Dual-Arm Space Robot On-Orbit Operation of Auxiliary Docking Prescribed Performance Impedance Control

Abstract

The impedance control of a dual-arm space robot in orbit auxiliary docking operation is studied. First, for the closed-chain hybrid system formed by the dual-arm space robot after capture operation, the dynamic equation of position uncontrolled and attitude controlled is established. The second-order linear impedance model and second-order approximate environment model are established for the problem of simultaneous output force/pose control of the end of the manipulator. Then, aiming at the transient performance control requirements of the dual-arm space robot auxiliary docking operation in orbit, a sliding mode controller with equivalent replacement of tracking errors is designed by introducing Prescribed Performance Control (PPC) theory. Next, Radial Basis Function Neural Networks (RBFNN) are used to accurately compensate for the modeling uncertainties of the system. Finally, the stability of the system is verified by Lyapunov stability determination. The simulation results show that the attitude control accuracy is better than $0.5°$, the position control accuracy is better than ${10}^{-3}$ m, and the output force control accuracy is better than 0.5 N when it reaches 30 N. It also indicated that the proposed control algorithm can limit the transient performance of the controlled system within the preset range and achieve high-precision force/pose control, which ensures a more stable on-orbit auxiliary docking operation of the dual-arm space robot.

1. Introduction

With the deepening of human space exploration, a large number of different types of aircraft have been launched into space. Therefore, how to properly dock the aircrafts that have been in orbit with the later launched aircrafts, so that they can enter at the right position and play their roles, has become a key technical problem to be solved [1,2,3,4]. Considering multiple aspects, the use of a dual-arm space robot to perform the auxiliary docking operation for the above in-orbit tasks provides a set of solutions that takes into account both economy and safety, and has high feasibility [5,6]. Therefore, it is of great significance to conduct research on dual-arm space robots [7,8,9].

At present, some achievements have been made in the research of dual-arm space robots. Guo et al. proposed an improved nonlinear model predictive control method introducing the idea of a sliding mode variable structure to solve the problem of trajectory tracking and the base attitude stability of a free-floating dual-arm space robot with parameter perturbation and external disturbance [10]. Based on a hybrid map, improved artificial potential field (IAPF) and adaptive coupling guidance, Hong et al. proposed a collision-free path planning approach for the mission manipulator of a dual-arm space robot [11]. Wang et al. proposed a force/pose control method based on reinforcement learning to solve the grasping problem of dual-arm space robots without grapple fixtures [12]. Stolfi et al. proposed an impedance control algorithm for dual-arm space manipulator systems to grasp space debris and evaluated the dynamic performance of the system by co-simulation [13]. However, it should be noted that, at present, researchers mainly focus on the on-orbit acquisition and trajectory planning of dual-arm robots, and the integrated operation technology of acquisition and docking is rarely included in the scope of research.

When the dual-arm space robot has completed the acquisition of the docking target spacecraft, it is assumed that the end of the dual-arm system is directly locked into the docking target, and a dual-arm closed-chain hybrid system is formed. In the ideal situation, the docking rod and docking slot can be replaced by the “plug-jack” model for the on-orbit auxiliary docking task operation of the dual-arm space robot. According to the different control purposes and force/pose control outputs, the operation process of the above on-orbit auxiliary docking task is segmented, and can be divided into three stages: (1) stabilization control of the dual-arm closed-chain hybrid system; (2) the end of two manipulators clamped to the target aircraft moving close to the docking slot; (3) and the dual-arm space robot performing on-orbit auxiliary docking operations. In the first stage, the dual-arm space robot needs to calm down quickly; in the second stage, the end of the two manipulators clamp to the docking plug contacts and collide with the slot; and in the third stage, the dual-arm space robot needs to stably and accurately control the output force of the end plug on the premise of ensuring accurate tracking of the end pose. Therefore, the control design for it is quite difficult and challenging.

For the stabilization control problem of the double-arm closed-chain hybrid system in the first stage, Dong et al. proposed a robust control algorithm for the unsteady motion of the dual-arm closed-chain hybrid system formed after the capture collision [14]. Cheng et al. studied the dynamic evolution process of satellite acquisition by a dual-arm space robot and proposed a stabilizing control strategy [15]. Aiming at the problem of contact collision between space robot and environment in the second stage, Wang and Katupitiya proposed a smooth quasi-continuous second-order sliding mode control strategy for two-arm space robots to capture non-cooperative targets and remove the chattering effects caused by the sliding mode control [16]. Yang et al. presented a coordinated control scheme for two capture tasks via a dual-arm space robot with its base on a flexible space structure, where the impact of structural vibration of the dual-arm system is counteracted by reactionless motion [17]. Xia et al. designed a new capture scheme for the dual-arm space robot to capture the moving target by using the spherical outer envelope of the dual-arm [18].

However, the above researchers have not paid much attention to the problem of precise output force/pose control of the manipulator end for dual-arm space robots performing on-orbit tasks. For the force/pose control problem of the end of manipulators in the third stage, impedance control theory provides strong support for solving this problem [19,20,21,22]. What is more, it should be noted that, considering the complexity of space missions and the diversity of spacecrafts, more thorough and detailed research work in this field is still needed at this stage. In addition, the transient performance of the controller, such as the error convergence rate and overshoot, also plays an important role in the successful completion of on-orbit docking missions. In order to solve this problem, an effective solution is to introduce Prescribed Performance Control (PPC) theory and make equivalent replacement for tracking errors within the preset range [23,24]. Therefore, the controller design for the dual-arm space robot on-orbit auxiliary docking needs to overcome these difficulties and ensure its transient performance at the same time, which is also a challenge.

According to the above discussion, in order to realize the high-precision force/pose control with ensured transient performance of the dual-arm space robot on-orbit operation of auxiliary docking with target spacecrafts, a PPC sliding mode impedance control strategy based on the compensation of adaptive RBFNN is proposed. The contributions are listed as follows:

(1) The closed-chain hybrid system dynamic model for a dual-arm space robot after capturing the target spacecraft is established by the Lagrange method, and the second-order linear impedance model and the second-order approximate environment model are established based on impedance control theory. (2) The proposed control algorithm uses adaptive RBFNN to fit the modeling uncertainties of the system, so that the controller has a good effect on the force/pose control accuracy of the space robot. The attitude control accuracy is better than $0.5°$, position control accuracy is better than ${10}^{-3}$ m, and end output force control accuracy is better than 0.5 N. (3) The simulation results show that the proposed control algorithm can stabilize the dual-arm closed-chain hybrid system quickly, and its transient performance is limited in the preset range, which provides a theoretical basis and data support for the stable and smooth execution of the on-orbit mission.

