Investigation on Micro-Vibration Test and Image Stabilization of a High-Precision Space Optical Payload

Abstract

With the advancement of space exploration and optical communication toward deep space, the high-precision evaluation and image stabilization of space optical payloads under micro-vibration have become increasingly critical. To address these challenges and ensure sub-micro-radian pointing accuracy for high-precision space optical payloads (HPSOPs), this paper proposes a high-precision micro-vibration testing scheme and a two-stage image stabilization system. The micro-vibration testing scheme is based on an automated quasi-zero stiffness suspension device (AQZSSD), which enhances testing sensitivity and environmental disturbance resistance, ensuring the accuracy of the results. The two-stage image stabilization system integrates three bipod vibration isolation legs (BVILs) and a decoupled fast steering mirror (FSM), extending control bandwidth and achieving comprehensive vibration suppression. Micro-vibration testing and image stabilization experiments were conducted under disturbances from multiple vibration sources. Experimental results demonstrate that the AQZSSD introduces disturbances below 0.4 Hz, confirming its quasi-zero stiffness characteristics in alignment with theoretical predictions. Furthermore, the line-of-sight (LOS) jitter root mean square (RMS) value is reduced from 1.253 μrad to 0.276 μrad, achieving sub-micro-radian stability. Additionally, due to the coupling effect of the micro-vibration response, the collaborative testing results were found to be lower than the linear superposition of individual sources. This work offers critical theoretical and technical support for the development of HPSOPs, with potential applications in future space missions and advanced optical technologies.

1. Introduction

As space exploration and optical communication advance into deep space, high-precision space optical payloads (HPSOPs) with sub-micro-radian precision have emerged as a focal point in space science and advanced optical research [1,2]. However, the line-of-sight (LOS) jitters induced by moving components, such as flywheels, control moment gyros (CMGs), and cryocoolers, may lead to image degradation and communication interruptions [3,4]. Consequently, conducting high-precision micro-vibration tests and image stabilization studies on these optical payloads has become both a challenging and imperative task [5,6,7].

The primary challenges associated with high-precision micro-vibration testing are the design of gravity unloading equipment with robustness against environmental disturbances and the accurate assessment of effects caused by various disturbance sources [3,8,9,10,11]. Gravity unloading equipment design can be categorized into four types based on differences in elastic elements, such as linear springs, disc springs, parallel mechanisms with positive and negative stiffness, and air springs. Among these, suspension systems with linear springs have been widely used in ground tests due to their safety, adjustability, and simplicity [9,12,13,14,15,16]. For instance, L. Li et al. proposed a quasi-zero stiffness suspension device (QZSSD) based on linear springs adjusted by bidirectional bolts to study the image point offset of a 0.1″ space pointing measuring instrument [9]. Nevertheless, achieving the desired accuracy remains a challenge. To address this, an automated quasi-zero stiffness suspension device (AQZSSD) adjusted by linear motors is proposed in this paper to enhance precision in gravity unloading. In addition, for analyzing the effects caused by different micro-vibration sources accurately, the test and mechanism research of independent disturbance sources has always been the focus in the past years [13,17,18,19,20]. Combining high-precision acceleration sensors, S. B. Chen et al. predicted the image offset (refers to the movements of the image point) caused by flywheels [13]. Y. Takei et al. found that the energy resolution of the detector would reduce because of the micro-vibration caused by cryocoolers during testing the energy changes in the detector on the HITOMI [20]. However, the LOS jitter (refers to the line-of-sight jitter, a different concept from image offset) caused by multiple disturbance sources may be coupled and nonlinearly amplified, simultaneously [9]. Some researchers have begun to study the micro-vibration response caused by multi-disturbance sources [3,9,21,22,23]. For example, M. Remedia studied the micro-vibration effects caused by the flywheel and antenna pointing mechanism on the satellite SSTL 300 S1 [22]. Yan Shen studied the micro-vibration energy transfer characteristics when two flywheels work simultaneously [23]. By contrast, few studies have reported on the micro-vibrations caused by the CMGs and cryocoolers together. To address this research gap, CMGs and cryocoolers were chosen as the micro-vibration sources for our study.

In addition to addressing the high-precision micro-vibration testing, the suppression of LOS jitters constitutes another core challenge, necessitating effective image stabilization strategies [3]. Owing to micro-vibration ranging from 0.01 Hz to kHz, the design of vibration isolation systems has always been critical in image stabilization [24]. Typically, micro-vibration isolation systems can be classified into passive isolation, active isolation, active–passive hybrid isolation, and semi-active isolation. Among these, passive isolation has garnered significant attention due to its high reliability and absence of energy transfer [24,25,26,27,28]. For example, C. Qin et al. designed a vibration isolator by optimizing the structure and introducing dampers to isolate the micro-vibration of the satellite platform above 10 Hz [27]. J. Tang et al. also proposed a 6-DOF isolation platform based on the quasi-zero-stiffness isolator to suppress disturbances above 10 Hz [28]. M. Safarabadi et al. proposed a high-damping viscoelastic material to reduce high-frequency disturbance of the reaction wheel on the camera [29]. However, traditional passive vibration isolation systems struggle to suppress low-frequency disturbances. Although the incorporation of active control can broaden the suppression bandwidth, as demonstrated by A. Xu et al. with a coarse-precision parallel vibration isolation pointing platform for disturbances ranging from 2 to 20 Hz, such systems often become overly complex, contradicting the trend toward lightweight designs for high-precision optical loads [30,31]. Conversely, integrating a fast steering mirror (FSM) to compensate for residual LOS jitter under low-frequency conditions has garnered increasing favor among researchers [7,32,33]. For instance, Y. Wu et al. studied a two-stage vibration isolation precision pointing system consisting of a Stewart platform and an FSM capable of suppressing micro-vibrations above 1 Hz [7]. Taking the large size of the high-precision space optical payload (HPSOP) in our study into account, a two-stage image stabilization system, composed of three bipod vibration isolation legs (BVILs) and a decoupled FSM, is introduced in this paper to achieve high-precision pointing accuracy.

