# Compound Attitude Maneuver and Collision Avoiding Control for a Novel Noncontact Close-Proximity Formation Satellite Architecture

Authors: He Liao; Jinjin Xie; Xiaodong Zhou; Chuang Yao; Zhongxing Tang; Yanbin Zhao; Jirong Qi
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

In order to achieve the attitude maneuver performance of the noncontact close-proximity formation satellite architecture, this paper presents a compound control strategy with variable-parameter sliding mode control and disturbance observer-based feedforward compensation. Firstly, the variable-parameter sliding mode control is proposed to guarantee the attitude maneuver performance of the payload module. Secondly, the collision avoiding control with disturbance observer-based feedforward compensation is proposed to guarantee the synchronization of the two separated modules within the small air clearance constraint of the noncontact Lorentz actuator. Finally, a physical air-floating platform is established to verify the effectiveness of the proposed approach.


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## Body

## 1. Introduction

With the development of the space camera technology, high-resolution imaging under attitude maneuver has been focused on recently [1–3]. However, the imaging results could be declined seriously by the huge flexible solar array and the vibration-generating devices [4–6]. For example, the huge flexible solar array could induce pointing overshoot, the high-frequent vibration could induce the imaging blurring, and the low-frequent vibration could induce the imaging warping [7, 8].The active, passive, and integral active-passive vibration isolations have been proposed to suppress the transmission of the above-mentioned disturbance. Nevertheless, due to their respective limitations, the impact of the above-mentioned disturbance on the control performance cannot fundamentally be rejected [9–12].A noncontact close-proximity formation satellite concept has been proposed to separate the conventional integrative-contact satellite into an ultraquiet payload module and a service module by noncontact Lorentz actuator [13]. A ground test system with a single rotation degree of freedom (1R-DOF) and two translation degrees of freedom (2T-DOF) by using planar air-bearing technology has been proposed to conduct several experiments, indicating that the isolation performance is improved by two orders of magnitude and not limited by sensor characteristics [14]. However, it is insufficient to verify the multidynamics coupling problem of the noncontact close-proximity formation satellite architecture. The three-axis coupling dynamic model of the noncontact close-proximity formation satellite has been conducted by Newton-Euler approach, indicating that the current driving accuracy of the noncontact Lorentz actuator plays the most important role in the whole architecture control [15–17]. The optimization approaches based on multiphysical fields have been proposed to resolve the contradiction between the high power and low ripple, leading to a resolution with ppm within its air clearance [18]. By taking the advantages of the noncontact Lorentz actuator, the advanced control algorithms, such as twistor-based synchronous sliding model control, and model predictive control, have been proposed to guarantee a good performance of the stability control. The development of the noncontact close-proximity formation satellite has been greatly promoted through the above-mentioned researches [19–24]. Also, due to the advantage of the noncontact close-proximity formation satellite, the payload module can be considered the conventional satellite and there are many related researches on controller design [25, 26]. It can be easily found that the recent literatures mainly focus on the theory and simulation analysis to improve the stability control performance of the noncontact close-proximity formation satellite.In view of the application, the noncontact architecture design demonstrates not only the structure separating but also inertia tensor separating, indicating that the attitude maneuver operation can be realized through two-echelon stratification. However, it is more difficult than stability control due to its close-proximity formation characteristics. The whole attitude maneuver control should be designed carefully within small air clearance of the noncontact Lorentz actuator. Otherwise, the separated payload module and service module could be collided. The collision may damage the actuators and the precision instruments of the noncontact close-proximity formation satellite. Therefore, in order to realize the attitude maneuver performance, it is necessary to study the collision avoidance control.A compound control with variable-parameter sliding mode control and disturbance observer-based feedforward compensation has been proposed to realize both the attitude maneuver performance and the synchronization of the two separated modules in this paper. Meanwhile, a physical air-floating platform with 3R-DOF and 2T-DOF is established to verify the effectiveness of the proposed approach. The main contributions of this paper can be outlined as follows:(1)
Compared with the previous researches on the noncontact close-proximity formation satellite in [19–24], this paper extends its advantage into the attitude maneuver operation by using compound control strategy(2)
Compared with the 1R-DOF ground test system established in [14], the physical air-floating platform with full 3R-DOF can effectively verify the overall performance of the noncontact close-proximity formation satellite with multidynamicsThe rest of this article is organized as follows. The dynamic modeling of the noncontact close-proximity formation satellite is given in Section2. And the compound controller design is introduced in Section 3. Section 4 details the physical air-floating platform and the experimental verification results of the proposed approach. Finally, Section 5 exhibits the conclusions of the whole paper.