2. Dynamics and Kinematics Modeling

2.1. System Dynamics Model

The process of a dual-arm space robot auxiliary docking with the target spacecraft in orbit is shown in Figure 1, which is taken as the research object. The global inertial coordinate system of the dual-arm space robot system $XOY$ is established by arbitrary selection of $O$ point as the reference origin.

The symbol definitions in Figure 1 are shown in

Table 1. Definition of system parameter.

Symbol	Definition
$O_0$	Carrier centroid
$x_0$, $y_0$	Coordinate of $O_0$ in $XOY$
$O_i(i=0,1,2,\dots ,6)$	Geometric center of the rotating joints
$O_{\mathrm{t}}$	Center of mass of the target spacecraft
$x_{\mathrm{t}}$, $y_{\mathrm{t}}$	The coordinate of $O_{\mathrm{t}}$ in $XOY$
$m_0,I_0$	Mass and moment of inertia of the carrier
$d_0$, $L_{\mathrm{B}}$	Length of $O_0{O}_1$, $O_0{O}_4$, and $O_1{O}_4$
${\psi }_1$, ${\psi }_2$	Angle between $O_0{O}_1$, $O_0{O}_4$, and $x_0$
$m_i,I_i,L_i(i=1,2,3)$	Mass, moment of inertia, and length of ith left arm
$m_i,I_i,L_i(i=4,5,6)$	Mass, moment of inertia, and length of ith right arm
$d_i(i=1,\dots ,6)$	Length between hinge center of ith joint and mass center of ith arm
$m_{\mathrm{t}},I_{\mathrm{t}},d_L,d_R$	Mass, moment of inertia, and radius of the target spacecraft
${\theta }_0,{\theta }_{\mathrm{t}}$	Carrier attitude angle and target spacecraft rotation angle
${\theta }_i(i=1,\dots ,6)$	Rotation angle of ith arm
$r_{\mathrm{c}}$	System total mass center radius vector in $XOY$
$r_0$	Carrier centroid radius vector in $XOY$
$r_i(i=1,2,\cdots ,6)$	Mass center of ith arm radius vector in $XOY$

.

Based on the Second Lagrange Equation, the system dynamics model of a dual-arm space robot with uncontrolled carrier position and controlled attitude before capture can be derived as follows:

$$M(q)\ddot{q}+C(q,\dot{q})\dot{q}=\tau +J^{\mathrm{T}}F,$$

(1)

where $q={[q_{\mathrm{B}}^{\mathrm{T}},{\theta }_{\mathrm{L}}^{\mathrm{T}},{\theta }_{\mathrm{R}}^{\mathrm{T}}]}^{\mathrm{T}}$, $q_{\mathrm{B}}={[x_0,y_0,{\theta }_0]}^{\mathrm{T}}$, ${\theta }_{\mathrm{L}}=[{\theta }_1,{\theta }_2,{\theta }_3{]}^{\mathrm{T}}$, ${\theta }_{\mathrm{R}}=[{\theta }_4,{\theta }_5,{\theta }_6{]}^{\mathrm{T}}$. $M(q)\in {\mathrm{R}}^{9\times 9}$ represents the generalized mass matrix of the dual-arm space robot system, $C(q,\dot{q})\dot{q}\in {\mathrm{R}}^{9\times 1}$ is the column vector including Coriolis force and centrifugal force. $\tau ={[{\tau }_{\mathrm{B}}^{\mathrm{T}},{\tau }_{\mathrm{L}}^{\mathrm{T}},{\tau }_{\mathrm{R}}^{\mathrm{T}}]}^{\mathrm{T}}$, ${\tau }_{\mathrm{B}}={[0,0,{\tau }_0]}^{\mathrm{T}}$ is the control torque of the carrier pose. ${\tau }_{\mathrm{L}}={[{\tau }_1,{\tau }_2,{\tau }_3]}^{\mathrm{T}}$ is the control torque of the left arm of each joint. ${\tau }_{\mathrm{R}}={[{\tau }_4,{\tau }_5,{\tau }_6]}^{\mathrm{T}}$ is the control torque of the right arm of each joint. $J\in {\mathrm{R}}^{6\times 9}$ is the motion Jacobian matrix corresponding to the collision points ${\mathrm{P}}_{\mathrm{L}}$ and ${\mathrm{P}}_{\mathrm{R}}$ at the end of the manipulator, which consists of the elements of the motion Jacobian matrix $J_{\mathrm{L}}$ and $J_{\mathrm{R}}$ at the ends of the left and right arms. $J_{\mathrm{L}}$, $J_{\mathrm{R}}\in {\mathrm{R}}^{3\times 6}$ are the motion Jacobian matrices corresponding to the collision points ${\mathrm{P}}_{\mathrm{L}}$ and ${\mathrm{P}}_{\mathrm{R}}$. $F$ represents the collision force at the end of the dual-arm space robot.

Similarly, the Lagrange modeling method is used to obtain the dynamics model of the target spacecraft system:

$$M_{\mathrm{t}}{\ddot{q}}_{\mathrm{t}}=J_{\mathrm{t}}^{\mathrm{T}}{F}^{\prime },$$

(2)

where $q_{\mathrm{t}}={[x_{\mathrm{t}},y_{\mathrm{t}},{\theta }_{\mathrm{t}}]}^{\mathrm{T}}$, $M_{\mathrm{t}}\in {\mathrm{R}}^{3\times 3}$ represents the generalized mass matrix of the target spacecraft system. $J_{\mathrm{t}}=[\begin{array}{cc}{J}_{\mathrm{tL}}^{\mathrm{T}}& J_{\mathrm{tR}}^{\mathrm{T}}\end{array}{]}^{\mathrm{T}}$, $J_{\mathrm{tL}}$, $J_{\mathrm{tR}}\in {\mathrm{R}}^{3\times 3}$ are the motion Jacobian matrices corresponding to the target spacecraft contact collision points ${\mathrm{P}}_{\mathrm{L}}^{\prime }$ and ${\mathrm{P}}_{\mathrm{R}}^{\prime }$ with respect to their centroids, respectively. $F^{\prime }$ is the collision force at the docking target.