In light of the aforementioned analysis, there are several obstacles to the micro-vibration test and image stabilization of the HPSOPs. First, due to the adjustment errors caused by the non-automatic gravity unloading, traditional micro-vibration test platforms cannot meet the higher precision measurement requirements of the micro-vibration on HPSOPs. Second, because of the nonlinear coupling and amplification, the linear superposition of independent disturbances cannot reflect the actual impact of composite micro-vibration, such as CMGs and cryocoolers. Last, owing to the wide frequency band cover of micro-vibration, it is difficult to achieve sub-micro-radian pointing accuracy by suppressing the disturbances appearing in the low frequency and mid-high frequency at the same time. To address the above problems, this paper proposes a micro-vibration test platform equipped with AQZSSD gravity unloading devices, and a two-stage image stabilization system incorporated with three BVILs and a decoupled FSM, bridging the gap between the concept and engineering implementation of high-precision micro-vibration measurement and sub-micro-radian LOS stabilization. This study is organized as follows. First, the detailed composition and experimental process of the micro-vibration test platform are introduced in Section 2. Then, the quasi-zero stiffness characteristic of an AQZSSD and the principle of calculating LOS jitter angles are discussed in Section 3. The disturbance suppression in the mid-high frequency and LOS jitter angle compensation in the low frequency of the two-stage image stabilization system are proved in Section 4. Further, experimental results with the analysis and discussion, are presented in Section 5. Finally, a conclusion is drawn in Section 6.

2. Micro-Vibration Test Scheme

As shown in Figure 1, the micro-vibration test platform proposed in this paper is composed of the light source, dot target, collimator, HPSOP, satellite platform, and AQZSSDs. The light source, dot target, and collimator are installed on the adjustment platforms to provide parallel light simulating deep space target incidents on the optical payload, where adjustment platforms 1 and 2 are used to align the light source with the collimator and ensure that the light emitted by the collimator is incident on the optical payload in parallel.

To minimize the impact of micro-vibrations from other moving parts and maximize the observation field, the optical payload is installed at the top center of the satellite platform via three BVILs. During the launch phase, the isolation struts are locked with the satellite platform through frangible bolts to prevent damage to the precision optical equipment caused by random vibrations and shocks. After entering orbit, the frangible bolts are exploded to ensure the execution of the payload observation imaging or laser communication mission. The FSM control system and the image acquisition system within the optical payload are used to decrease LOS jitter and acquire the image point coordinates of the incident light focused on the charge-coupled device (CCD), respectively. Additionally, CMGs and cryocoolers on the satellite platform generate micro-vibration disturbances to the optical payload when they are working.

According to Equation (1), the vibration equation of the multi-degree-of-freedom damping system, it is necessary to introduce an AQZSSD to reduce the disturbances caused by the unloading equipment since the frequency range of micro-vibration is 0.01 Hz to kHz [24].

$$M\ddot{x}+C\dot{x}+Kx=0$$

(1)

As shown in Figure 2, the AQZSSD1 and AQZSSD2 proposed in this paper are used for gravity unloading and horizontal alignment adjusting of the satellite platform and HPSOP, respectively. Each device includes a crane hitch, a sling, a spreader, an adjustable elastic cord set, and a connecting fixture. Notably, the adjustable elastic cord set contains a linear motor, a dynamometer (Type: LH-S02, Liheng Sensing Technology Co., Ltd. in Shanghai, China), and a set of elastic ropes.

The connecting fixtures connect the optical payload and the satellite platform to the adjustable elastic cord set through screws and lifting rings to achieve gravity unloading without deformation. The adjustable elastic cord set driven by the linear motors is connected to the connecting fixture through a hook to ensure precise adjustment. The feedback values of the four dynamometers here are indicators for evaluating whether the gravity is unloaded. The crane hook above the AQZSSD lifts the center ring of the sling to bear the gravity of the system to be tested and the suspension tool.

Before the test, the gravity of the satellite platform and the optical payload were unloaded through the AQZSSD, and the parallelism and position of the optical payload were adjusted in sequence. The detailed automatic adjustment mechanism based on AQZSSD is presented as follows (Figure 3): Step 1, to unload the payload’s gravity, make parallel adjustments to the linear motors until the dynamometer value $F_D$ is equal to the payload gravity $G_P$; Step 2, to prevent imbalance caused by different deformations among the elastic ropes, adjust each linear motor independently until the theodolite measured value ${\theta }_m$ is smaller than the parallelism tolerance ${\theta }_t$; Step 3, to avoid material creep and relaxation of the suspension ropes during experiments, take stress relaxation; and Step 4, to achieve precise alignment and installation of optical payloads and satellite platforms, adjust the crane until the laser rangefinder measure ${\mathrm{T}}_m$ is equal to the position tolerance ${\mathrm{T}}_t$. Then, the adjustment platforms 1 and 2 were adjusted in sequence to make the parallel light imaging at the CCD center. During the test, the CMGs and cryocoolers were turned on by the disturbance source control system to generate micro-vibrations at first, and then the BVILs and FSM were unlocked in sequence to achieve image point stabilization of the HPSOP.