## 2. Dynamic Modeling

In this section, the noncontact close-proximity formation satellite architecture is introduced, and the dynamic modelling is established by using Newton-Euler approach.


### 2.1. Hierarchical Architecture

The noncontact close-proximity formation satellite is divided the structure into an ultraquiet payload module and a service module by noncontact Lorentz actuator, as shown in Figure1.Figure 1
Hierarchical architecture.The payload module is used to install the ultraquiet devices, such as the star sensor and the coil component of the noncontact Lorentz actuator. Since the huge flexible solar panel and the vibration-generating devices are not equipped within the payload module, it can be regarded as a rigid body. The service module is used to install the control moment gyroscope, thruster, solar panel, and the permanent magnet component of the noncontact Lorentz actuator. Meanwhile, the wireless communication and the power supply is becoming more mature and reliable gradually, suggesting that the wireless technologies can be adopted to avoid the issues due to the cables [27, 28].The noncontact Lorentz actuator is a magnetic levitation actuator which is driven by the current with linear force output. Compared with the control moment gyroscope, the accuracy of the noncontact Lorentz actuator is more precise without unbalanced vibration [29].


### 2.2. Payload Module Dynamics

According to the angular momentum theorem, the attitude dynamics of the payload module has the following expression:(1)Jpω̇p+ωp×Jpωp=Tcp+Tdp,where ωp represents the angular velocity vector of the payload module; Jp represents the inertia tensor of the payload module; Tcp represents the control torque by the noncontact Lorentz actuators; and Tcp represents the sum of the external disturbance torque to the payload module.The attitude kinematics of the payload module is established based on the quaternion method, as shown in the following expression:(2)q̇=q̇0q̇1q̇2q̇3=120−ωpx−ωpy−ωpzωpx0ωpz−ωpyωpy−ωpz0ωpzωpzωpy−ωpx0q0q1q2q3.In matrix form,(3)q̇0=−12qTωp,q̇=12q×+q0I3×3ωp,where q×∈ℝ3×3 is the vector cross-product operator of a skew-symmetric matrix:
(4)q×=0−q3q2q30−q1−q2q10.In this paper, the controller design will be based on the state error. Therefore, defining the desired attitude quaternionqt=qt0qtT=qt0qt1qt2qt3T and the desired attitude angular velocity ωpt=ωptxωptyωptzT. The error attitude quaternion qe=qe0qeT=qe0qe1qe2qe3T and the error angular velocity ωpe=ωpexωpeyωpezT can be solved by the following expression:
(5)qe=qt∗⊗q,ωpe=ωp−ωpt,where qt∗=qt0−qtT represents the conjugate quaternion and the symbol “⊗” represents the multiplication of quaternions.


