Enhancing Long-Term Robustness of Inter-Space Laser Links in Space Gravitational Wave Detection: An Adaptive Weight Optimization Method for Multi-Attitude Sensors Data Fusion

Abstract

The stable and high-precision acquisition of attitude data is crucial for sustaining the long-term robustness of laser links to detect gravitational waves in space. We introduce an effective method that utilizes an adaptive weight optimization approach for the fusion of attitude data obtained from charge-coupled device (CCD) spot-positioning-based attitude measurements, differential power sensing (DPS), and differential wavefront sensing (DWS). This approach aims to obtain more robust and lower-noise-level attitude data. A system is designed based on the Michelson interferometer for link simulations; validation experiments are also conducted. The experimental results demonstrate that the fused data exhibit higher robustness. Even in the case of a single sensor failure, valid attitude data can still be obtained. Additionally, the fused data have lower noise levels, with root mean square errors of 9.5%, 37.4%, and 93.4% for the single CCD, DPS, and DWS noise errors, respectively.

1. Introduction

Gravitational wave detection is at the forefront of international scientific research, opening a new window for understanding the universe [1]. Currently, many countries are actively pursuing space-based gravitational wave detection projects, such as the Laser Interferometer Space Antenna (LISA) [2], Taiji [3], and Tianqin [4]. These projects aim to deploy spaceborne Michelson interferometers to detect gravitational wave signals in the frequency range of 0.1 mHz to 1 Hz. A critical aspect of these missions is establishing and maintaining stable inter-satellite laser links [5], which is typically achieved through acquisition, tracking, and pointing (ATP) techniques [6,7]. Given the low-frequency gravitational wave signals, especially those near 0.1 mHz with periods on the order of hours, ensuring long-term robustness of these laser links is essential for successful detection [8].

In practice, various uncontrollable factors—such as minor meteoroid impacts, satellite disturbances [9], laser phase loss, errors in the point ahead angle mechanism (PAAM) [10], or issues with the optical assembly tracking mechanism (OATM)—may prevent one or more attitude sensors from obtaining reliable data [11,12,13]. This could lead to disruptions in the inter-space laser links. The lowest frequency gravitational wave signals targeted by space missions, around 0.1 mHz, correspond to periods of approximately 10,000 s (2.78 h). To detect these signals effectively, the laser link must remain stable for several cycles (often exceeding 10 h). While rapid reacquisition techniques are helpful, they cannot recover data from interrupted gravitational wave signals [14,15]. Addressing this challenge requires minimizing the risk of link interruptions, which primarily involves improving actuators, control algorithms, and the robustness of attitude data acquisition. This study focuses on enhancing the long-term robustness of attitude data acquisition systems.

Numerous studies have explored methods for establishing and reacquiring laser links. Cirillo, for instance, proposed a multi-stage acquisition scheme for LISA, using a single sensor at each stage while analyzing pointing errors [12]. GRACE-Follow-On employed bilateral scanning for laser link establishment, sharing some similarities with space gravitational wave detection [16]. Ground validation experiments by Danielle M. R. Wuchenich for GRACE-Follow-On reduced the scan area during reacquisition, achieving a laser link within 7 s [17]. For the Quantum Science Satellite (Mozi), Bai Shuai utilized an N-point finite memory filtering algorithm to achieve a pointing prediction accuracy of 0.1° despite a 5-s link interruption [18]. However, the key factor contributing to laser link interruption risk is the inability to consistently obtain high-precision attitude data.

Research on attitude data acquisition has also been extensive. Meshksar analyzed the accuracy and range of DWS and DPS technologies, concluding that, while DWS offers higher precision, DPS provides a broader measurement range [19]. Duan developed analytical and numerical models of DWS technology, exploring how beam clipping impacts DWS non-linearity [20]. Gao, as part of the Taiji project, proposed a high-precision centroid algorithm for weak-light conditions during CCD acquisition, achieving a centering accuracy of 0.0582 pixels at an optical intensity of 100 pW [21]. While these works have contributed to laser link establishment and reacquisition techniques, they focus on single-sensor solutions, leaving a gap in addressing the long-term robustness of laser links in space gravitational wave detection, particularly through multi-sensor data fusion.

To address this gap, this study proposes an adaptive weight optimization method for fusing attitude data from multiple sensors. We conducted experiments by combining attitude data from CCD (position-based attitude measurement), DPS, and DWS sensors. Our method not only provides effective attitude data in cases where a single sensor fails but also simplifies control systems, reducing the risk of laser link interruptions. By adopting a data fusion approach, we reduce noise in attitude data, offering a more reliable data source for the comprehensive control of laser links across all operational stages.