3. Micro-Vibration Test Principle

3.1. The Quasi-Zero Stiffness Characteristics Analysis

There are two gravity unloading methods: rigid support and flexible suspension. Both states will introduce additional perturbation, which will affect the micro-vibration test accuracy. Compared with rigid support, the disturbance peak caused by the flexible suspension system appears at a low frequency, close to 0 Hz [34]. Therefore, this paper adopted a set of adjustable elastic ropes and a high-stiffness sling to achieve flexible suspension. In addition, during the actual measurement, traction ropes were added on both sides of the load. As studies have shown in the past years, the suspension system can be simplified as a simple pendulum system or a double pendulum system to make the stiffness study [5,9,35]. Although there may be a risk of forward and backward torsion on the pendulum, due to the strict symmetry of the disturbance source and the introduction of the traction rope, this problem is negligible compared to the longitudinal vibration along the X direction and simple pendulum motion. After analysis, the AQZSSD suspending the HPSOP as shown in Figure 4a can be simplified to a simple pendulum system as in Figure 4b.

Combining the kinetic energy equation $T={\displaystyle \frac{1}{2}}{m}_{OL}\left({\dot{x}}^2+{\left(l_0+x\right)}^2{\dot{\theta }}^2\right)$ and the potential energy equation $V={\displaystyle \frac{1}{2}}{k}_C{x}^2+m_{OL}g(l_0+x)(1-cos\theta )$, the dynamic equation of the simple pendulum system can be expressed as follows:

$$\left\{\begin{array}{c}\ddot{x}+{\displaystyle \frac{k_C}{m_{OL}}}x+g(1-cos\theta )-(l_0+x){\dot{\theta }}^2=0\\ \ddot{\theta }+{\displaystyle \frac{g}{l_0+x}}sin\theta +{\displaystyle \frac{2}{l_0+x}}\dot{x}\dot{\theta }=0\end{array}\right.$$

(2)

where x stands for the changes in an elastic rope; $\theta$ represents the swing angle of the simple pendulum system; $m_{OL}$ expresses the mass of the HPSOP and AQZSSD equivalent to 300 kg ($m_{OL}$ = $m_{HP}$ + $m_{AQ}$, where $m_{HP}$ and $m_{AQ}$, measured by the electronic hanging scale, indicate the mass value of the HPSOP and AQZSSD, respectively); $k_C$ indicates the comprehensive stiffness of the parallel suspended system composed of 16 elastic ropes (since the stiffness value test of each elastic rope made of latex is equal to 130 N/m, it can be obtained that $k_C$ = 16 × 130 N/m from the stiffness calculation theory of spring parallel systems); and $l_0$ represents the initial length 3.5 m of the latex elastic rope without the optical payload.

Since the swing angle of the suspension system changes caused by micro-vibrations is little, Equation (2) can be simplified as:

$$\left\{\begin{array}{c}\ddot{x}+{\displaystyle \frac{k_C}{m_{OL}}}x+{\displaystyle \frac{g}{2}}{\theta }^2-(l_0+x){\dot{\theta }}^2=0\\ \ddot{\theta }+{\displaystyle \frac{2}{l_0+x}}\dot{x}\dot{\theta }+{\displaystyle \frac{g}{l_0+x}}\theta =0\end{array}\right.$$

(3)

Obviously, the vertical eigenfrequency $f_v={\displaystyle \frac{1}{2\pi }}\sqrt{{\displaystyle \frac{k_C}{m_{OL}}}}\approx 0.42$ Hz can be obtained by solving $\ddot{x}+{\displaystyle \frac{k}{m}}x=0$, an equation simplified by Equation (3) while only considering the vertical vibration of the simple pendulum system corresponding to $\theta =0$. Similarly, the horizontal eigenfrequency $f_{\mathrm{h}}={\displaystyle \frac{1}{2\pi }}\sqrt{{\displaystyle \frac{g}{l_0+m_{OL}g/k_C}}}\approx 0.22$ Hz can also be obtained by solving $\ddot{\theta }+{\displaystyle \frac{g}{l_0+x}}\theta =0$, another equation simplified by Equation (3) while only considering the horizontal vibration of the simple pendulum system corresponding to $x\approx {\displaystyle \frac{m_{OL}g}{k_C}}$.

3.2. The Image Point Offset Extraction and LOS Jitter Calculation of the HPSOP

The LOS jitter and image point offset caused by micro-vibration have a significant impact on the detection clarity and communication stability of the HPSOP. Therefore, the image point offset extraction and the LOS jitter calculation are crucial in evaluating the pointing accuracy.

3.2.1. Image Point Offset Extraction

As shown in Figure 5a, during the micro-vibration test, the image point of the dot target imaged by the optical system will shift on the CCD. Therefore, we plan to evaluate the effect of micro-vibration on the optical system based on the offset of the image point. However, affected by the micro-vibration and non-ideal optical system, the image usually appears to be a light spot rather than an ideal point, as shown in Figure 5b.

Typically, to reflect the overall offset of the light spot, the grayscale centroid method is usually used to calculate the energy center position of the light spot. As shown in Equation (4), the position of the pixel with the maximum grayscale value is taken as the center, and the grayscale subdivision is performed within an n × n window. This approach allows the measurement accuracy to reach the sub-pixel level.

$$\left\{\begin{array}{c}{u}_{cm}={\displaystyle \frac{{\sum }_{j=1}^{16}{\sum }_{i=1}^{16}{G}_{ij}^m·u_i}{{\sum }_{j=1}^{16}{\sum }_{i=1}^{16}{G}_{ij}^m}}\\ v_{cm}={\displaystyle \frac{{\sum }_{j=1}^{16}{\sum }_{i=1}^{16}{G}_{ij}^m·v_j}{{\sum }_{j=1}^{16}{\sum }_{i=1}^{16}{G}_{ij}^m}}\end{array}\right.$$

(4)

where $u_{cm}$ and $v_{cm}$ express the gray value centroid of the m-th frame image along the U and V coordinates, respectively; $G_{ij}^m$ represents the gray value of the pixel at the i-th row and j-th column; and $u_i$ and $v_i$ indicate the energy center of the pixel at the i-th row and j-th column.