### 2.3. Service Module Dynamics

The control moment gyroscope, flexible solar panel, and other parts with large vibration are installed in the service module; the attitude dynamics of the service module has the following expression:(6)ḣs+ωs×hs+Ca1q¨a1+Ca2q¨a2=Tcs+Tds+Tps,where hs=Jsωs+hc represents the total angular momentum of the service module; Js represents the inertia tensor of the service module; ωs represents the angular velocity vector of the service module; hc represents the angular momentum of the control moment gyroscopes mounted on the service module; ωs represents the angular velocity vector; Caiandqai (i=1,2) represent the coupling matrix and modes of the solar panels, as shown in Equations (7) and (8); Tcs represents the control torque of the control moment gyroscopes; Tns represents the sum of the external disturbance torque to the service module; and Tps represents the reaction torque of the noncontact Lorentz actuators to the service module, as shown in Equation (9).
(7)q¨a1+2ζΛaq̇a1+Λa2qa1+Ca1Tω̇b=0,(8)q¨a2+2ζΛaq̇a2+Λa2qa2+Ca2Tω̇b=0,where ζ represents the damping coefficient of the solar panels and Λa represents the modal frequency of the diagonal matrix.
(9)Tps=−CspTTcp,where Csp represents the coordinate transformation matrix of the service module relative to the payload module.Defining the angular velocity of the service module relative to the payload module asωsp, thus, the following relationship has the following expression:
(10)ωsp=ωs−CspTωp.According to Equations (1) and (6), the relative attitude dynamics equation of the two modules can be obtained by derivative of the above equation:
(11)ω̇sp=Js‐1Tcs+Tds+Tps−Ca1q¨a1−Ca2q¨a2−Js‐1ωs×Jsωs+hc+ωsp×CspTωp−CspTJp‐1Tcp+Tdp−ωp×Jpωp.


## 2.1. Hierarchical Architecture

The noncontact close-proximity formation satellite is divided the structure into an ultraquiet payload module and a service module by noncontact Lorentz actuator, as shown in Figure1.Figure 1
Hierarchical architecture.The payload module is used to install the ultraquiet devices, such as the star sensor and the coil component of the noncontact Lorentz actuator. Since the huge flexible solar panel and the vibration-generating devices are not equipped within the payload module, it can be regarded as a rigid body. The service module is used to install the control moment gyroscope, thruster, solar panel, and the permanent magnet component of the noncontact Lorentz actuator. Meanwhile, the wireless communication and the power supply is becoming more mature and reliable gradually, suggesting that the wireless technologies can be adopted to avoid the issues due to the cables [27, 28].The noncontact Lorentz actuator is a magnetic levitation actuator which is driven by the current with linear force output. Compared with the control moment gyroscope, the accuracy of the noncontact Lorentz actuator is more precise without unbalanced vibration [29].


## 2.2. Payload Module Dynamics

According to the angular momentum theorem, the attitude dynamics of the payload module has the following expression:(1)Jpω̇p+ωp×Jpωp=Tcp+Tdp,where ωp represents the angular velocity vector of the payload module; Jp represents the inertia tensor of the payload module; Tcp represents the control torque by the noncontact Lorentz actuators; and Tcp represents the sum of the external disturbance torque to the payload module.The attitude kinematics of the payload module is established based on the quaternion method, as shown in the following expression:(2)q̇=q̇0q̇1q̇2q̇3=120−ωpx−ωpy−ωpzωpx0ωpz−ωpyωpy−ωpz0ωpzωpzωpy−ωpx0q0q1q2q3.In matrix form,(3)q̇0=−12qTωp,q̇=12q×+q0I3×3ωp,where q×∈ℝ3×3 is the vector cross-product operator of a skew-symmetric matrix:
(4)q×=0−q3q2q30−q1−q2q10.In this paper, the controller design will be based on the state error. Therefore, defining the desired attitude quaternionqt=qt0qtT=qt0qt1qt2qt3T and the desired attitude angular velocity ωpt=ωptxωptyωptzT. The error attitude quaternion qe=qe0qeT=qe0qe1qe2qe3T and the error angular velocity ωpe=ωpexωpeyωpezT can be solved by the following expression:
(5)qe=qt∗⊗q,ωpe=ωp−ωpt,where qt∗=qt0−qtT represents the conjugate quaternion and the symbol “⊗” represents the multiplication of quaternions.