2. Optimization Method

In current space gravitational wave detection projects, such as LISA/Taiji, the inter-space laser link is primarily divided into three stages: establishment, scientific measurement, and reacquisition. A system control flowchart of the inter-space laser link is shown in Figure 1. During the laser link establishment stage, the system primarily employs a star tracker (STR) and CCD for attitude measurement, adjusting the far-end incident laser direction using satellite micronewton propulsion to create interference with the local laser on the quadrant photodiode (QPD) surface. In the scientific measurement stage, the control system utilizes DWS to measure the attitude and suppresses the laser pointing jitter noise [6,22]. When a laser link interruption occurs, the control system enters the reacquisition stage. In this stage, the system makes judgments based on the data returned by various sensors and executes laser link reacquisition from different stage nodes. Notably, when the laser pointing angle is within the measurement range of the attitude sensors, these sensors can simultaneously provide valid attitude data for use.

From Figure 1, it is evident that the control system of the laser link requires the use of different sensors for attitude adjustments at different stages. It complicates the control system and also necessitates the design of a powerful decision-making system. Therefore, we simplified the control flow by designing an attitude data fusion processor. As shown in Figure 2, the modified control system acquires attitude data solely from the attitude data fusion processor, which can be applied to control all stages of the satellite, thereby effectively simplifying the control system. There are several methods, such as optimizing sensors, control algorithms, optical design, and data acquisition, to enhance the robustness of experimental systems for laser links. Therefore, we designed an attitude data fusion processor to improve the robustness of the laser link experimental system from the perspective of attitude data acquisition, while simplifying the control system.

The control flow shown in Figure 1 relies on the attitude data from a single attitude sensor at each stage. The scheme makes the control system complex and imposes high requirements on the performance and robustness of attitude sensors. In contrast, the attitude data fusion processor in the improved scheme can fuse data from multiple sensors. In addition to simplifying the control system, it effectively enhances the robustness of the attitude data acquisition and reduces the risk of laser link interruptions. Backup devices are commonly used in traditional designs to enhance robustness, and we have considered this. The attitude data fusion processor that we intend to design is not meant to be a substitute for the three detectors (DPS, DWS, and CCD). Instead, the novelty of this work lies in optimizing using software algorithms without adding to or modifying the hardware design.

In the scientific measurement stage of space gravitational wave detection, a DWS has been proposed for attitude measurement. Although DWS offers high precision, its measurement range is limited, which increases the risk of laser link interruptions. Research indicates that a satellite system based on a single QPD can simultaneously employ both the DWS and DPS to acquire two different types of attitude measurement data without introducing additional hardware resources [23,24]. Although DPS has a lower accuracy than DWS, it has a larger measurement range, contributing to reduced link interruption risk and improved robustness of the inter-space laser link [19,25,26]. Therefore, in designing the attitude data fusion processor, we incorporated the DPS angle measurement technology in addition to the existing DWS to obtain more robust attitude data.

In summary, we aim to improve the robustness of inter-space laser links by designing an attitude data fusion processor (limited to current technology). This paper focuses on the study of CCD, DPS, and DWS. Figure 3 describes the improved closed-loop control system of the inter-space laser link, where CCD represents attitude data measurement using the CCD sensor for spot positioning, and DPS and DWS represent differential power sensing and differential wavefront sensing, respectively, based on the QPD. We introduced a data fusion processor into the control system to fuse attitude data from the sensors at different stages into a set of attitude data. Attitude data can be efficiently acquired through this processor, even in the case of partial sensor failure, thereby enhancing the robustness of the control system, reducing attitude measurement noise, and simplifying the satellite control system. The following sections provide a detailed explanation of the design of the attitude data fusion processor.

3. Optimization Design

The diagram in Figure 4 illustrates the basic framework of the attitude data fusion processor. The sensor data, labeled as D1, D2, D3, …, Dn, represent attitude data from n sensors. The processor’s output consists of the weight parameters for each sensor. Initially, the attitude data from the sensors must undergo preprocessing to ensure they form valid attitude data vectors. These vectors are then processed using machine learning algorithms, with the goal of minimizing either the root mean square (RMS) or variance of the fused data to calculate the optimal weight parameters. The study proceeds by exploring the fusion of data from three types of sensors: CCD, DPS, and DWS.