However, the image point extracted by Equation (4) is not only determined by the micro-vibration but also influenced by the electronic noise caused by the CCD [5]. To solve this problem, an N-th power-weighted grayscale value solution method with a threshold $T_h$ is proposed to reduce the grayscale value error, as shown below:

$$\left\{\begin{array}{c}{u}_{cm}^{\prime }={\displaystyle \frac{{\sum }_{j=1}^{16}{\sum }_{i=1}^{16}{\left(G_{ij}^m-T_h\right)}^N·u_i}{{\sum }_{j=1}^{16}{\sum }_{i=1}^{16}{G}_{ij}^m}}\hfill \\ \\ v_{cm}^{\prime }={\displaystyle \frac{{\sum }_{j=1}^{16}{\sum }_{i=1}^{16}{\left(G_{ij}^m-T_h\right)}^N·v_j}{{\sum }_{j=1}^{16}{\sum }_{i=1}^{16}{G}_{ij}^m}}\hfill \end{array}\right.$$

(5)

where $T_h=kmax\left(G_{ij}^m\right)$, and k indicates the scaling factor changing with the image quality (0 < k < 1). Studies have shown that when $T_h$ is the background gray and N is 2, the improved centroid accuracy has an extreme point.

3.2.2. LOS Jitter Calculation

The LOS jitter caused by micro-vibration can be divided into translation and angular jitter. Compared with angular jitters, the effects caused by translational jitters can be ignored in deep space. Moreover, the angular micro-vibration around the Z-axis has little effect on image stabilization because of the rotational symmetry of the primary mirror (PM) and the second mirror (SM). Hence, this paper pays more attention to the LOS jitter around the non-optical axes.

As shown in Figure 6, with the optical payload jitter around the non-optical axes causing the jitter of the incident light, the image point will be offset on the CCD. In Figure 6, $P_0$, P, and $P^{\prime }$ stand for the object point, the image point without jitter, and the image point with jitter.

The LOS jitter angles of the HPSOP around the X-axis and Y-axis are as follows:

$$\left\{\begin{array}{c}{\theta }_x={\displaystyle \frac{L_v·f}{f^2+y^2}}\\ {\theta }_y={\displaystyle \frac{L_u·f}{f^2+x^2}}\end{array}\right.$$

(6)

where ${\theta }_x$ and ${\theta }_y$ indicate the jitter angles around the X-axis and the Y-axis; x and y are the image point coordinates before micro-vibration; $L_u$ and $L_v$ represent the image point translation caused by the micro-vibration; and f expresses the focal length.

Notably, Equation (6) can be simplified to Equation (7) when x and y are both equal to 0.

$$\left\{\begin{array}{c}{\theta }_x={\displaystyle \frac{L_y}{f}}\\ {\theta }_y={\displaystyle \frac{L_x}{f}}\end{array}\right.$$

(7)

Because the image point is usually corrected to the center of the CCD when there is no micro-vibration, this paper uses Equation (7) to calculate the LOS jitter angles of HPSOP.

4. Micro-Vibration Suppression and Image Stabilization Principle

The mid-high disturbances and low-frequency LOS jitter can be suppressed by combining the first-order vibration suppression by passive BVILs and the second-order vibration suppression controlled by the FSM, respectively. As shown in Figure 7, the offset amplitude of the image points decreases obviously after being suppressed by the two-stage image stabilization system.

4.1. The First-Order Vibration Suppression

As shown in Figure 8a, the first-order vibration suppression is composed of three BVILs. Most dynamic characteristics of the BVIL, modeled in 2D, can be carried over to 3D directly. To simplify the analysis, a 2D model of the BVIL is built to analyze the dynamic response of the first-order vibration suppression in this paper. As shown in Figure 8b, the connection between the optical payload and satellite platform is achieved by BVILs with S-joints and U-joints. Each BVIL has a stiffness k, damping c, and total length L. The angle between the two legs is 90°. Moreover, the distance between the upper platform, carrying an equivalent payload $m_{pe}$, and the U-joint, connecting with the lower base, is L. The distance between the center of the upper connecting rod, with mass $m_u$, and the U-joint of the lower base is $L_u$. The distance between the center of the lower connecting rod, with mass $m_b$, and the U-joint of the lower base is $L_b$. The moment inertia of the rod groups, connected to the HPSOP and satellite platform, are $I_u$ and $I_b$, respectively.

To facilitate the dynamic analysis, a local coordinate system is built on the support center as shown in Figure 8b. Notably, the angle between the disturbance $F_d$ and the x-axis is $\alpha$. In particular, the dynamic equation of the BVIL when $\alpha$ = 0° is shown in Equation (8):

$$\left\{\begin{array}{c}{m}_{dl}\ddot{x}-F_l-F_r-F_d=0\\ I_{dr}{\ddot{\beta }}_r+L(m_{pe}+m_u)\ddot{x}-L(F_l+F_d)=0\end{array}\right.$$

(8)

where $F_l$ and $F_r$ are the reaction forces of the left and right legs, respectively. x denotes the displacement of the BVIL equivalent mass along the x-axis of the local coordinate system; however, in conjunction with the small angle approximation, $y=0$. $m_{dl}$, which can be expanded to $m_{dl}=m_{pe}+m_u+{\displaystyle \frac{L_u}{L}}{m}_u+{\displaystyle \frac{L_b}{L}}{m}_b$, is the mass of the entire BVIL equivalent to the left leg. $I_{dr}$, which can be expanded to $I_{dr}=I_u+I_b+m_u{L}_u^2+m_b{L}_b^2$, indicates the moment of inertia of the right leg about the U-joint.