## 2.3. Service Module Dynamics

The control moment gyroscope, flexible solar panel, and other parts with large vibration are installed in the service module; the attitude dynamics of the service module has the following expression:(6)ḣs+ωs×hs+Ca1q¨a1+Ca2q¨a2=Tcs+Tds+Tps,where hs=Jsωs+hc represents the total angular momentum of the service module; Js represents the inertia tensor of the service module; ωs represents the angular velocity vector of the service module; hc represents the angular momentum of the control moment gyroscopes mounted on the service module; ωs represents the angular velocity vector; Caiandqai (i=1,2) represent the coupling matrix and modes of the solar panels, as shown in Equations (7) and (8); Tcs represents the control torque of the control moment gyroscopes; Tns represents the sum of the external disturbance torque to the service module; and Tps represents the reaction torque of the noncontact Lorentz actuators to the service module, as shown in Equation (9).
(7)q¨a1+2ζΛaq̇a1+Λa2qa1+Ca1Tω̇b=0,(8)q¨a2+2ζΛaq̇a2+Λa2qa2+Ca2Tω̇b=0,where ζ represents the damping coefficient of the solar panels and Λa represents the modal frequency of the diagonal matrix.
(9)Tps=−CspTTcp,where Csp represents the coordinate transformation matrix of the service module relative to the payload module.Defining the angular velocity of the service module relative to the payload module asωsp, thus, the following relationship has the following expression:
(10)ωsp=ωs−CspTωp.According to Equations (1) and (6), the relative attitude dynamics equation of the two modules can be obtained by derivative of the above equation:
(11)ω̇sp=Js‐1Tcs+Tds+Tps−Ca1q¨a1−Ca2q¨a2−Js‐1ωs×Jsωs+hc+ωsp×CspTωp−CspTJp‐1Tcp+Tdp−ωp×Jpωp.


## 3. Compound Controller Design

In this section, the attitude maneuver operation of the payload module with variable-parameter sliding mode control and the collision avoiding of the service module with disturbance observer-based feedforward compensation is proposed to realize both the attitude maneuver performance and the synchronization of the two separated modules.


### 3.1. Block Diagram

The block diagram of the compound attitude maneuver and collision avoiding control for noncontact close-proximity formation satellite control system is shown in Figure2. There are two control loops, including the active control of the payload module and the cooperative control of the service module. The variable-parameter sliding mode control algorithm is adopted to guarantee the active attitude maneuver control for payload module, while the disturbance observer-based feedforward compensation is adopted to guarantee the cooperative synchronization for service module within the small air clearance constraint.Figure 2
Block diagram of noncontact close-proximity formation satellite control system.


### 3.2. Attitude Maneuver Control of Payload Module

The payload module could be affected by external interference. In order to improve the robustness of the system, the article adopts the variable-parameter sliding mode control to design the active attitude maneuver control for payload module. And the sliding mode control is based on the exponential reaching law [28–32].Defining the sliding surface as follows:(12)s=kpqe+ωpe,where kp represents the three-dimensional diagonal matrix and its elements are positive real numbers.According to the exponential approach law, the following equation has the following expression:(13)ṡ=−εs−λsats.Bothε and λ are the three-dimensional diagonal matrixes, whose elements are positive real numbers.Combining the dynamics and kinematics models of the payload module, the attitude maneuver control law of the payload module has the following expression:(14)Tcp=12Jpkpqe×+qe0Iωpe+ωpe×Jpωpe+εs+λsats.In order to realize the attitude maneuver control of the payload module, the idea of the variable-parameter control is adopted, and different control parameterskp and ε are selected for the different attitude angle errors. As the attitude angle increases, the parameters will increase accordingly to achieve the purpose of quickly tracking large-angle attitude errors during maneuvering. On the other hand, when the angle error is small, the parameters will decrease accordingly to achieve high precision control of the payload module.