3.1. Data Fusion Processor Design

The diagram in Figure 5 represents a flowchart of the adaptive weight optimization method for multi-attitude sensors, corresponding to the attitude data fusion processor shown in Figure 3.

First, attitude data were obtained using the CCD, DPS, and DWS. The CCD images capture the laser spot emitted by a distant satellite, allowing us to determine the spot’s position within the image and measure its pixel offset from the reference standard position. By employing the pixel/angle conversion coefficients, the angular deviation of the satellite laser pointing at the yaw and pitch angles can be obtained. DPS is a traditional positioning method based on the intensity of the laser on the QPD photosensitive surface, whereas DWS utilizes the phase difference of the heterodyne interference light on different photosensitive surfaces of the QPD for positioning.

Subsequently, the three sets of acquired attitude data were downsampled, that is, frame-rate synchronization was performed. The measurement bandwidths of the CCD, DPS, and DWS are determined by the frame rates of the camera, voltage signal output by the DAC, and the phase meter, respectively. Generally, the frame rates of these three sensors are not the same. Therefore, downsampling is necessary to unify their effective data frame rates and satisfy the system requirements.

The data validity was then assessed. In the scientific measurement mode, DWS signals are most prone to failure because of their limited effective measurement range, typically in the micro-radian range. Therefore, assessing the validity of the measurement data can effectively prevent erroneous data from being used for attitude adjustments. The data validity assessment method is illustrated in Figure 6. When the CCD, DPS, and DWS data are input, the first step is to check whether the intensity of the spot imaged by the CCD exceeds a threshold. If the spot intensity is below the threshold, all data are deemed invalid; otherwise, the subsequent step is performed. The second step assesses whether the attitude data provided by the CCD fall within the measurement range of the DPS; if not, the DPS and DWS data are deemed invalid, and the CCD data are valid. The third step checks whether the attitude data measured by the DPS are within the measurement range of the DWS; if not, the DWS data are invalid, and the CCD and DPS data are valid. Otherwise, all data are considered valid.

The three datasets were then subjected to time synchronization. As different attitude sensors have different data processing times, there may be a time difference between them. This time difference constitutes a system error, and its value can be determined by pre-calibration and adjustment during data time synchronization. The calibration method involves inputting a step change in the attitude angle variation to the system after the system is installed and tuned, obtaining the time difference between the detected signals of the different attitude sensors, and correcting it during data time synchronization.

Finally, using the adaptive weight optimization method for multi-attitude sensors, the attitude data from multiple sensors are fused to obtain the optimal attitude data for the output. The method employs a machine learning algorithm to obtain the most robust attitude data possible through multiple sensor data and adaptively adjusts the weight parameters of different sensors. The following subsection provides a detailed description of the proposed method.

3.2. Data Fusion Algorithm

The primary objective of a data fusion processor is to obtain data with higher robustness. Therefore, we aimed to combine the data from multiple sensors using different weights to calculate the most robust attitude data. We utilized a machine learning algorithm for weight calculation [27], and the calculation method is as follows:

Let the data fusion weight equation be

$$S_f=\sum _{i=1}^N{W}_i·S_i$$

(1)

where N indicates the number of sensors used, $W_i$ indicates the weight parameter for each sensor, $S_i$ indicates the attitude data acquired by each sensor, and $S_f$ indicates the result after data fusion.

Based on practical requirements, the boundary conditions for obtaining weight parameters are

$$\sum _{i=1}^N{W}_i=1$$

(2)

$$0\le W_i\le 1,i\in \left\{1,2,3,\dots ,N\right\}$$

(3)

Since the output values of the sensors are all attitude data, in radians, the weights should sum up to one to ensure the correctness of the fused data. Constraining the weight coefficients $W_i$ between 0 and 1 aims to prevent the generation of negative weight values.

The cost function of the system determines the direction of the parameter optimization, and two optimization directions are provided. The RMS reflects the amplitude and fluctuation of the attitude data, and the variance reflects the dispersion of the data. Therefore, the cost functions with the RMS and variance as the optimization directions are

$$J_{rms}\left(W_i\right)=\sqrt{{\displaystyle \frac{{\sum }_{d=1}^L{\left({\sum }_{i=1}^N{W}_i·S_{i\_d}\right)}^2}{L}}}=\sqrt{{\displaystyle \frac{{\sum }_{d=1}^L{S}_{f\_d}^2}{L}}}$$

(4)

$$J_{var}\left(W_i\right)={\displaystyle \frac{1}{L}}\sum _{d=1}^L{\left(\sum _{i=1}^N{W}_i·S_{i\_d}-\mu \right)}^2={\displaystyle \frac{1}{L}}\sum _{d=1}^L{(S_{f\_d}-\mu )}^2$$

(5)

In the equations, L represents the sampling depth of the data, N indicates the number of sensors, $W_i$ corresponds to the adjustable weight parameter, $S_{i\_d}$ represents the values measured by attitude sensors, and $\mathsf{\mu }$ denotes the mean value of this set of attitude data.