Since the rotation angle of the vibration isolation leg under micro-vibration is not large, $\ddot{x}$ can be approximately equal to the product of L and ${\ddot{\beta }}_r$. Then, Equation (9) can be obtained by combining Equation (8) and the reaction force of the left leg.

$$\left\{\begin{array}{c}{F}_l=-kx-c\dot{x}\\ F_r=\gamma (F_l+F_d)\end{array}\right.$$

(9)

where $\gamma ={\displaystyle \frac{m_{dl}{L}^2}{I_{dr}+\left(m_{pe}+m_u\right)L^2}}-1$.

Furthermore, a more meaningful analysis can be found by performing the Laplace transform on Equations (8) and (9). It is as follows:

$$\left\{\begin{array}{c}T{F}_a={\displaystyle \frac{-\left(\gamma +1\right)\left(cs+k\right)}{m_{dl}{s}^2+\left(\gamma +1\right)(cs+k)}}\\ T{F}_s={\displaystyle \frac{\gamma m_{dl}{s}^2}{m_{dl}{s}^2+\left(\gamma +1\right)(cs+k)}}\end{array}\right.$$

(10)

where $T{F}_a$ and $T{F}_s$ are the axial and tangential reaction forces of BVIL, respectively.

To objectively reflect the mechanical response of BVIL under different disturbances, the transfer function, Equation (11), of any angle $\alpha$ can be derived from Equation (10).

$$\left\{\begin{array}{c}T{F}_l={\left[\begin{array}{cc}T{F}_{a}cos\alpha & T{F}_{s}sin\alpha \end{array}\right]}^T\\ T{F}_r={\left[\begin{array}{cc}T{F}_{s}cos\alpha & T{F}_{a}sin\alpha \end{array}\right]}^T\end{array}\right.$$

(11)

Compared with the optical payload, the mass of BVIL can be negligible. Each isolation leg can be regarded as a 1D vibration isolator without mass ($m_u$ = $m_b$ = $I_u$ = $I_b$ = 0). Then, the results $\gamma$ = 0 and $T{F}_s$ = 0 can be obtained.

Taking $\alpha$ = 0° as an example, the disturbance suppression curve of a BVIL with the structural parameters listed in

Table 1. Parameters of the BVIL.

Parameters	Value	Unit
L	0.8	m
k	15,800	N/m
c	280	Ns/m
$m_{pe}$	100	kg

can be extracted from the Lissajous ellipse of the vector $T{F}_l$ + $T{F}_r$, as shown in Figure 9. It is not difficult to see that disturbances above 5 Hz will be suppressed significantly, but there is a peak around 2 Hz.

4.2. The Second-Order Vibration Suppression

The second-order vibration suppression is a decoupled FSM driven by piezoelectric actuators.

As shown in Figure 10a, it is composed of a mirror, a tip–tilt mechanism about two axes, and piezoelectric actuators. In general, the LOS jitter is corrected by rotating the mirror around the X-axis and Y-axis through the tip–tilt mechanism. The rotation of the X-axis is taken as an example, as shown in Figure 10b. The piezoelectric ceramics produce elongation and shortening deformations, respectively. Then, the deformations are amplified by the amplification mechanism and transmitted to the flexible support. Finally, the flexible support pushes and pulls the mirror frame to rotate the mirror around the X-axis. Notably, to guarantee structural rigidity and motion decoupling, the FSM adopts an integrated design. This results in the rigidity being too high to achieve a large rotation angle. To overcome this shortcoming, a two-stage amplification mechanism is proposed in this paper, as shown in Figure 10c.

Moreover, the image stabilization process is shown in Figure 11. Affected by micro-vibration disturbances, optical components, such as the primary mirror (PM) and the secondary mirror (SM), undergo displacement relative to the calibration state, resulting in the LOS jitter and image offset as shown in Figure 11a. Then, the FSM is controlled to correct the optical axis to the CCD center, as shown in Figure 11b. The control structure is shown in Figure 11c, where $R\left(s\right)$ and $Y\left(s\right)$ are the LOS jitter and target position, respectively; $e^{-\tau s}$ expresses the time delay characteristics of CCD; $C_p\left(s\right)$ indicates the controller on position; $G_{FSM}\left(s\right)$ is the transfer function of the FSM; and $D_{Com}\left(s\right)$ represents the LOS jitter caused by the jitter on the PM and SM.