According to the force analysis of the above system based on Newton’s third law, it is obvious that $F$ and $F^{\prime }$ interact with each other, which can be described as:

$$F+F^{\prime }=0^{6\times 1}.$$

(3)

With reference to Formula (3), Formula (2) is deformed as:

$$F^{\prime }={\left(J_{\mathrm{t}}^{\mathrm{T}}\right)}^{+}{M}_{\mathrm{t}}{\ddot{q}}_{\mathrm{t}}+F_{\mathrm{I}},$$

(4)

where ${\left(J_{\mathrm{t}}^{\mathrm{T}}\right)}^{+}$ is the pseudo-inverse of $J_{\mathrm{t}}^{\mathrm{T}}$, ${\left(J_{\mathrm{t}}^{\mathrm{T}}\right)}^{+}{M}_{\mathrm{t}}{\ddot{q}}_{\mathrm{t}}$ is the operating force for the auxiliary docking of the dual-arm space robot in orbit, and $F_{\mathrm{I}}$ is the internal force exerted by the dual-arm space robot on the target spacecraft during docking, considering that the internal force of $F_{\mathrm{I}}$ occurs in the zero space of $J_{\mathrm{t}}^{\mathrm{T}}$, namely: $J_{\mathrm{t}}^{\mathrm{T}}{F}_{\mathrm{I}}=0^{6\times 1}$.

By substituting Formula (2)–(4) into Formula (5), the dynamic model of a dual-arm space robot capturing the target spacecraft can be described as:

$$M\ddot{q}+C\dot{q}=\tau -J^{\mathrm{T}}{\left(J_{\mathrm{t}}^{\mathrm{T}}\right)}^{+}{M}_{\mathrm{t}}{\ddot{q}}_{\mathrm{t}}-J^{\mathrm{T}}{F}_{\mathrm{I}}.$$

(5)

It is assumed that after the capture collision, the end effector of a dual-arm space robot is locked with the target spacecraft, and a dual-arm closed-chain hybrid system is formed. By analyzing the motion of the left capture point ${\mathrm{P}}_{\mathrm{L}}({\mathrm{P}}_{\mathrm{L}}^{\prime })$, the motion constraints can be described as:

$$J_{\mathrm{L}}{\dot{q}}_{\mathrm{L}}=J_{\mathrm{tL}}{\dot{q}}_{\mathrm{t}},$$

(6)

where $q_{\mathrm{L}}={[x_0,y_0,{\theta }_0,{\theta }_{\mathrm{L}}^{\mathrm{T}}]}^{\mathrm{T}}$.

For the deformation of Formula (6), there is:

$${\dot{q}}_{\mathrm{t}}=J_{\mathrm{L}}^{-1}{J}_{\mathrm{tL}}{\dot{q}}_{\mathrm{L}}.$$

(7)

Taking the derivative of Equation (7), there is:

$${\ddot{q}}_{\mathrm{t}}=J_{\mathrm{L}}^{-1}({\dot{J}}_{\mathrm{tL}}{\dot{q}}_{\mathrm{L}}+J_{\mathrm{tL}}{\ddot{q}}_{\mathrm{L}})-J_{\mathrm{L}}^{-1}{\dot{J}}_{\mathrm{L}}{J}_{\mathrm{tL}}{\dot{q}}_{\mathrm{L}}.$$

(8)

When the dual-arm closed-chain hybrid system after collision is formed, the geometric constraints and motion constraints of the system can be described as:

$$\dot{q}={{\rm Z}}^{\mathrm{T}}{\dot{q}}_{\mathrm{L}},$$

(9)

where ${\rm Z}=[E_{6\times 6},U_1^{\mathrm{T}}]$, $U_1=[0_{3\times 3},J_{\mathrm{OR}}^{-1}{J}_{\mathrm{OL}}]$, $J_{\mathrm{OL}}={[J_{\mathrm{L}}^{\mathrm{T}},O_{3\times 1}]}^{\mathrm{T}}$, $J_{\mathrm{OR}}={[J_{\mathrm{R}}^{\mathrm{T}},O_{3\times 1}]}^{\mathrm{T}}$.

For Equation (9), taking the derivative of time, the generalized acceleration is given as:

$$\ddot{q}={\dot{{\rm Z}}}^{\mathrm{T}}{\dot{q}}_{\mathrm{L}}+{{\rm Z}}^{\mathrm{T}}{\ddot{q}}_{\mathrm{L}}.$$

(10)

By analyzing the dynamic of the capture collision process, the following assumptions were obtained: the time of the capture collision process $\mathsf{\Delta }t$ is very short, and the generalized position of the system can be approximated as unchanged; only the generalized velocity and generalized acceleration change. If the above analysis process is described in mathematical language, there are:

$$\left\{\begin{array}{l}\mathsf{\Delta }t=O\left(\vartheta \right)\hfill \\ \ddot{q},{\ddot{q}}_{\mathrm{t}}=O\left(1/\vartheta \right), \vartheta \ll 1\hfill \\ q,\dot{q},q_{\mathrm{t}},{\dot{q}}_{\mathrm{t}}=O\left(1\right)\hfill \end{array}\right.,$$

(11)

In the process of capture collision, obviously, the impact force is much larger than the internal force $F_{\mathrm{I}}$ in order of magnitude. Therefore, the term of $F_{\mathrm{I}}$ can be ignored during the calculation. Then, substituting Equations (4), (8), and (10) into Equation (1), the arrangement is as follows:

$$M_{\mathrm{h}}{\ddot{q}}_{\mathrm{L}}+C_{\mathrm{h}}{\dot{q}}_{\mathrm{L}}={\tau }_{\mathrm{h}},$$

(12)

where ${\tau }_{\mathrm{h}}={\rm Z}\tau$, $C_{\mathrm{h}}=[{\rm Z}(C{{\rm Z}}^{\mathrm{T}}+M{\dot{{\rm Z}}}^{\mathrm{T}})+{\rm Z}{J}^{\mathrm{T}}{(J_{\mathrm{t}}^{\mathrm{T}})}^{+}{M}_{\mathrm{t}}{J}_{\mathrm{tL}}^{-1}({\dot{J}}_{\mathrm{L}}-{\dot{J}}_{\mathrm{tL}}{J}_{\mathrm{tL}}^{-1}{J}_{\mathrm{L}})]$, $M_{\mathrm{h}}=A+{\rm Z}{J}^{\mathrm{T}}{(J_{\mathrm{t}}^{\mathrm{T}})}^{+}{M}_{\mathrm{t}}{J}_{\mathrm{tL}}^{-1}{J}_{\mathrm{L}}$, $A={\rm Z}M{{\rm Z}}^{\mathrm{T}}$.