Using the optimization algorithm, we obtain the values of the weight parameters $W_i$ that minimize the cost function J and satisfy the boundary conditions:

$$arg\underset{W_i}{min}{J}_{rms}\left(W_i\right)=arg\underset{W_i}{min}\sqrt{{\displaystyle \frac{{\sum }_{d=1}^L{S}_{f\_d}^2}{L}}}$$

(6)

$$arg\underset{W_i}{min}{J}_{\nu ar}\left(W_i\right)=arg\underset{W_i}{min}({\displaystyle \frac{1}{L}}\sum _{d=1}^L{(S_{f\_d}-\mu )}^2)$$

(7)

Notably, if the data of a sensor are determined to be invalid during data acquisition, its corresponding weight value will also be set to zero. Several optimizers are suitable for the optimization algorithm. The adaptive moment estimation (Adam) optimizer was used in this study. It employs a stochastic gradient descent to determine the minimum cost function with an adaptive learning rate. It can automatically adjust the learning rate based on its performance during training, making it better suited for adapting to the changing speeds of different parameters, which is suitable for the data fusion requirements of our experiment [28].

In this experiment, we utilized a CCD, DPS, and DWS for attitude data acquisition. Therefore, as an example, we present a data fusion method for these three sensors, with the goal of minimizing the RMS value. Similar data fusion algorithms for incorporating more sensors can be derived as follows.

Let the data fusion weight equation be

$$S_f=W_1·S_{ccd}+W_2·S_{dps}+W_3·S_{dws}$$

(8)

where $W_1$, $W_2$, and $W_3$ represent the weight parameters for CCD data, DPS data, and DWS data, respectively, and $S_f$ is the result after data fusion.

The boundary conditions for the weight parameters are

$$W_1+W_2+W_3=1$$

(9)

$$0\le W_1,W_2,W_3\le 1$$

(10)

The cost function of the system is

$$\begin{array}{c}\hfill J_{rms}(W_1,W_2,W_3)=\sqrt{{\displaystyle \frac{{\sum }_{d=1}^L{(W_1·S_{ccd\_d}+W_2·S_{dps\_d}+W_3·S_{dws\_d})}^2}{L}}}=\sqrt{{\displaystyle \frac{{\sum }_{d=1}^L{S}_{f\_d}^2}{L}}}\end{array}$$

(11)

In the equations, L represents the sampling depth of the data, $W_1$, $W_2$, and $W_3$ are the adjustable weight parameters, and $S_{ccd\_d}$, $S_{dps\_d}$, and $S_{dws\_d}$ denote the angle measurement values of CCD, DPS, and DWS, respectively.

We calculate the values of $W_1$, $W_2$, and $W_3$ using the optimization algorithm when the cost function $J_{rms}$ is minimized and satisfies the boundary conditions:

$$\begin{array}{c}\hfill arg\underset{W_1,W_2,W_3}{min}{J}_{rms}(W_1,W_2,W_3)=arg\underset{W_1,W_2,W_3}{min}\sqrt{{\displaystyle \frac{{\sum }_{d=1}^L{S_{f\_d}}^2}{L}}}\end{array}$$

(12)

Thus, the optimal values of the weight parameters $W_1$, $W_2$, and $W_3$ can be obtained using the Adam optimizer.

The ultimate goal of the proposed algorithm is to obtain the weight parameters of multiple sensors. The weight parameters of this algorithm change in real time based on the measurement data; thus, the algorithm exhibits excellent real-time adjustment characteristics. At the same time, the higher the algorithm complexity, the longer the computation time required, which is not conducive to the real-time attitude adjustment of the space satellite payload. This reduces the control system bandwidth, affects the payload attitude adjustment and attitude noise suppression, and poses challenges to satellite computational resources.

Therefore, the specific advantages of our proposed algorithm are as follows: (a) real-time adaptable parameters; (b) low algorithm complexity, high computation speed, and minimal impact on subsequent attitude control; (c) minimal design changes are required to provide a weight parameter interface, with parameter computations performed by either ground-based computers or satellite computers; (d) introduction of DPS as a source of attitude data that does not require additional hardware; (e) high generalizability of the algorithm. Although the experiments used only a CCD, DPS, and DWS, they can further integrate additional sensors, such as gyroscopes, star trackers, and multiple primary and backup QPDs.