To obtain the plant transfer function of position $G_p\left(s\right)=e^{-\tau s}{G}_{FSM}\left(s\right)$, frequency sweep experiments on FSM are conducted in this paper, and the results after eliminating the low-frequency measurement noise are shown in Figure 12. Then, $G_p\left(s\right)$ can be obtained through parameter identification, as shown in Figure 12. According to the fitting curve, the transfer functions around the X-axis and Y-axis can be obtained as follows:

$$G_{FSM-X}\left(s\right)=e^{(-0.0013s)}{\displaystyle \frac{N_{FSM-X}}{D_{FSM-X}}}$$

(12)

where $N_{FSM-X}=8.374\times {10}^{36}{s}^4+9.86\times {10}^{38}{s}^3+9.523\times {10}^{43}{s}^2+5.694\times {10}^{45}s+2.674\times {10}^{50}$ and $D_{FSM-Y}=2.589\times {10}^{27}{s}^7+4.626\times {10}^{30}{s}^6+4.674\times {10}^{34}{s}^5+6.62\times {10}^{37}{s}^4+2.664\times {10}^{41}{s}^3+2.882\times {10}^{44}{s}^2+4.782\times {10}^{47}s+3.628\times {10}^{50}$.

$$G_{FSM-Y}\left(s\right)=e^{(-0.0014s)}{\displaystyle \frac{N_{FSM-Y}}{D_{FSM-Y}}}$$

(13)

where $N_{FSM-Y}=2.245\times {10}^{36}{s}^4+1.09\times {10}^{39}{s}^3+2.182\times {10}^{43}{s}^2+5.146\times {10}^{45}s+5.136\times {10}^{49}$ and $D_{FSM-Y}=1.135\times {10}^{27}{s}^7+1.86\times {10}^{30}{s}^6+1.74\times {10}^{34}{s}^5+2.373\times {10}^{37}{s}^4+8.66\times {10}^{40}{s}^3+9.654\times {10}^{43}{s}^2+1.4\times {10}^{47}s+1.24\times {10}^{50}$.

Then, to satisfy the design requirements of a phase margin greater than 45° and a gain margin greater than 6 dB, controllers $C_{P-X}\left(s\right)$ and $C_{P-Y}\left(s\right)$ can be designed by the pole-zero cancellation according to $G_{P-X}\left(s\right)$ and $G_{P-Y}\left(s\right)$. The forms of the controllers are as follows:

$$C_{P-X}\left(s\right)=74750{\displaystyle \frac{(1+0.32s)}{s(1+110s)}}$$

(14)

$$C_{P-Y}\left(s\right)=62973{\displaystyle \frac{(1+0.99s)}{s(1+130s)}}$$

(15)

Furthermore, the disturbance attenuation function $S_x={\displaystyle \frac{1}{1+G_{FSM-X}·C_{P-X}}}$ can be obtained by combining Equations (12) and (14), and $S_y={\displaystyle \frac{1}{1+G_{FSM-Y}·C_{P-Y}}}$ can be obtained by combining Equations (13) and (15). As shown in Figure 13, the bandwidth of the disturbance attenuation function $S_x$ and $S_y$ are 18.7 Hz and 22.1 Hz, respectively.

To evaluate the impact of controller parameters on the disturbance rejection performance related to different operating conditions of the FSM, Bode response curves corresponding to five different gain values K are plotted, as shown in Figure 14. A comparative analysis reveals that as the gain K increases, the control bandwidth correspondingly expands. Moreover, when K = 74,750, the corresponding control bandwidth reaches its maximum while still satisfying the stability margin requirements. Furthermore, to analyze the disturbance rejection characteristics of the FSM in the time domain, a sinusoidal excitation with an amplitude of A = 0.01 and a frequency of f = 2 Hz is used to simulate the micro-vibration disturbance input. The corresponding closed-loop response curve is obtained, as shown in Figure 14b. Under the FSM closed-loop condition, the micro-vibration disturbance input is significantly attenuated, which corresponds to the Bode response curves in Figure 14a.

5. Experiments and Discussion

5.1. Micro-Vibration Test and Coupling Analysis

To test the LOS jitters caused by multiple micro-vibration sources, a test platform with AQZSSDs was built in this paper, as shown in Figure 15a. The composition of the platform and the test process were described in Section 2, where the focal length and aperture of the collimator are 10 m and 600 mm, respectively. To prevent material creep and relaxation of the suspension ropes during long-term experiments, we conducted stress relaxation after installing the suspension system and fine-tuned the rope assembly based on the dynamometer readings before each experiment. In addition, another micro-vibration test platform with a rigid fixed gravity unloading tool is also built to test the micro-vibration, as shown in Figure 15b. The satellite platform carrying the optical payload is fixedly connected to the L-shaped fixed tooling through rigid legs. Before the test, the high-low point adjustment screw of the L-shaped tooling needs to be adjusted to ensure the light enters HPSOP parallelly.

According to the micro-vibration measurement systems shown in Figure 15, the micro-vibration tests of the HPSOP, unloaded by a flexible suspension or a rigid fixed tool, respectively, were carried out in this paper. Similar to JWST’s deep space exploration, there is a need to turn on the CMGs and cryocoolers at the same time when the HPSOP performs space steady-state observation missions. CMGs are used to maintain the attitude stability of space payloads, while cryocoolers are used to avoid the influence of temperature noise on high-precision imaging. (Note: the rotational speed of CMGs is 7000 r/min, while the cryocooler operates at a rotational speed of 15,000 r/min). The test results are shown in Figure 16. Note that the micro-vibration source consists of CMGs and cryocoolers.

As shown in Figure 16, while the gravity unloading tool is in a rigidly fixed state, the image point movements of the HPSOP along the X-axis are equal to the Y-axis, and the peak frequencies of the LOS jitter are both 2.4 Hz, represented by $F_{R2}$. When the gravity unloading tool is in a flexible suspension state, compared with the rigidly fixed state, the image point movements are significantly reduced along the Y-axis but without change on the X-axis. The reason is that while the suspension system avoids the high-frequency resonance of the fixed system, it introduces a new peak response at a lower frequency. Compared to the peaks that appeared in the fixed state, the new response peak reduced along the Y-axis but changed little along the X-axis, as shown in Figure 16b,c. In addition, for the flexible suspension state, the first peak frequencies of the LOS jitter along the X-axis and Y-axis are both 0.24 Hz, corresponding to $F_{F1}$, and the second peak frequency $F_{F2}$ appearing on the X-axis but not Y-axis is 0.39 Hz. Compared with the peak frequencies caused by the unloading tool in a rigidly fixed state, the peak frequencies appearing on the flexible suspension state are close to 0 Hz, proving the quasi-zero stiffness of the AQZSSD. Notably, for the micro-vibration test of the HPSOP with AQZSSD, the LOS jitter angles about the X-axis and the Y-axis are 1.253 μrad and 0.625 μrad, respectively.