As shown in Equation (12), the dynamic model of a dual-arm closed-chain hybrid system is an underdriven model, which is not conducive to controller design. Considering that the first two columns of matrix $C_{\mathrm{h}}$ are zero and ${[{\ddot{q}}_{\mathrm{L}}(1),{\ddot{q}}_{\mathrm{L}}(2)]}^{\mathrm{T}}$ can be obtained by using the inverse solution, then Equation (12) can be rewritten into a full-drive dynamic model in the following form:

$$M_{\mathrm{c}}{\ddot{q}}_{\mathrm{c}}+C_{\mathrm{c}}{\dot{q}}_{\mathrm{c}}={\tau }_{\mathrm{c}},$$

(13)

where $M_{\mathrm{c}}=M_{\mathrm{h}22}-M_{\mathrm{h}21}{M}_{\mathrm{h}11}^{-1}{M}_{\mathrm{h}12}$, $C_{\mathrm{c}}=C_{\mathrm{h}22}-M_{\mathrm{h}21}{M}_{\mathrm{h}11}^{-1}{C}_{\mathrm{h}12}$, $q_{\mathrm{c}}={[{\theta }_0,{\theta }_{\mathrm{L}}^{\mathrm{T}}]}^{\mathrm{T}}$, ${\tau }_{\mathrm{c}}={[{\tau }_0,{\tau }_{\mathrm{L}}^{\mathrm{T}}]}^{\mathrm{T}}$. $M_{\mathrm{h}11}\in {\mathrm{R}}^{2\times 2}$, $M_{\mathrm{h}12}\in {\mathrm{R}}^{2\times 4}$, $M_{\mathrm{h}21}\in {\mathrm{R}}^{4\times 2}$, $M_{\mathrm{h}22}\in {\mathrm{R}}^{4\times 4}$ is the submatrix of $M_{\mathrm{h}}$ after block processing; and $C_{\mathrm{h}11}\in {\mathrm{R}}^{2\times 2}$, $C_{\mathrm{h}12}\in {\mathrm{R}}^{2\times 4}$, $C_{\mathrm{h}21}\in {\mathrm{R}}^{4\times 2}$, $C_{\mathrm{h}22}\in {\mathrm{R}}^{4\times 4}$ is the submatrix of $C_{\mathrm{h}}$ after block processing.

2.2. Impedance Modeling

In order to realize the precise position and attitude control of the docking target plug during the on-orbit auxiliary docking operation, the kinematic Jacobian relationship between the $B_t$ point of the plug end and the carrier should be established. By projecting the position of point $B_t$ relative to $r_t$ onto the coordinate system $x_0{O}_0{y}_0$, the relationship can be deformed as:

$$\left\{\begin{array}{c}{x}_{\mathrm{Bt}}=x_{\mathrm{O}1}+L_1\sin {\theta }_1+L_2\sin ({\theta }_1+{\theta }_2)+L_{3\mathrm{t}}\sin {\theta }_{\mathrm{Bt}}\\ y_{\mathrm{Bt}}=y_{\mathrm{O}1}+L_1\cos {\theta }_1+L_2\cos ({\theta }_1+{\theta }_2)+L_{3\mathrm{t}}\cos {\theta }_{\mathrm{Bt}}\end{array}\right.,$$

(14)

where $(x_{\mathrm{O}1},y_{\mathrm{O}1})$ represents the coordinates of point $O_1$ in coordinate system $x_0{O}_0{y}_0$, $L_{3\mathrm{t}}=L_3+L_{\mathrm{t}}$, $L_{\mathrm{t}}=d_L$, ${\theta }_{\mathrm{Bt}}={\theta }_1+{\theta }_2+{\theta }_3$.

According to Equation (14), the kinematic Jacobian relationship can be obtained as follows:

$$\dot{X}=J_{\mathrm{Bt}}{\dot{q}}_{\mathrm{c}},$$

(15)

where $X={[{\theta }_0,x_{\mathrm{Bt}},y_{\mathrm{Bt}},{\theta }_{\mathrm{Bt}}]}^{\mathrm{T}}$, ${\theta }_{\mathrm{Bt}}={\theta }_1+{\theta }_2+{\theta }_3$, $J_{\mathrm{Bt}}\in R^{4\times 4}$ is the augmented motion Jacobian matrix of point $B_t$ relative to the carrier.

For the sake of preventing contact and collision between the space robot and the carrier, it is necessary to accurately control the attitude and position of the carrier. The impedance control puts force and attitude in the same framework of the impedance model. By adjusting the impedance parameters, the dynamic relationship between force and attitude is controlled to maintain an ideal condition between target attitude and environmental contact force. In general, the mathematical model of the front impedance relationship of the docking device can be expressed as a second-order differential equation, and the environmental model can be approximated as a second-order nonlinear function:

$$\begin{array}{l}{F}_{\mathrm{Bt}}=M_{\mathrm{Bt}}({\ddot{X}}_{\mathrm{d}}-\ddot{X})+B_{\mathrm{Bt}}({\dot{X}}_{\mathrm{d}}-\dot{X})+K_{\mathrm{Bt}}(X_{\mathrm{d}}-X)\\ F_{\mathrm{e}}=B_{\mathrm{e}}(\dot{X}-{\dot{X}}_{\mathrm{e}})+K_{\mathrm{e}}(X-X_{\mathrm{e}})\end{array},$$

(16)

where $X_{\mathrm{d}}$, $X_{\mathrm{e}}$ are the expected pose and reference pose, respectively. $M_{\mathrm{Bt}}\in R^{4\times 4}$, $B_{\mathrm{Bt}}\in R^{4\times 4}$, $K_{\mathrm{Bt}}\in R^{4\times 4}$ are the inertia matrix, damping matrix, and stiffness matrix of the mechanical arm, respectively; $B_{\mathrm{e}}\in R^{4\times 4}$, $K_{\mathrm{e}}\in R^{4\times 4}$ are the damping matrix and stiffness matrix of the environment, respectively; and $F_{\mathrm{Bt}}\in R^{4\times 1}$, $F_{\mathrm{e}}\in R^{4\times 1}$ are the plug output force/moment and contact force/moment, respectively.