3.3. Algorithmic Mathematical Analysis

We conducted a mathematical analysis to demonstrate the effectiveness of the data fusion algorithm. Since the steady-state measurements of commonly used attitude sensors approximate Gaussian white noise, we consider N sets of Gaussian white noise signals $x_i\left(t\right)$ with the same mean $\mu$ but different variances ${\sigma }_i^2$, where $i=1,2,\dots ,N$. For simplicity, we set their mean $\mu =0$. Each signal $x_i\left(t\right)$ is assigned a weight $w_i$, and the weighted sum of the signals is $z\left(t\right)$. We aim to find the weights $w_i$ that minimize the RMS value of $z\left(t\right)$.

The fused signal $z\left(t\right)$ is expressed as

$$z\left(t\right)=\sum _{i=1}^N{w}_i{x}_i\left(t\right)$$

(13)

Since the signals $x_i\left(t\right)$ are independent, the variance of $z\left(t\right)$ is given by

$${\sigma }_z^2=\mathrm{Var}\left(z\left(t\right)\right)=\mathrm{Var}\left(\sum _{i=1}^N{w}_i{x}_i\left(t\right)\right)$$

(14)

Assuming the signals are independent, we obtain

$${\sigma }_z^2=\sum _{i=1}^N{w}_i^2{\sigma }_i^2$$

(15)

Since $z\left(t\right)$ represents a Gaussian white noise signal with a mean $\mu$ of zero, its standard deviation is its RMS value:

$$\sigma =\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{(x_i-\mu )}^2}=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{x}_i^2}=\mathrm{RMS}$$

(16)

Thus, the RMS of $z\left(t\right)$ is

$${\mathrm{RMS}}_z=\sqrt{{\sigma }_z^2}=\sqrt{\sum _{i=1}^N{w}_i^2{\sigma }_i^2}$$

(17)

To ensure the effectiveness of the fusion, we impose the constraint

$$\sum _{i=1}^N{w}_i=1$$

(18)

We seek to minimize the RMS of $z\left(t\right)$:

$$\underset{w_i}{min}{\mathrm{RMS}}_z=\underset{w_i}{min}\sqrt{\sum _{i=1}^N{w}_i^2{\sigma }_i^2}$$

(19)

To simplify the problem, we minimize the squared term:

$$\underset{w_i}{min}\sum _{i=1}^N{w}_i^2{\sigma }_i^2$$

(20)

We use the Lagrange multiplier method to solve this constrained optimization problem. Introducing a Lagrange multiplier $\lambda$, we construct the Lagrangian function:

$$\mathcal{L}(w_1,w_2,\dots ,w_N,\lambda )=\sum _{i=1}^N{w}_i^2{\sigma }_i^2+\lambda \left(\sum _{i=1}^N{w}_i-1\right)$$

(21)

Taking the partial derivatives with respect to $w_i$ and $\lambda$ and setting them to zero, we obtain:

$${\displaystyle \frac{\partial \mathcal{L}}{\partial w_i}}=2w_i{\sigma }_i^2+\lambda =0$$

(22)

$${\displaystyle \frac{\partial \mathcal{L}}{\partial \lambda }}=\sum _{i=1}^N{w}_i-1=0$$

(23)

From Equation (22), we obtain:

$$w_i=-{\displaystyle \frac{\lambda }{2{\sigma }_i^2}}$$

(24)

Substituting this result into the constraint ${\sum }_{i=1}^N{w}_i=1$ yields

$$\sum _{i=1}^N-{\displaystyle \frac{\lambda }{2{\sigma }_i^2}}=1$$

(25)

$$\lambda =-{\displaystyle \frac{2}{{\sum }_{i=1}^N\frac{1}{{\sigma }_i^2}}}$$

(26)

Substituting $\lambda$ back into the expression for $w_i$:

$$w_i={\displaystyle \frac{\frac{1}{{\sigma }_i^2}}{{\sum }_{i=1}^N\frac{1}{{\sigma }_i^2}}}$$

(27)

Thus, we obtain the weights $w_i$ that minimize the RMS value. These weights are inversely proportional to the variances of the signals: i.e., the larger the variance, the smaller the weight, and vice versa.