### 3.3. Collision Avoiding Control of Service Module

According to the function of the noncontact close-proximity formation satellite, the service module should be tracked the payload module within the small air clearance of the noncontact Lorentz actuator to avoid collision. Therefore, the service module needs to track the movement of the payload module synchronously.According to Equation (6), it can be seen that the control torque by noncontact Lorentz actuators act on the payload module produce reaction torques to the service module at the same time. Since the noncontact Lorentz actuator is a magnetic levitation actuator with active current driving, its reaction torque to the service module can be obtained and compensated by the control law through Equation (9).Meanwhile, the parameters such as the flexible accessory modal matrix of the solar panel and the coupling coefficient matrix can be precalibrated; the control law thus can be designed to compensate the coupling torques of the flexible accessory. Therefore, the control law has the following expression:(15)Tcs=−KpΘsp−Kdωsp−Tps+Ca1q¨a1+Ca2q¨a2,where Θsp=φsp,θsp,ψspT represents the attitude angle error angle of the service module relative to the three-axis of the payload module.Substituting Equation (15) into Equation (11), the relative attitude dynamics equation has the following expression:
(16)ω̇sp=Js−1−KpΘsp−Kdωsp+Tds−ωs×Jsωs+hc+ωsp×CspTωp−CspTJp‐1Tcp+Tdp−ωp×Bpωp.Since the unmeasurable terms, complex high-order terms, and other unknown random disturbance moments could affect the control performance, the disturbance observer is designed to guarantee the synchronicity of the service module relative to the payload module within the small air clearance constraint. Therefore, Equation (16) has the following expression:
(17)Θ̇sp=ωsp,ω̇sp=Js−1−KpΘsp−Kdωsp+ωsp×CspTωp−Js−1ωs×Jsωs+hc+Js−1Tds.It can also be written as(18)JsΘ¨sp+GΘsp,Θ̇sp+T=Tds,GΘsp,Θ̇sp=KpΘsp+KdΘ̇sp−JsΘ̇sp×CspTωp,T=ωs×Jsωs+hc.The form of the disturbance observer has the following expression:(19)ż=−LΘsp,Θ̇spz+LΘsp,Θ̇spGΘsp,Θ̇sp+T−pΘsp,Θ̇sp,T^ds=z+pΘsp,Θ̇sp,where z,LΘsp,Θ̇sp,pΘsp,Θ̇sp represents the internal state variable of the disturbance observer and LΘsp,Θ̇sp has the following relationship with pΘsp,Θ̇sp:
(20)LΘsp,Θ̇spJsΘ¨sp=∂pΘsp,Θ̇sp∂Θsp∂pΘsp,Θ̇sp∂Θ̇spΘ̇spΘ¨sp.The internal state variablepΘsp,Θ̇sp of the disturbance observer has the following expression:
(21)pΘsp,Θ̇sp=cφ̇spφ̇sp+θ̇spφ̇sp+θ̇sp+ψ̇sp,where c∈R+ represents the observer gain.From Equations (20) and (21), we can get
(22)LΘsp,Θ̇spJsΘ¨sp=∂pΘsp,Θ̇sp∂Θsp∂pΘsp,Θ̇sp∂Θ̇spΘ̇spΘ¨sp=c100110111Θ¨sp.


## 3.1. Block Diagram

The block diagram of the compound attitude maneuver and collision avoiding control for noncontact close-proximity formation satellite control system is shown in Figure2. There are two control loops, including the active control of the payload module and the cooperative control of the service module. The variable-parameter sliding mode control algorithm is adopted to guarantee the active attitude maneuver control for payload module, while the disturbance observer-based feedforward compensation is adopted to guarantee the cooperative synchronization for service module within the small air clearance constraint.Figure 2
Block diagram of noncontact close-proximity formation satellite control system.