Based on the analysis, the error between the plug output force/moment $F_{\mathrm{Bt}}$ and the contact force/moment $F_{\mathrm{e}}$ can be calculated as:

$${\tau }_{\mathrm{t}\mathrm{e}}=F_{\mathrm{Bt}}-F_{\mathrm{e}}.$$

(17)

To facilitate the controller design, the system dynamics model based on joint space should be rewritten as the following form based on the inertial space:

$$M_{\mathrm{X}}\ddot{X}+C_{\mathrm{X}}\dot{X}={\tau }_{\mathrm{X}}+{\tau }_{\mathrm{t}\mathrm{e}},$$

(18)

where $M_{\mathrm{X}}=J_{\mathrm{Bt}}^{-\mathrm{T}}{M}_{\mathrm{c}}{J}_{\mathrm{Bt}}^{-1}$, $C_{\mathrm{X}}=J_{\mathrm{Bt}}^{-\mathrm{T}}(C_{\mathrm{c}}-M_{\mathrm{c}}{J}_{\mathrm{Bt}}^{-1}{\dot{J}}_{\mathrm{Bt}})J_{\mathrm{Bt}}^{-1}$, ${\tau }_{\mathrm{X}}=J_{\mathrm{Bt}}^{-\mathrm{T}}{\tau }_{\mathrm{c}}$.

3. Controller Design

First, the system position tracking errors are defined as:

$$e=X-X_{\mathrm{d}},$$

(19)

In order to improve the transient performance of the control system and ensure the control effect of the dual-arm space robot in the auxiliary on-orbit docking task, the tracking error should be constrained in the predetermined range and the controller should be designed under the preset performance framework. The following positive definite decreasing smooth function is chosen as the prescribed performance function:

$$\mu (t)=({\mu }_0-{\mu }_{\infty })e^{-\alpha t}+{\mu }_{\infty },$$

(20)

where ${\mu }_0>{\mu }_{\infty },\alpha >0$ are the default performance parameters, ${\mu }_0$ is the default performance function initial value, and ${\mu }_{\infty }$ is the stable value. $\alpha$ is related to the convergence rate of $\mu (t)$.

Obviously, the following properties can be obtained from Equation (20):

$$\left\{\begin{array}{l}\underset{t\to 0}{\lim }\mu (t)={\mu }_0\&\underset{t\to \infty }{\lim }\mu (t)={\mu }_{\infty }\\ \mu (0)=({\mu }_0-{\mu }_{\infty })e^{-\alpha t}+{\mu }_{\infty }={\mu }_0\end{array}\right.,$$

(21)

Then, based on the prescribed performance function, the system tracking errors are restricted to the prescribed upper and lower boundary with a set of inequalities, which can be written as:

$$-\underset{\_}{\kappa }\mu (t)<e_i(t)<\overline{\kappa }\mu (t),\forall t\ge 0,$$

(22)

where $\kappa >0$ is the overshoot boundary parameter, $i=1,2,3,4$.

Combining Equations (20)–(22), the system tracking errors can be limited to the prescribed boundary with the parameters of ${\mu }_0,{\mu }_{\infty },\alpha ,\kappa$. In terms of prescribed performance controller design, it should be pointed out that the tracking errors and the restriction condition need to be transformed into a nonrestrictive case to facilitate the controller design. Then, a homeomorphic mapping function is used to achieve peer conversion from performance constrained space to unconstrained space. According to the analysis, Equation (22) can be written as:

$$e_i(t)=\mu (t){\nu }_i(t),$$

(23)

where $\nu \left(t\right)$ is the equivalent tracking errors and $-\underset{¯}{\kappa }<{\nu }_i(t)<\overline{\kappa }$, $i=1,2,3,4$.

The following tangent function is chosen for homeomorphic mapping, which can be written as:

$$Th({\nu }_i)={\displaystyle \frac{\overline{\kappa }{e}^{{\nu }_i}-\underset{¯}{\kappa }{e}^{-{\nu }_i}}{e^{{\nu }_i}+e^{-{\nu }_i}.}}$$

(24)

Based on Equation (24), the switching errors $e_{\delta }$ can be calculated as:

$$e_{\delta }={\displaystyle \frac{1}{2}}\ln \left({\displaystyle \frac{\underset{\_}{\kappa }+v}{\overline{\kappa }+v}}\right).$$

(25)

According to Equation (25), it can be calculated with the original system tracking errors that:

$$e^{2e_{\delta }}={\displaystyle \frac{\underset{\_}{\kappa }+v}{\overline{\kappa }+v}}.$$

(26)

Furthermore, by combining Equations (22)–(26), it can be obtained that:

$$-\underset{\_}{\kappa }<{\displaystyle \frac{e^{-2e_{\delta i}}\overline{\kappa }-\underset{\_}{\kappa }}{e^{-2e_{\delta i}}+1}}=\underset{\_}{\nu }\le {\nu }_i\le \overline{\nu }={\displaystyle \frac{e^{2e_{\delta i}}\overline{\kappa }-\underset{\_}{\kappa }}{e^{2e_{\delta i}}+1}}<\overline{\kappa }.$$

(27)

Equation (27) shows that the switching errors $e_{\delta }$ can be limited in the prescribed boundary by designing the controller under the prescribed performance control theory. Then, the sliding mode surface is formed as follows:

$$\left\{\begin{array}{l}S=\Lambda e_{\delta }+{\dot{e}}_{\delta }\\ \dot{S}=\Lambda {\dot{e}}_{\delta }+{\ddot{e}}_{\delta }\end{array}\right.,$$

(28)

where $\Lambda >0$ is the positive constant matrix.