Subsequently, we need to prove that the RMS value of the weighted signal $z\left(t\right)$ is less than that of any single signal $x_i\left(t\right)$, i.e., ${\mathrm{RMS}}_z<{\mathrm{RMS}}_{x_i}$. To verify this, we compare the variance of $z\left(t\right)$ with the variance of $x_i\left(t\right)$:

$${\sigma }_z^2=\sum _{i=1}^N{w}_i^2{\sigma }_i^2$$

(28)

Substituting the optimal weights $w_i$, we get

$${\sigma }_z^2=\sum _{i=1}^N{\left({\displaystyle \frac{\frac{1}{{\sigma }_i^2}}{{\sum }_{i=1}^N\frac{1}{{\sigma }_i^2}}}\right)}^2{\sigma }_i^2={\displaystyle \frac{{\sum }_{i=1}^N\frac{1}{{\sigma }_i^2}}{{\left({\sum }_{i=1}^N\frac{1}{{\sigma }_i^2}\right)}^2}}={\displaystyle \frac{1}{{\sum }_{i=1}^N\frac{1}{{\sigma }_i^2}}}$$

(29)

Since the denominator ${\sum }_{i=1}^N{\displaystyle \frac{1}{{\sigma }_i^2}}$ is the sum of positive terms, we have

$$\sum _{i=1}^N{\displaystyle \frac{1}{{\sigma }_i^2}}>{\displaystyle \frac{1}{{\sigma }_i^2}}$$

(30)

Therefore:

$${\displaystyle \frac{1}{{\sum }_{i=1}^N\frac{1}{{\sigma }_i^2}}}<{\displaystyle \frac{1}{\frac{1}{{\sigma }_i^2}}}={\sigma }_i^2$$

(31)

Taking the square root of both sides yields

$$\sqrt{{\displaystyle \frac{1}{{\sum }_{i=1}^N\frac{1}{{\sigma }_i^2}}}}<\sqrt{{\sigma }_i^2}$$

(32)

Thus:

$${\mathrm{RMS}}_z<{\mathrm{RMS}}_{x_i}$$

(33)

This shows that the RMS value of the weighted signal $z\left(t\right)$ is always less than the RMS value of any single signal $x_i\left(t\right)$ (the same applies to variance). Therefore, using weighted summation reduces the RMS value of the fusion signal compared to any individual signal in the set. Additionally, as the number of sensors N increases, the value of ${\sum }_{i=1}^N{\displaystyle \frac{1}{{\sigma }_i^2}}$ increases, further decreasing ${\mathrm{RMS}}_z$. In conclusion, in practical engineering applications, the measurement noise in attitude data can be effectively reduced by using a weighted summation of multiple sensors. However, the theoretical weight parameters are overly idealized and difficult to achieve optimal results in practice. Therefore, using the algorithm we proposed in Section 3.2 can obtain the optimal weight values, effectively reducing attitude measurement noise.

3.4. Simulation Analysis

We validated the results of the mathematical analysis using MATLAB2020b. Suppose that the input signals are three sets of Gaussian white noise, denoted as A, B, and C. All three datasets have a mean of zero. The variance of A remains constant at one, whereas the variances of B and C increase incrementally from 0.1 to 2 with a step size of 0.1. We calculated the weight parameters using the algorithm described in Section 3.2 and performed data fusion, resulting in a fused signal, termed Fusion. We conducted a simulation analysis of the RMS values and variances of the data, as shown in Figure 7. From Figure 7c,d, it is evident that the variance and RMS values of the fused data are consistently lower than those of any single signal, thereby effectively validating the correctness of the mathematical analysis.

Figure 8 illustrates the changes in the variance and RMS values during the exhaustive computation. The solid blue line represents the minimum variance (Figure 8a) and minimum RMS (Figure 8b) among datasets A, B, and C. The solid red line represents the difference between the variance of Fusion and the minimum variance among A, B, and C (Figure 8a) and the difference between the RMS of Fusion and the minimum RMS among A, B, and C (Figure 8b). The solid red line is always greater than zero, indicating that the RMS and variance of Fusion are consistently lower than the minimum RMS and variance among data A, B, and C. This confirms that the fused data are superior to any single dataset, which is consistent with the mathematical analysis results (Equation (33)).

4. Experimental System Design

To validate the effectiveness of the attitude data fusion processor and to assess the robustness and noise level of the fused attitude data, we designed a link simulation system and conducted performance tests on key metrics.

