The gravitational wave spacecraft is a complex multiinput multioutput dynamic system. The gravitational wave detection mission requires the spacecraft to achieve single spacecraft with two laser links and highprecision control. Establishing one spacecraft with two laser links, compared to one spacecraft with a single laser link, requires an upgraded decoupling algorithm for the link establishment. The decoupling algorithm we designed reassigns the degrees of freedom and forces in the control loop to ensure sufficient degrees of freedom for optical axis control. In addressing the distinct dynamic characteristics of different degrees of freedom, a transfer function compensation method is used in the decoupling process to further minimize motion coupling. The openloop frequency response of the system is obtained through simulation. The upgraded decoupling algorithms effectively reduce the openloop frequency response by 30 dB. The transfer function compensation method efficiently suppresses the coupling of lowfrequency noise.
In 2016, the LIGO team in the United States announced the detection of gravitational waves. It was the first direct detection of gravitational waves in human history. To detect gravitational waves at lower frequencies, researchers have proposed space gravitational wave detection missions such as the Laser Interferometer Space Antenna (LISA) [
In these spacebased gravitational wave detection missions, the laser acquisition should be achieved prior to scientific measurement. Various acquisition schemes for different conditions are proposed [
The optical axis is defined by the optical bench (OB), which is fixed with respect to the OB without considering pointahead angle mechanism and deformation. Thus there are 2 optical axes on a spacecraft. In the laser acquisition (
In Taiji program, the interspacecraft uncertain halfangle is 24
More researches are required to upgrade from the single laserlink to multi laserlinks. Decoupling algorithms are necessary here. Due to complex spacecraft dynamics, each actuator leads to the motion of multiple degrees of freedom. Therefore, a decoupling module should be included in the control system to realize singleDOF motion by converting the motion demand of a single DOF into control signals to multiple actuators. In previous work, the 19 DOFs were assigned to three control loops as follows: dragfree control, spacecraft attitude control, and electrostatic suspension control. Each control loop contained an equal number of DOFs and driving forces (torques) [
The decoupling algorithm designed in this paper further addresses two problems. One is the degree of freedom problem. During the laser acquisition, the optical axes of two OBs on a spacecraft need to be controlled continuously and independently. 4 DOFs are needed for the two optical axes, while there are 3 DOFs in the spacecraft attitude control loop. Therefore, the spacecraft attitude control cannot achieve independent control of the optical axes. In previous work, the control and decoupling of the optical axis were only considered for the spacecraft’s attitude. Therefore, the earlier decoupling algorithms did not meet the requirements of degrees of freedom. Considering the entire MOA can be rotated by the OATM, the DOFs of the optical axis motion can be increased. Therefore the rotational torque of the spacecraft and the driving torque of OATM should be used to achieve the optical axis control.
The second is the motion coupling problem. Coupling causes noise transfer between different DOFs, so decoupling should minimize coupling between different DOFs. When a OATM operates, a torque is applied to the target MOA and a reaction torque is applied to the rest of the spacecraft. This can cause coupling problems between the optical axis motions. The rotational torque of the spacecraft can be used to offset the reaction torque of the OATM to reduce the coupling. Thus, the rotational torque of the spacecraft and the driving torque of the OATM should be put into a control loop for decoupling.
In response to the requirements of one spacecraft with two laser links in gravitational wave detection, this paper proposes the upgraded decoupling algorithms. The three control loops of the spacecraft control system are reassigned. Unlike previous work, optical axis control and decoupling in this study are not only focused on the spacecraft’s attitude but also involve Optical Assembly Tracking Mechanism. The driving torque of the OATM and the rotational torque of the spacecraft are assigned to the optical axis control loop, to achieve independent control of optical axes. The transfer function compensation method is proposed to reduce the coupling by analyzing the linear dynamics model of the spacecraft. The decoupling effect is verified by the openloop frequency response of the decoupled system. The openloop performance also affects the subsequent closedloop control performance.
In
The spacecraft’s control system is shown in the red dashed box in
CCD and Quadrant photodiodes (differential wavefront sensing, effective after attitude adjustment) [
During the laser acquisition, if one spacecraft has already established one laser link, the established laser link is kept while establishing another laser link, and then the other optical axis is controlled for scanning or alignment. The optical axes of both OBs of one spacecraft needed to be controlled independently (4 DOFs are required). The DOFs controlled by the optical axis control loop should include 3 DOFs of the spacecraft attitude and the rotation angle of the MOA. The driving torque of the optical axis control loop should include the spacecraft rotational torque and the driving torque of the OATM. The spacecraft rotational torque and the driving torque of the OATM need to be collaboratively controlled and jointly decoupled in the optical axis control loop. Therefore, the spacecraft control system is divided into three control loops: dragfree control, optical axis control and electrostatic suspension control. The DOFs and driving forces and torques in each control loop are listed in the
Drag free control  Degrees of freedom  The position of the TM in the direction of the sensitive axis and the position of TM1 perpendicular to the solar panel 
Driving forces and torques  The translational forces 