Based on Equations (24) and (25), by taking the first derivative of time with respect to $e_{\delta }$ and substituting the results into Equation (28), then we have:

$$\dot{S}=(\Lambda \xi +\dot{\xi })\left(\dot{e}-{\displaystyle \frac{e\dot{\mu }}{\mu }}\right)+\xi \left[\ddot{e}-{\displaystyle \frac{\dot{e}\dot{\mu }}{\mu }}-{\displaystyle \frac{e\ddot{\mu }}{\mu }}+{\displaystyle \frac{e{\dot{\mu }}^2}{{\mu }^2}}\right],$$

(29)

where $\xi$ is a bounded variable, which satisfies $0<\xi <{\xi }_m$, $\xi$ and $\dot{\xi }$ can be written as follows:

$$\left\{\begin{array}{l}\xi ={\displaystyle \frac{1}{2\mu }}\left({\displaystyle \frac{1}{e_{\delta }+\underset{¯}{\kappa }}}-{\displaystyle \frac{1}{e_{\delta }-\overline{\kappa }}}\right)\\ \dot{\xi }=-{\displaystyle \frac{\dot{\mu }}{2{\mu }^2}}\left({\displaystyle \frac{1}{e_{\delta }+\underset{¯}{\kappa }}}-{\displaystyle \frac{1}{e_{\delta }-\overline{\kappa }}}\right)-{\displaystyle \frac{\dot{e}\mu -e\dot{\mu }}{2{\mu }^3}}\left[{\left({\displaystyle \frac{1}{e_{\delta }+\underset{\_}{\kappa }}}\right)}^2-{\left({\displaystyle \frac{1}{e_{\delta }-\overline{\kappa }}}\right)}^2\right]\end{array}\right.,$$

(30)

For the dynamic model represented by Equation (18), it is assumed that its uncertainty term can be expressed in the following form:

$$\left\{\begin{array}{l}{M}_{\mathrm{X}}={\widehat{M}}_{\mathrm{X}}+\Delta M_{\mathrm{X}}\\ C_{\mathrm{X}}={\widehat{C}}_{\mathrm{X}}+\Delta C_{\mathrm{X}}\end{array}\right.,$$

(31)

where $\Delta M_{\mathrm{X}},\Delta C_{\mathrm{X}}$ are the uncertain part of the dynamic modeling of $M_{\mathrm{X}},C_{\mathrm{X}}$, respectively.

Then, Equation (18) can be rewritten as follows:

$${\widehat{M}}_{\mathrm{X}}\ddot{X}+{\widehat{C}}_{\mathrm{X}}\dot{X}+f_{\mathrm{X}}={\tau }_{\mathrm{X}}+{\tau }_{\mathrm{t}\mathrm{e}},$$

(32)

where $f_{\mathrm{X}}=\Delta M_{\mathrm{X}}\ddot{X}+\Delta C_{\mathrm{X}}\dot{X}$.

Clearly, the above dynamic model satisfies the following properties:

Property 1.

The matrix of ${\dot{M}}_{\mathrm{X}},C_{\mathrm{X}}$ is oblique symmetry; that is, for any $z\in R^{4\times 1}$, there are $(z^{\mathrm{T}}{\dot{M}}_{\mathrm{X}}z)/2-z^{\mathrm{T}}{C}_{\mathrm{X}}z=0$.

Property 2.

The uncertain part of the system represented by Equation (32) is bounded, and its upper bound is $f_{\mathrm{X}}\left(t\right)\le f_{\max }$, $f_{\max }\in R^{4\times 1}$.

Assumption 1.

Aiming at using an RBF neural network to approximate the uncertain dynamic modeling factors, if there is $f_{\mathrm{X}}=W^{\ast \mathrm{T}}\phi (y)+\epsilon$, where $W^{\ast }$ is the matrix of the ideal weight matrix of the network and $\epsilon$ is the approximation error, the approximation error $\epsilon$ is bounded, namely $‖\epsilon ‖\le \overline{\epsilon }$, $\overline{\epsilon }>0$.

For the dual-arm closed-chain hybrid system represented by Equation (32), a controller of the following form is designed:

$$\begin{array}{l}{\tau }_{\mathrm{X}}={\tau }_0+{\tau }_1\\ {\tau }_0={\widehat{M}}_{\mathrm{X}}\left({\ddot{X}}_{\mathrm{d}}+{\displaystyle \frac{\dot{e}\dot{\mu }}{\mu }}+{\displaystyle \frac{e\ddot{\mu }}{\mu }}-{\displaystyle \frac{e{\dot{\mu }}^2}{{\mu }^2}}-{\displaystyle \frac{1}{\xi }}\left[\left(\Lambda \xi +\dot{\xi }\right)\left(\dot{e}-{\displaystyle \frac{e\dot{\mu }}{\mu }}\right)+KS\right]\right)+{\widehat{C}}_{\mathrm{X}}\dot{X}\\ {\tau }_1={\widehat{f}}_{\mathrm{X}}\end{array},$$

(33)

where ${\tau }_0$ is the control torque of the certain part of system modeling and ${\tau }_1$ is the compensate torque of the uncertain part of dynamic system modeling; $K>0$ is the matrix of the control gain coefficient; ${\widehat{f}}_{\mathrm{X}}={\widehat{W}}^{\mathrm{T}}\phi (y)$ is the compensate torque of the RBFNN compensation item; $\widehat{W}$ is its estimated value matrix of $W^{\ast }$; $y={[e^{\mathrm{T}},{\dot{e}}^{\mathrm{T}},X_{\mathrm{d}},{\dot{X}}_{\mathrm{d}},{\ddot{X}}_{\mathrm{d}}]}^{\mathrm{T}}$ is the input of the RBFNN; $\phi (y)$ are the Gaussian base functions and ${\phi }_j(y)={\mathrm{e}}^{-{‖y-c_j‖}^2/(2b_j^2)},(j=1,2,\cdot \cdot \cdot ,p)$; p is the number of neurons in a hidden layer of the network; and $c_j$, $b_j$ are the center value and width of the Gaussian base function.

The neural network update rate is designed using the following form:

$${\dot{\widehat{W}}}_i={\Lambda }_i(S_i{\phi }_i-{\beta }_i{\widehat{W}}_i),$$

(34)

where ${\Lambda }_i>0$ and ${\beta }_i>0$, $\tilde{W}$ are the matrices of error between the ideal weight and the estimated weight, which is defined as $\tilde{W}=W^{\ast }-\widehat{W}$.

Theorem 1.

For a given system dynamics model Equation (18), if Assumption 1 holds, then the control rate shown in Equation (33) and the update rate of the neural network shown in Equation (16) are adopted, the system convergence can be guaranteed.

Proof. 