Figure 9 depicts the structure of the experimental system. In the experiment, we utilized an Nd:YAG laser with a wavelength of 1064 nm as the laser source. The emitted laser beam passed through a 50:50 fiber splitter (FS) and split into two sub-beams with basically the same characteristics. The two laser beams in the experimental system employed heterodyne interference with the frequency difference generated by the acousto-optical modulator (AOM) set to 1.6 MHz. Following frequency shifting, the two laser beams were emitted through a fiber collimator and reflected off the surfaces of the fast steering mirrors (FSM1 and FSM2). FSM1 simulated a laser emitted by a distant satellite, whereas FSM2 simulated a laser emitted by a local satellite. The reflected laser beams were directed to a CCD for position detection to obtain the position information of the spots. The first beam splitter (BS1) reflected the laser to the CCD for position detection, and BS2 aligned the two laser beams. The aligned laser beams passed through BS3, generating heterodyne interference on the surfaces of QPD1 and QPD2. QPD1 and QPD2, utilizing a phase meter (PM) through DWS technology, provided the attitude data. Additionally, the laser pointing deviation angle can be calculated using DPS technology.

The key equipment used in the experiment includes the following:

• FSM: S-330 piezo steering mirror by Physik Instrumente (PI) in Germany, with an angular resolution of 100 nrad and a motion range of 5 mrad;
• CCD: SH640 small high-performance shortwave infrared camera produced by China TEKWIN SYSTEMS; it features a resolution of 640 × 512 pixels and a pixel pitch of 15 $\mathsf{\mu }$m;
• QPD: GD4542-20MHZ-12K photodetector receiving module developed by the 44th Research Institute of China Electronics Technology Group Corporation, with a noise-equivalent power (NEP) better than 4.5 $pW/\sqrt{Hz}$;
• PM: A phase meter developed by the Institute of Mechanics, Chinese Academy of Sciences, with a phase measurement accuracy of up to 2 $\pi \mu rad/\sqrt{Hz}$ [29,30].

Based on the experimental system structure shown in Figure 9, we constructed an experimental test bench in a laboratory environment, as shown in Figure 10. In the experimental system, we simulated a far-field Gaussian flat-top laser using a beam expander and an iris diaphragm. Two FSMs were used to assist optical alignment during the setup [31].

5. Results

Based on the experimental system shown in Figure 10, we collected the attitude data from the CCD, DPS, and DWS for 250 s. However, the data obtained by the attitude sensors include the amplitude data of the signals, which have a fixed proportional relationship with the pointing angle of the actual laser beam (since the beam deflection angle is only in the micro-radian range, it can be assumed that the sensor signal amplitude is linearly related to the actual laser pointing angle). The calibration of this proportional factor can be derived and calculated using the angular measurement formula of the sensor or calibrated through the experimental system. In our experiment, a system calibration method was employed, which offered a more realistic and reliable calibration factor, ensuring the reliability of the experimental data.

The calibration principle of the experimental system involved the use of an FSM to generate a continuous triangular wave with angular deflection in the beam. The angular measurement coefficient of the sensor was determined by comparing the amplitude data obtained using the sensor with the actual deflection angle. In the experiment, we applied a deflection angle of 500 $\mathsf{\mu }$rad with a triangular wave period of 10 s. Following calibration, the angular measurement coefficients were determined as follows:

• $K_{CCD}=3.75\times {10}^{-5}$ rad/pixel
• $K_{DPS}=1.28\times {10}^{-3}$ rad/V
• $K_{DWS}=6.04\times {10}^{-4}$ rad/rad

Figure 11 displays the attitude data measured by the CCD, DPS, and DWS, with Fusion representing the attitude data obtained using the attitude data fusion processor from Section 3. The proposed adaptive weight optimization algorithm features weights that can be updated in real time. These weights are calculated using proprietary Python software (version: Python 3.9) code executed on the VS code. During the experimental calculations, the weight parameters were computed within a few seconds, demonstrating a high optimization efficiency. Consequently, online optimization can be achieved in which the weight parameters are transmitted in real time to the data fusion processor for application.

The calculated weight parameters $W_1$, $W_2$, and $W_3$ for Fusion are 0.012, 0.124, and 0.864, respectively. We utilized the RMS value for data evaluation to assess the effectiveness of the data fusion method. The RMS values for CCD, DPS, DWS, and Fusion were found to be 2.34 × ${10}^{-5}$ rad, 5.97 × ${10}^{-6}$ rad, 2.39 × ${10}^{-6}$ rad, and 2.23 × ${10}^{-6}$ rad, respectively. Therefore, it can be concluded that the Fusion data obtained through attitude data fusion processor have the optimal RMS value. Moreover, as described in Section 3, data fusion only requires data weighting, and the computational load is minimal, enabling real-time calculations. The weight parameters can be computed using other high-performance computers and then transmitted to the satellite computing unit, effectively reducing the computational load on a satellite.