## 3.2. Attitude Maneuver Control of Payload Module

The payload module could be affected by external interference. In order to improve the robustness of the system, the article adopts the variable-parameter sliding mode control to design the active attitude maneuver control for payload module. And the sliding mode control is based on the exponential reaching law [28–32].Defining the sliding surface as follows:(12)s=kpqe+ωpe,where kp represents the three-dimensional diagonal matrix and its elements are positive real numbers.According to the exponential approach law, the following equation has the following expression:(13)ṡ=−εs−λsats.Bothε and λ are the three-dimensional diagonal matrixes, whose elements are positive real numbers.Combining the dynamics and kinematics models of the payload module, the attitude maneuver control law of the payload module has the following expression:(14)Tcp=12Jpkpqe×+qe0Iωpe+ωpe×Jpωpe+εs+λsats.In order to realize the attitude maneuver control of the payload module, the idea of the variable-parameter control is adopted, and different control parameterskp and ε are selected for the different attitude angle errors. As the attitude angle increases, the parameters will increase accordingly to achieve the purpose of quickly tracking large-angle attitude errors during maneuvering. On the other hand, when the angle error is small, the parameters will decrease accordingly to achieve high precision control of the payload module.


## 3.3. Collision Avoiding Control of Service Module

According to the function of the noncontact close-proximity formation satellite, the service module should be tracked the payload module within the small air clearance of the noncontact Lorentz actuator to avoid collision. Therefore, the service module needs to track the movement of the payload module synchronously.According to Equation (6), it can be seen that the control torque by noncontact Lorentz actuators act on the payload module produce reaction torques to the service module at the same time. Since the noncontact Lorentz actuator is a magnetic levitation actuator with active current driving, its reaction torque to the service module can be obtained and compensated by the control law through Equation (9).Meanwhile, the parameters such as the flexible accessory modal matrix of the solar panel and the coupling coefficient matrix can be precalibrated; the control law thus can be designed to compensate the coupling torques of the flexible accessory. Therefore, the control law has the following expression:(15)Tcs=−KpΘsp−Kdωsp−Tps+Ca1q¨a1+Ca2q¨a2,where Θsp=φsp,θsp,ψspT represents the attitude angle error angle of the service module relative to the three-axis of the payload module.Substituting Equation (15) into Equation (11), the relative attitude dynamics equation has the following expression:
(16)ω̇sp=Js−1−KpΘsp−Kdωsp+Tds−ωs×Jsωs+hc+ωsp×CspTωp−CspTJp‐1Tcp+Tdp−ωp×Bpωp.Since the unmeasurable terms, complex high-order terms, and other unknown random disturbance moments could affect the control performance, the disturbance observer is designed to guarantee the synchronicity of the service module relative to the payload module within the small air clearance constraint. Therefore, Equation (16) has the following expression:
(17)Θ̇sp=ωsp,ω̇sp=Js−1−KpΘsp−Kdωsp+ωsp×CspTωp−Js−1ωs×Jsωs+hc+Js−1Tds.It can also be written as(18)JsΘ¨sp+GΘsp,Θ̇sp+T=Tds,GΘsp,Θ̇sp=KpΘsp+KdΘ̇sp−JsΘ̇sp×CspTωp,T=ωs×Jsωs+hc.The form of the disturbance observer has the following expression:(19)ż=−LΘsp,Θ̇spz+LΘsp,Θ̇spGΘsp,Θ̇sp+T−pΘsp,Θ̇sp,T^ds=z+pΘsp,Θ̇sp,where z,LΘsp,Θ̇sp,pΘsp,Θ̇sp represents the internal state variable of the disturbance observer and LΘsp,Θ̇sp has the following relationship with pΘsp,Θ̇sp:
(20)LΘsp,Θ̇spJsΘ¨sp=∂pΘsp,Θ̇sp∂Θsp∂pΘsp,Θ̇sp∂Θ̇spΘ̇spΘ¨sp.The internal state variablepΘsp,Θ̇sp of the disturbance observer has the following expression:
(21)pΘsp,Θ̇sp=cφ̇spφ̇sp+θ̇spφ̇sp+θ̇sp+ψ̇sp,where c∈R+ represents the observer gain.From Equations (20) and (21), we can get
(22)LΘsp,Θ̇spJsΘ¨sp=∂pΘsp,Θ̇sp∂Θsp∂pΘsp,Θ̇sp∂Θ̇spΘ̇spΘ¨sp=c100110111Θ¨sp.