The Lyapunov function is selected as follows:

$$V={\displaystyle \frac{1}{2}}{S}^{\mathrm{T}}S+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^4{\tilde{W}}_i^{\mathrm{T}}}{\Lambda }_i^{-1}{\tilde{W}}_i,$$

(35)

□

By differentiating Equation (35), we can obtain:

$$\begin{array}{ll}\dot{V}& =S^{\mathrm{T}}\dot{S}+{\displaystyle \sum _{i=1}^4{\tilde{W}}_i^{\mathrm{T}}}{\Lambda }_i^{-1}{\dot{\tilde{W}}}_i\\ & =S^{\mathrm{T}}(-KS+\xi ({\widehat{W}}^{\mathrm{T}}\phi (y)-{\rho }_{\mathrm{X}}))+{\displaystyle \sum _{i=1}^4{\tilde{W}}_i^{\mathrm{T}}}{\Lambda }_i^{-1}{\dot{\tilde{W}}}_i\\ & =S^{\mathrm{T}}(-KS+\xi {\tilde{W}}^{\mathrm{T}}\phi (y))-{\displaystyle \sum _{i=1}^4{\tilde{W}}_i^{\mathrm{T}}}(S_i{\phi }_i-{\beta }_i{\widehat{W}}_i)\\ & =-S^{\mathrm{T}}KS-\xi S^{\mathrm{T}}{\tilde{W}}^{\mathrm{T}}\phi (y)-S^{\mathrm{T}}{\tilde{W}}^{\mathrm{T}}\phi (y)+{\beta }_i{\displaystyle \sum _{i=1}^4{\tilde{W}}_i^{\mathrm{T}}}{\widehat{W}}_i\\ & \le -S^{\mathrm{T}}KS-\xi S^{\mathrm{T}}\epsilon -S^{\mathrm{T}}\epsilon +{\beta }_i{\displaystyle \sum _{i=1}^4{\tilde{W}}_i^{\mathrm{T}}}\widehat{W}\end{array}.$$

(36)

According to the norm inequality properties, Equation (36) can be scaled as the following form:

$$\begin{array}{ll}\dot{V}& \le -S^{\mathrm{T}}KS-{\xi }_{\mathrm{m}}‖S^{\mathrm{T}}\overline{\epsilon }‖-‖S^{\mathrm{T}}\overline{\epsilon }‖-{\displaystyle \sum _{i=1}^4(\frac{{\beta }_i}{2}{\tilde{W}}_i^{\mathrm{T}}{\tilde{W}}_i)}+{\displaystyle \sum _{i=1}^4(\frac{{\beta }_i}{2}{W}_i^{\ast \mathrm{T}}{W}_i^{\ast })}\\ & \le -S^{\mathrm{T}}(K-{\xi }_{\mathrm{m}}-{\displaystyle \frac{1}{2}}E)S-{\displaystyle \frac{{\beta }_i}{2}}{\displaystyle \sum _{i=1}^4({\tilde{W}}_i^{\mathrm{T}}{\tilde{W}}_i)}+({\xi }_{\mathrm{m}}+{\displaystyle \frac{1}{2}}E){\overline{\epsilon }}^{\mathrm{T}}\overline{\epsilon }+{\displaystyle \frac{{\beta }_i}{2}}{\displaystyle \sum _{i=1}^4(W_i^{\ast \mathrm{T}}{W}_i^{\ast })}\\ & \le -\rho V+C\end{array},$$

(37)

where $\rho =\min \left(2(K_i-{\xi }_{\mathrm{m}})-1,\underset{i=1,2,3,4}{\min }({\displaystyle \frac{{\beta }_i}{{\Lambda }_i^{-1}}})\right)$, $C=({\xi }_{\mathrm{m}}+{\displaystyle \frac{1}{2}}){\overline{\epsilon }}^{\mathrm{T}}\overline{\epsilon }+{\displaystyle \frac{{\beta }_i}{2}}{\displaystyle \sum _{i=1}^4(W_i^{\ast \mathrm{T}}{W}_i^{\ast })}$.

Therefore, since ${\Lambda }_i>0$, ${\beta }_i>0$ and $K_i>{\xi }_{\mathrm{m}}+{\displaystyle \frac{1}{2}}$, if and only if $S=0$, $\dot{V}=0$. According to LaSalle’s invariance principle, the closed-loop system is asymptotically stable; that is, when $t\to \infty$, $S\to 0$.

4. Simulation Analysis

The dual-arm space robot shown in Figure 1 was taken as an example for simulation analysis. Its system dynamics parameters are listed as follows:

$$\begin{array}{l}{m}_0=200 \mathrm{kg},I_0=193.75 \mathrm{kg}\cdot {\mathrm{m}}^2,L_0=1.064 \mathrm{m},L_B=2 \mathrm{m};\\ m_i=10 \mathrm{k}\mathrm{g},I_i=5 \mathrm{k}\mathrm{g}\cdot {\mathrm{m}}^2,L_i=2 \mathrm{m},d_i=1 \mathrm{m} (i=1,2,4,5);\\ m_i=2.5 \mathrm{k}\mathrm{g},I_2=2 \mathrm{k}\mathrm{g}\cdot {\mathrm{m}}^2,L_i=0.5 \mathrm{m},d_i=0.25 \mathrm{m} (i=3,6);\\ {\psi }_1=2.791 \mathrm{rad},{\psi }_2=0.349 \mathrm{rad}.\end{array}$$

The initial configuration of the dual-arm space robot system before capture is as follows: $q={[0.3 \mathrm{m}, 0.3 \mathrm{m}, 10°, 120°, -60°, -60°, 60°, 60°, 60°]}^{\mathrm{T}}$. After the capture is completed, the desired position of the hybrid system in the stabilization control phase is set as $q_{\mathrm{d}}={[120°, -60°, -60°, 60°, 60°, 60°]}^{\mathrm{T}}$.

The dynamics parameters of the docking target spacecraft are described as follows:

$$m_t=75 \mathrm{kg},I_t=10 \mathrm{kg}\cdot {\mathrm{m}}^2,d_L=0.5 \mathrm{m},d_R=0.5 \mathrm{m}.$$

The prescribed performance function is chosen as $\mu (t)=\left(0.5-0.01\right)e^{-5t}+0.01$; the upper and lower bounds are set as $\overline{\kappa }={[1,1,1,1]}^T$ and $\underset{\_}{\kappa }={[1,1,1,1]}^T$.

The control gains are $K=\mathrm{diag}\left(\right[15,15,15,15\left]\right)$, $\Lambda =\mathrm{diag}\left(\right[5,5,5,5\left]\right)$, $\beta =[1,1,1,1]$.

The center value and width of the neural network are, respectively, $c=[-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8]$, $b=3$.