6. Discussion

The adaptive weight optimization method for multi-attitude sensors reduces noise in the attitude measurement of the system and effectively enhances the robustness of the control system. Therefore, we selected two of the three attitude sensors to conduct attitude data fusion tests to simulate operational scenarios in real space conditions, where a single sensor may fail.

Figure 12 shows the data fusion results for CCD and DPS, with a CCD weight parameter of 0.058 and a DPS weight parameter of 0.942. Figure 13 presents the data fusion results for CCD and DWS, with a CCD weight parameter of 0.014 and a DWS weight parameter of 0.986. Figure 14 illustrates the data fusion results for DPS and DWS, with a DPS weight parameter of 0.126 and a DWS weight parameter of 0.874. We further evaluated the effectiveness of data fusion using the RMS, variance, and amplitude range of the data, as described in

Table 1. Comparison table of data fusion results for different sensors (CCD: charge coupled device measurement, DPS: differential power sensing, DWS: differential wavefront sensing, CCD&DPS: CCD and DPS fusion results, CCD&DWS: CCD and DWS fusion results, DPS&DWS: DPS and DWS fusion results, CCD&DPS&DWS: CCD, DPS, and DWS fusion results, RMS: root mean square).

	CCD	DPS	DWS	CCD&DPS	CCD&DWS	DPS&DWS	CCD&DPS&DWS
RMS (rad)	$2.34\times {10}^{-5}$	$5.79\times {10}^{-6}$	$2.39\times {10}^{-6}$	$5.81\times {10}^{-6}$	$2.36\times {10}^{-6}$	$2.25\times {10}^{-6}$	$2.23\times {10}^{-6}$
Variance (rad2)	$5.31\times {10}^{-10}$	$3.56\times {10}^{-11}$	$4.42\times {10}^{-12}$	$3.35\times {10}^{-11}$	$4.46\times {10}^{-12}$	$4.15\times {10}^{-12}$	$4.17\times {10}^{-12}$
Amplitude (rad)	$1.06\times {10}^{-4}$	$2.95\times {10}^{-5}$	$1.15\times {10}^{-5}$	$2.78\times {10}^{-5}$	$1.15\times {10}^{-5}$	$1.06\times {10}^{-5}$	$1.09\times {10}^{-5}$

.

From Table 1, it is evident that the attitude data obtained through the adaptive weight optimization method for multi-attitude sensors can be further improved. The RMS errors of the fused data from the CCD, DPS, and DWS were 9.5%, 37.4%, and 93.4%, respectively, compared to the single sensor noise error. In addition, multi-sensor data fusion provides more robust attitude data for satellites. In the event of a single sensor failure, stable attitude data can still be provided for satellite adjustment, effectively enhancing the robustness of the inter-space laser links.

7. Conclusions

We innovatively proposed an adaptive weight optimization method for multi-attitude sensors and designed an attitude data fusion processor. We conducted mathematical and simulation analyses of the algorithm, and the results show that the attitude data processed by this method are superior to any single set of attitude data. To verify the effectiveness of the method, we built an experimental system based on a Michelson interferometer and fused the CCD, DPS, and DWS data to obtain effective attitude data, considering scenarios where a single sensor fails. The experimental results demonstrate that the robustness of the attitude data obtained by fusing data from multiple sensors is increased, and attitude data noise errors are significantly reduced. The RMS errors of the fused attitude data from the CCD, DPS, and DWS were 9.5%, 37.4%, and 93.4% of the noise error of a single CCD, DPS, and DWS, respectively.

This study provides valuable insights for improving the long-term robustness and attitude accuracy of inter-satellite laser links for low-frequency gravitational wave detection in space. In future research, the adaptive weight optimization method for multi-attitude sensors proposed in this paper can be extended to fuse different types of sensors (such as gyroscopes, star trackers, and multiple primary and backup QPDs) and adaptively adjust their weight parameters based on different sensors. Through the proposed method, more robust and less noisy attitude data can be obtained, thereby enhancing the robustness of the laser link, simplifying the design of satellite control systems, and ensuring the successful implementation of low-frequency gravitational wave detection in scientific mode. In addition, this study has significant implications for long-period deep-space signal detection, improving multi-sensor control system robustness, and suppressing sensor measurement noise.