Purpose  Control spacecraft translation, maintain the relative position of electrode cage and TM  
Optical axis control  Degrees of freedom  The attitude of the spacecraft 
Driving forces and torques  Torque of micro thruster 

Purpose  Control the optical axes of both OBs for scanning or alignment with the incident laser  
Electrostatic suspension control  Degrees of freedom  The attitude of the TM 
Driving forces and torques  Electrostatic torque of electrode cage 

Purpose  Control the attitude and position of TM relative to the electrode cage (excluding three directions in drag free control) 
In this paper, we use MATLAB Simscape to build the nonlinear dynamics model of the spacecraft in
For the dynamics model, the input vector
Forces and torques  Description  Dimensionality 

The combined force of the microthrusters acting on the spacecraft center of mass  3  
The combined torque of the microthrusters acting on the spacecraft center of mass  3  
Torque of OATM1 to MOA1  1  
Torque of OATM2 to MOA2  1  
The electrostatic force applied to TM1 by electrode cage 1  3  
The electrostatic torque applied to TM1 by electrode cage 1  3  
The electrostatic force applied to TM2 by electrode cage 2  3  
The electrostatic torque applied to TM2 by electrode cage 2  3 
Degrees of freedom  Description  Dimensionality 

Position of spacecraft on the inertial reference frame  3  
Attitude of spacecraft on the inertial reference frame  3  
Turning angle of MOA1 relative to the nominal position on spacecraft reference frame  1  
Turning angle of MOA2 relative to the nominal position on spacecraft reference frame  1  
Position of TM1 relative to electrode cage 1  3  
Attitude of TM1 relative to electrode cage 1  3  
Position of TM2 relative to electrode cage 2  3  
Attitude of TM2 relative to electrode cage 2  3 
Therefore, a 20 * 20 MultipleIn MultipleOut (MIMO) system is used in this paper. After the start of the dragfree control, only 3 of the 6 DOFs translational forces on the TM will be used, and the other 3 directions will be tracked by driving the spacecraft translations without applying electrostatic forces.
After obtaining the the spacecraft nonlinear dynamics model, the state space equation can be obtained by the linearization at the working point, and the initial working point is selected as
The state space equation is obtained as
Decomposing
It can be written as two equations,
Substituting
Perform Laplace transform,
The transfer function of driving forces (torques) to DOFs is
For the decoupler, the input is the control signal of DOFs
The transfer function of the decoupler and spacecraft dynamics model is
It can be seen that the transfer function compensation is performed when use
In the previous decoupling design, there are only scale factors in the transfer function of the decoupler,
The above
Each decoupling algorithm can be divided into two parts. One is the transfer function, which can be
Decoupling algorithm I: Only one driving force (torque) will be used to control one DOF. The input (DOF to be controlled) and output (driving force and torque) distribution of the decoupler can be represented in (a) in
Decoupling algorithm II: Decoupling is performed within each control loop so that the driving forces (torques) within corresponding control loop are used to control one DOF. The input and output distribution of the decoupler is represented in (a) in
Decoupling algorithm III: Decoupling is performed within each control loop (
The decoupling effect can be evaluated by analyzing the openloop frequency response of the decoupler and the spacecraft dynamics model represented in
dB  1 Hz  10 Hz  0.01 Hz 

Decoupling strategy I  −29.1  −87.4  0.2 
Decoupling strategy II  −76.2  −128.4  30.1 
Decoupling strategy III  −88.1  −128.7  −34.8 
The frequency response at 1 Hz is analyzed first (
At the higher frequency point of 10 Hz (
In addition, the transfer function compensation method is also meaningful in cases where the output of the driver is the angle of rotation rather than the torque. This method can also be used to solve the problem of different units of input variables.
During the laser acquisition, the optical axes on a spacecraft need to be controlled independently. This imposes different decoupling requirements on the control system. The decoupling algorithm is divided into two parts, control loop assignment and transfer function design. In order to achieve independent control of the optical axes, the DOFs and the driving forces are reassigned to the three control loop. The sufficient freedom of optical axis motion is achieved by introducing the driving torque of OATM. The coupling problem due to reaction torque is reduced by decoupling in one control loop. The transfer function compensation method is used to reduce the coupling due to the different dynamics of the drivers. The openloop simulation demonstrates that the decoupling algorithm significantly reduces the coupling of the optical axis. The decoupling algorithm addressed the motion coupling issue from a single laser link to a dual laser link on one spacecraft.
The authors are thankful to the anonymous reviewers for improving this article.
This work was supported by the National Key Research and Development Program of China (2022YFC2203700).
The authors confirm contribution to the paper as follows: analysis and discussion: Xue Wang, Xingguang Qian, Jinke Yang; study conception and design: Xue Wang; draft manuscript preparation: Xue Wang, Yikun Wang, Weizhou Zhu, Zhao Cui; management and coordination: Jianjun Jia. All authors reviewed the results and approved the final version of the manuscript.
The data and related programs are available from the first and corresponding authors upon reasonable request.
The authors declare that they have no conflicts of interest to report regarding the present study.