The control algorithm proposed in this paper realizes the force/pose control of the end of the two-arm space robot simultaneously through the outer loop impedance module, and the realization of this function depends on the tracking error between the actual and expected state of the end of the robot and the impedance parameters. The output forces depend on the interaction of the state tracking errors and impedance parameters. Among them, if the impedance parameter is too large, it will increase the intensity of the collision and reduce the flexibility of the docking moment. Therefore, the impedance parameters need to be adjusted to an appropriate size, and the precision of the pose control may be slightly decreased. Then, the impedance parameters are: $M_{\mathrm{Bt}}=\mathrm{diag}([10,10,10,10\left]\right)$, $B_{\mathrm{Bt}}=\mathrm{diag}([1500,1500,1500,1500\left]\right)$, $K_{\mathrm{Bt}}=\mathrm{diag}([5800,5800,5800,5800\left]\right)$.

It is assumed that each joint motor starts up after 1.5 s of the collision. The simulation time is set as 25 s, and the step size is set as 0.001 s.

In order to ensure the precise force/pose control of the operation of on-orbit auxiliary docking, the mission is carried out in three steps:

Stage 1 ($0\le t<5$): Stabilization stage. At this stage, the control purpose is to quickly calm the dual-arm closed-chain hybrid system so that it can perform subsequent on-orbit operations. Furthermore, the impedance module of the controller is closed, and the corresponding expected output force and expected trajectory of the end of the dual-arm space robot are designed as follows:

$$\begin{gathered} X_{\mathrm{d}}={[10, 0, 2.9289, 0]}^{\mathrm{T}}, (0\le t<5), \\ {F}_{y\mathrm{d}}=0, (0\le t<5) \end{gathered}$$

where the first element of $X_{\mathrm{d}}$ is the expected attitude of the carrier, and the second to fourth elements are the expected trajectory and attitude of the end point of the target plug. The unit of carrier and plug attitude is °, the unit of plug position is m, and the unit of plug output force is N.

Stage 2 ($5\le t<15$): Closing stage. The control purpose of this stage is to make the axis of the docking plug and the axis of the jack coincide as much as possible, so as to reduce the severity of the contact collision between them. The corresponding terminal expected output force and expected trajectory are designed as follows:

$$\begin{gathered} X_{\mathrm{d}}={[10, 0, 2.929-\left(\right(2.164(\mathrm{t}-5\left)\right)/10), 0]}^{\mathrm{T}}, (5\le t<15), \\ {F}_{y\mathrm{d}}=\left\{\begin{array}{ll}0,& (5\le t<10)\\ 0.008{(t-10)}^4,& (10\le t<15)\end{array}\right.. \end{gathered}$$

Stage 3 ($15\le t<25$): Docking stage. At this stage, the expected output force at the end of the space robot and the expected trajectory are designed as follows:

$$\begin{gathered} X_{\mathrm{d}}={[10, 0, 0.7646-\left(\right(0.4(\mathrm{t}-15\left)\right)/10), 0]}^{\mathrm{T}}, (15\le t\le 25), \\ {F}_{y\mathrm{d}}=80(0.7646-X_{\mathrm{d}}\left(3\right)), (15\le t\le 25). \end{gathered}$$

The simulation results are shown in Figure 2, Figure 3, Figure 4 and Figure 5. The solid red line represents the result of the PPC method proposed by this paper, and the dashed blue line represents the result of the proportional-derivative (PD) method from the literature [25].

According to Figure 2a–d, the proposed control algorithm can realize a smooth on-orbit auxiliary docking in the three stages with good stability. From the local magnification results in Figure 2, it can be seen that the attitude control accuracy is better than $0.5°$ and the position control accuracy is better than ${10}^{-3}$ m. The verification results comparison of the PPC method with the PD method illustrates that the control group method cannot complete the stabilization control of the closed-chain hybrid system in the planned time. From the perspective of a real mission, its docking operation is a failure.

Figure 3 shows the joint trajectory tracking curves of dual-arm space robot.

As can be seen from Figure 4a, the proposed impedance control strategy can realize real-time tracking of the output force, and the control accuracy is better than 0.5 N when the output force reaches more than 30 N. Compared with the verification results of the control group, it is obvious that PD control alone cannot guarantee the accurate end force/pose control of the dual-arm space robot at the same time, and too large an output force may lead to blocking during on-orbit docking, resulting in the failure of the on-orbit mission and even the destruction of hardware equipment and spacecraft. Figure 4b–d shows the control torques of the base and each joint of the dual-arm system.

Figure 5a shows the tracking errors of the manipulator end of the proposed algorithm, and Figure 5b shows the tracking errors of control group algorithm. The comparison results show that the proposed algorithm can effectively limit the tracking error within the preset range and ensure the transient performance of the controlled system, which is conducive to the successful implementation of the auxiliary docking operation of the dual-arm space robot in orbit.

5. Conclusions

In this paper, the impedance control problem of the on-orbit auxiliary docking operation of the dual-arm space robot is studied. In order to achieve accurate control of force/pose at the same time and ensure the transient performance of the dual-arm closed-chain hybrid system, a prescribed performance impedance control strategy based on RBF neural network compensation is proposed. The main conclusions are as follows:

(1)

A second-order linear impedance control model is adopted to control the output force of the space robot during the on-orbit insertion and extraction operation, which achieves accurate control of the output force with an accuracy of 0.5 N.

(2)

Based on RBFNN compensation of the uncertain dynamic modeling, the controller has good control accuracy of the robot; the attitude control accuracy reaches 0.5°, and the position control accuracy reaches 10⁻³ m.

(3)

The controller design, based on a PPC framework, can keep the transient performance of the controlled system within the preset range, which can effectively ensure the success rate of an on-orbit auxiliary docking operation.

(4)

The overall planning for the on-orbit auxiliary docking operation of the dual-arm space robot is carried out, and the control system is designed based on the segmented control strategy, which effectively standardizes the on-orbit operation process and ensures the smooth completion of the on-orbit docking task to a certain extent.

In addition, in the face of the diversity of docking targets, especially some non-cooperative targets, the influence of the gravity gradient effect cannot be ignored. Next, we will carefully estimate the robustness and effectiveness of the control strategy under slower maneuvers. More importantly, this study and future studies will collect enough data to provide a theoretical basis and data support for the stable and smooth implementation of on-orbit missions. When conditions are ripe, further consideration will be given to semi-physical verification on the ground and cross-comparison with real mission data in space.