light ray vector containing the three unknown misalignment parameters. This vector is combined with the reference vector to establish Equation (17). Multiple target stars are selected for joint solving. The results are shown in the table below, where the preset misalignment angles are solved using three target stars. Increasing the number of target stars can further improve solving accuracy.

To compare the influence of the number of selected star points on the solution results and to highlight the sufficiency of star point selection, the misalignment angles were calculated by tracing a single light ray corresponding to a single star point (Star 2) under the same misalignment condition, and the results were compared with those obtained from joint solutions using three and four star points. In the case of the four-star joint solution, in addition to Star 1 through Star 3, Star 4 was also included, whose right ascension and declination are (271°, −88.5°). According to the calculation results shown in

Table 5. Three-axis misalignment of the scanning mirror.

	X-Axis	Y-Axis	Z-Axis
Preset Misalignment	0.015°	0.020°	−0.023°
Computed Misalignment of 3 Stars	0.01503390°	0.01990194°	−0.02311112°
Computed Misalignment of 4 Stars	0.01503133°	0.01998487°	−0.02313011°
Computed Misalignment of 1 Star	0.11843954°	3.37806788°	3.46244602°

, the improvement in accuracy from using four stars compared to three is very limited. In fact, due to slight errors, the Z-axis misalignment obtained from the four-star solution exhibits an additional error of −0.00002° compared to the three-star solution. The solution obtained using a single star differs from the preset misalignment by several orders of magnitude, indicating that due to degeneracy and coupling with unknown parameters, it is completely impossible to resolve the error using a single star. Therefore, considering both computational complexity and result accuracy, three stars are selected as the basic number of stars for the solution.

When the mirror exhibits misalignment, the theoretical and actual imaging positions of the stars are as follows (Figure 12,

Table 6. Changes in imaging pixel positions under misalignment.

Misaligned Axis	Misalignment Amount	Star Point 1	Star Point 2	Star Point 3
None	null	(0.0, −25,189.6)	(348.6, −9977.3)	(−65.3, −2491.9)
X-Axis	20″	(0.0, −25,218.2)	(348.6, −10,005.2)	(−65.3, −2519.6)
Y-Axis	20″	(−16.5, −25,189.6)	(333.8, −9977.4)	(−79.4, −2491.9)
Z-Axis	20″	(15.1, −25,189.6)	(363.0, −9977.3)	(−51.3, −2491.9)

).

From Table 6, it can be inferred that a 20″ misalignment along the X-axis causes the image points to shift vertically, a 20″ misalignment along the Y-axis causes horizontal shift, and a 20″ misalignment along the Z-axis also results in horizontal shift. Specifically, a Z-axis misalignment causes a 15.1-pixel deviation in the ground-pointing direction, corresponding to a 52.85 m line-of-sight deviation at a 500 km orbit.

To analyze the impact of mirror axis errors on pointing accuracy, simulations were conducted with misalignment angles set as ${\delta }_x$(X-axis), ${\delta }_y$(Y-axis), and ${\delta }_z$(Z-axis). Forty virtual star points with right ascension fixed at 266° and declination ranging from −90° to −50°, corresponding to side-swing angles of 0° to 20°, were analyzed for angle errors. First, we compute the three angular components between each selected virtual star reference vector and the three coordinate axes, and similarly compute the three angular components between the outgoing ray’s direction vector—obtained by back-tracing from the image-plane star coordinates—and the three axes. Second, we regard the reference vector as the ideal angle in the absence of misalignment, and the back-traced outgoing ray—which includes the misalignment—as the actual angle. Finally, by subtracting the three angular components of the actual outgoing ray from those of the ideal outgoing ray for each axis, we obtain the change in each coordinate axis angle when a given misalignment exists solely along one direction. The resulting data are then analyzed. The variation curves of the three component differences with respect to the coordinate axes, for independent misalignment angles of 20″ and 40″ in each direction, are shown below (Figure 13, Figure 14 and Figure 15).

By analyzing the impact of misalignment in different axes on angle errors it is observed that when only the X-axis misalignment angle is present then the angle with the X-axis shows no numerical change, indicating that X-axis misalignment only affects the Y and Z components. Conversely, when Y or Z-axis misalignment is present the X component of the angle error is most significantly affected. This is because, according to the law of reflection $k_{out}=k_{in}-2\left(n·k_{in}\right)n$, the initial normal vector n₀, and the incident light ray vector k_in, the first-order perturbation of the normal vector due to misalignment in the three directions is expressed as follows:

$$\delta n=\left(\begin{array}{c}{\delta }_x\\ {\delta }_y\\ {\delta }_z\end{array}\right)\times n_0={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}-{\delta }_y+{\delta }_z\\ {\delta }_x\\ -{\delta }_x\end{array}\right)$$

(26)

The X component $k_{out,x}$ of the reflected ray is as follows:

$$k_{out,x}=k_{in,x}-2\left(n·k_{in}\right)n_x$$

(27)

Since $n_x=n_{0,x}+\delta n_x=\delta n_x$, the perturbation is approximately:

$$\delta k_{out,x}\approx -2\left(n_0·k_{in}\right)\delta n_x={\delta }_y-{\delta }_z$$

(28)

Thus, any angular deviation in the Y or Z direction causes changes in the X component of the angle error (Figure 16, Figure 17 and Figure 18).

When the misalignment increases from 20″ to 40″, the error exhibits a linear growth trend, indicating that the method maintains high calibration accuracy over a wide misalignment range. However, in practical applications, measurement errors in star positions (e.g., star map identification accuracy) and mirror encoder errors may introduce cumulative effects on the results.

5. Discussion

Compared with traditional ground-based calibration or attitude–orbit coupled solution methods, the proposed approach does not rely on any external measurement units. It only requires reference star data captured during the scanning imaging process of the rotating mirror to complete the calibration.

Under the predefined X/Y/Z axis misalignment angles of 0.015°, 0.02°, and −0.023°, single-star calibration results show significant deviations from the preset values. In contrast, the error between the results of the joint calculation of three stars and the preset value is controlled in the order of 0.0001°, which verifies the high-precision performance of the method under the constraint of multiple stars. The analysis of the error propagation law in different rotation directions shows obvious asymmetry as misalignment about the X-axis primarily affects the pointing accuracy in the Y and Z directions, whereas misalignments about the Y and Z axes have a dominant influence on the X direction error.

However, this method also has some limitations. When the selected star geometry is approximately collinear the vector equations may degenerate, and the angular distance constraint strategy needs to be preset for constraint optimization. In addition, the current model assumes mirror normal perturbation as the sole error source and does not consider the coupling effect of error terms such as attitude drift. In the future, the universality of the model can be further enhanced through error propagation modeling.

6. Conclusions

This study addresses the on-orbit calibration challenge for rotating mirror-based wide-field space cameras by proposing a dynamic calibration method based on pointing vector equations. By establishing models for reference star pointing vectors and actual star point pointing vectors, high-precision calibration of the three-axis misalignment of the rotating mirror is achieved. Furthermore, by analyzing the influence patterns of misalignment in different axes on pointing errors, it is found that X-axis misalignment primarily affects the Y and Z directional components, while Y- and Z-axis misalignments significantly impact the X directional component. This finding provides a theoretical basis for subsequent error compensation. This study not only offers a novel technical pathway for autonomous on-orbit calibration of rotating mirror-based space cameras but also provides valuable reference for calibration issues in other space optical systems with moving components.