Furthermore, as shown in Figure 17, for the collaborative operation of CMGs and cryocoolers, the LOS jitters may not equal the superposition of the micro-vibration caused by CMGs and cryocoolers independently. In this figure, $J_1$ is the LOS jitter appearing when operating the CMGs, $J_2$ is the LOS jitter of the cryocoolers, and $\alpha$ represents the coupling coefficient. Therefore, for guiding the improvement of the HPSOP’s pointing accuracy, it is of great significance to evaluate the coupling micro-vibration between CMGs and cryocoolers. To address this problem, we conduct the micro-vibration test while operating the CMGs and cryocoolers independently to analyze the coupling of micro-vibration responses under the steady-state observation missions.

Since the environmental impact on the micro-vibration cannot be ignored at low frequencies, this paper only examines the LOS jitters above 5 Hz, as shown in Figure 18.

As shown in Figure 18, the peak frequencies of the LOS jitter caused by CMGs and cryocoolers are approximately within $5\sim 100$ Hz. Since the LOS jitter caused by the same disturbance is definite in the frequency domain, a linear superposition of the micro-vibration response in the frequency domain was obtained through Equation (16) to compare with the collaborative test, as shown in Figure 19.

$$\left\{\begin{array}{c}L{S}_x=\sum FFT(J_{xi},N_{xi})/L_{xi}\\ L{S}_y=\sum FFT(J_{yi},N_{yi})/L_{yi}\end{array}\right.$$

(16)

where ${LS}_x$ and ${LS}_y$ are the superposition results of the LOS jitter around the X-axis and the Y-axis in the frequency domain; $J_{xi}$ and $J_{yi}$ express the LOS jitter measured in the time domain, where i is the symbol that varies with CMGs or cryocoolers; $L_{xi}$ and $L_{yi}$ are the data lengths of $J_{xi}$ and $J_{yi}$; and $N_{xi}=2^{nextpow\left(L_{xi}\right)}$ and $N_{yi}=2^{nextpow\left(L_{yi}\right)}$, where $nextpow(·)$ is used to calculate the smallest power of 2 that is greater than or equal to $L_{xi}$ or $L_{yi}$.

As shown in Figure 19, the results of the collaborative test are lower than the linear superposition, which means that the coupling effect may result in LOS jitter reduction.

Table 2. Peak points obtained by linear superposition and collaborative test.

LOS Jitter Around the X-Axis (Hz, μrad)		LOS Jitter Around the Y-Axis (Hz, μrad)
Linear Superposition	Collaborative Test	Linear Superposition	Collaborative Test
(8.74, 0.267)	(8.74, 0.038)	(5.18, 0.164)	(5.18, 0.088)
(16.53, 0.133)	(16.53, 0.038)	(6.30, 0.177)	(6.30, 0.079)
(16.85, 0.060)	(16.85, 0.045)	(8.05, 0.194)	(8.05, 0.057)
(75.54, 0.026)	(75.54, 0.012)	(16.53, 0.171)	(16.53, 0.032)
		(21.61, 0.234)	(21.61, 0.115)
		(75, 0.058)	(75, 0.011)

lists the typical responses of the LOS jitter peaks. It is not difficult to see that the LOS jitter of the linear superposition is higher than the collaborative test. For the LOS jitter around the X-axis, the results superimposed linearly are about eight times and four times the collaborative test at 8.74 Hz and 16.53 Hz. And for the LOS jitter around the Y-axis, the results superimposed linearly are more than four times the collaborative test at 8.05 Hz and 16.53 Hz. Notably, the LOS jitters collected in the collaborative test still satisfy the sub-micro-radian pointing accuracy requirement, while the results of linear superposition have been far more than 0.1 μrad. Therefore, it is necessary to adopt a collaborative test, rather than linear superposition, to evaluate the LOS jitter of the optical payload with sub-micro-radian pointing accuracy before launch.

5.2. Micro-Vibration Suppression and Image Stabilization Analysis

As can be seen from Section 5.1, the LOS jitter caused by the CMGs and cryocoolers together is more than 1 μrad. Hence, it is necessary to introduce a vibration reduction system to suppress the micro-vibration and stabilize the image point. Section 4 proposes a two-stage image stabilization system composed of BVILs and the FSM to reduce the image offset. During the test, the BVILs are unlocked to suppress the disturbances in the mid-high frequency, and then the FSM is collaboratively controlled to compensate for the LOS errors existing in low frequency. The image point offsets under different states are as follows:

As shown in Figure 20, the image offsets increase by about 2 pixels along the X-axis, decreasing by 0.5 pixels along the Y-axis after unlocking BVILs in the first-order vibration suppression. Then, the image point offsets along the X-axis dropped to 2.31 pixels from 9.59 pixels, and the offsets along the Y-axis dropped to 3.63 pixels from 4.63 pixels after the FSM was in closed-loop control. According to the analysis in Section 4.1, it can be seen that BVILs will introduce the amplification of low-frequency responses around 2 Hz. The pixel offset becoming larger after unlocking BVILs may be the result of the amplification in the low-frequency being higher than the reduction in the mid-high frequency. On the contrary, the offset amplitude of the image decreased significantly because of the compensation for the LOS jitter in the low frequency when the FSM is controlled.

Furthermore, the amplitude–frequency response of the LOS jitter after two-stage stabilization is shown in Figure 21. After unlocking BVILs, the amplitude response above 4 Hz decreased, but the LOS jitter near 2 Hz rose a lot due to the amplification characteristics of BVILs in low frequency. Notably, the amplification in the low frequency is larger than the attenuation in the mid-high frequency, which explains the increase in the pixel offset in Figure 20 after the passive vibration isolation in the first stage. Fortunately, the LOS jitters, no matter in the mid-high frequency above 4 Hz or in the low frequency below 4 Hz, are both decreased significantly by opening the FSM and unlocking BVILs together. This shows the possibility of integrating the passive vibration isolation platform and FSM to achieve wide-band image stabilization.

Finally, to evaluate the LOS jitter of the HPSOP quantitatively, the root mean square error (RMS) of the jitter angles along the X-axis and the Y-axis are summarized in

Table 3. J_RMS value after the vibration suppression under different orders.

Type	JRMS Around the X-Axis (µrad)	JRMS Around the Y-Axis (µrad)
No vibration suppression	1.253	0.625
First-order vibration suppression	1.312	0.631
Two-stage vibration suppression	0.276	0.331

. Compared with the case without vibration reduction, after unlocking the first-stage BVILs, the LOS jitters in both directions increase due to the amplification caused by the vibration isolation legs. On the contrary, the RMSs in the two directions were reduced to 0.257 μrad and 0.317 μrad by the two-stage image stabilization platform with BVILs and FSM. The experiment in this paper demonstrates the feasibility of building HPSOPs that require sub-micro-radian pointing accuracy.

In addition, to further compare the superiority of the proposed solution “FSM + VILs”, we compared it with the image stabilization research of dual FSMs conducted by Xie et al. [36], and the results are shown in the following table (

Table 4. Comparison of the two J_RMS schemes.

Scheme	JRMS Around the X-Axis (µrad)	JRMS Around the Y-Axis (µrad)
FSM + VILs	0.276	0.331
dual FSMs [36]	0.513	0.653

).

Obviously, as shown in Table 4 compared with the dual FSM scheme, the two-stage vibration isolation and image stabilization based on FSM and VILs performs better.

5.3. Error Analysis and Calibration

During micro-vibration testing, the noise, which is generated by the external mechanical environment and the electronic components inside the system, is the main factor affecting the accuracy of HPSOP testing. According to the distribution characteristics, the errors in micro-vibration testing could be divided into systematic errors and random errors. For systematic errors, the gray value deviation errors of the CCD pixel and the suspension apparatus (AQZSSD) are the main parts, where the errors of CCD could be eliminated by image processing algorithms like the grayscale centroid method, and the errors caused by the low-frequency disturbance peak could be calibrated by using a high-pass filter algorithm or LOS stabilization equipment. For random errors, mechanical noise generated by complex the mechanical environments and the noise generated by electronic components such as CCD are the main components. Compared with systematic errors, random errors are usually difficult to eliminate because of the characteristics of randomness and wide bandwidth. Therefore, to evaluate the random errors, it is necessary to conduct noise testing before micro-vibration testing and calculate the random errors by subtracting systematic errors from the CCD responses. For the micro-vibration test system used in this paper, as mentioned before, the systematic errors appear at 0.4 Hz, while the RMS value of systematic errors is less than the pointing accuracy of 0.1 μrad, according to the static test results.

6. Conclusions

In this paper, micro-vibration tests and the two-stage image stabilization of an HPSOP are carried out under disturbances caused by CMGs and cryocoolers. To avoid the disturbance result of simulated flight devices on micro-vibration tests during ground tests, this paper proposes an AQZSSD adjusted by linear motors. The results indicate that the disturbance introduced by the AQZSSD is below 0.4 Hz, which is close to quasi-zero stiffness. Notably, the collaborative test results of CMGs and cryocoolers are lower than the linear superposition due to the coupling of the micro-vibration response. Moreover, this paper also proposes a two-stage image stabilization system, consisting of three passive BVILs and an FSM driven by piezoelectric ceramics, to address the difficulty of image stabilization due to wide-band micro-vibration disturbances. Three passive BVILs and an FSM are used to accomplish the first-order vibration suppression and the second-order vibration suppression. As can be seen from the theoretical analysis, the first-order vibration suppression and the second-order vibration suppression can realize image stabilization by suppressing LOS jitters in the mid-high frequency and low frequency, respectively. The experimental results show that the LOS jitter peak of the HPSOP reduced to 0.276 μrad from 1.253 μrad, corresponding to the image point offset amplitude decreasing about 5 pixels (pixel size: 5.5 μm × 5.5 μm).

The micro-vibration test platform and two-stage image stabilization system proposed in this paper provide feasible solutions for high-precision micro-vibration testing and image stabilization. Yet, there are still some limitations of the proposed system. For example, since the test object used is a dot target, it is not convenient to evaluate the system’s image quality of the complex targets. In addition, by comparing the LOS jitter changes before and after the two-stage image stabilization operation, there is still room for improvement in the jitter suppression. To address the above challenges, future work will concentrate on the following: (a) To evaluate the impact of micro-vibration on the system imaging quality, replace the dot target with an extended target during the micro-vibration test. (b) For higher image stabilization requirements, dual FSMs can be introduced to achieve three-stage image stabilization combined with the vibration isolation legs. (c) During the micro-vibration test, it will be possible and meaningful to restore image details and edges by incorporating machine learning to avoid the risk of image degradation.