## 4. Experimental Verification

In this section, the experimental verification by using the physical air-floating platform is introduced as first. Meanwhile, a PD controller is conducted to demonstrate that the collision avoiding control has to be considered while attitude maneuver operating. Finally, the feasibility and effectiveness of the proposed compound control are successfully verified to guarantee both attitude maneuver performance and collision avoiding performance for noncontact close-proximity formation satellite architecture.


### 4.1. Experimental System

In order to verify the proposed compound attitude maneuver and collision avoiding control of the noncontact close-proximity formation satellite, the schematic diagram of the physical air-floating platform with 3R-DOF and 2T-DOF is shown in Figure3. The system is composed of two air-floating platforms, in which the left air-floating platform is used to simulate the service module, the right air-floating platform is used to simulate the payload module, and eight noncontact Lorentz actuators are installed between the two modules.Figure 3
Schematic diagram of test system.The payload module is equipped with the batteries, DC-DC, industrial computer, fiber optic gyros, camera, etc., and the service module is equipped with the batteries, DC-DC, industrial control computer, eddy current sensors, magnetic suspension control moment gyroscopes, and flexible solar panel.The control system is adopted as the MATLAB/Simulink XPC system. The wide-range angle measurement system and fiber optic gyros are adopted to obtain the current attitude information of the payload module. The relative attitude and position between the service module and payload module are measured and calculated by eight eddy current sensors fixed with the noncontact Lorentz actuators. The attitude information of the experimental system is sent to the industrial computer through RS422 communication. The industrial computer is used to calculate the command input through the attitude control algorithm according to the difference between the current attitude information and the expected attitude and to drive the noncontact Lorentz actuators and control moment gyroscopes to output the command input. The under-stage computer and the on-stage industrial computer are used to upload the control program and to download the telemetry information through the wireless network.According to the above-mentioned design, the physical air-floating platform of the noncontact close-proximity formation satellite is shown in Figure4, in which the related parameters are shown in Table 1. Combining with the placement position in the laboratory, the reference coordinate system is established in the east-north-sky direction. Taking the center of mass of the service module as the coordinate origin, the body coordinate system OsXsYsZs of the service module is established along the three directions of the inertia axis. In the same way, the payload module body coordinate system OpXpYpZp of the payload module is established along the three directions of the inertia axis with the center of mass as the coordinate origin.Figure 4
Physical system of the noncontact close-proximity formation satellite.Table 1
Related parameters of physical system platform.
ParameterValueCapacity of air-floating platform1000-2000 kgRotation range of air-floating platform±25° (X-axis)Centroid fine adjustment error of air-floating platform≤0.2 kgmCentroid coarse adjustment error of air-floating platform≤5 gcmSystem total interference torque of air-floating platform≤0.1 NmDynamic range of wide-range angle measurement±30°Accuracy of wide-range angle measurement1″ (3σ)Angular momentum of control moment gyroscope≤210 NmsOutput torque of control moment gyroscope≤55 NmFrame angular velocity stability of control moment gyroscope3.9×10−3°/sOutput force of noncontact Lorentz actuator≤15 NAir-gap constraint of noncontact Lorentz actuator±0.9°Due to the gravity constraint, the physical system of the noncontact close-proximity formation satellite is carried out alongX-axis to verify the attitude maneuver control of the payload module and cooperative control of the service module. Setting the initial attitude angle of the two modules as -15°, the desired attitude angle as +15°, and attitude maneuver control as 30°/10s in a sinusoidal manner; therefore, the maneuver angle, angular velocity, and angular acceleration planning command curve can be obtained as follows:
(23)φ=3t−15πsinπ5t,ωx=3−3cosπ5t,t∈t0,t0+10,ω̇x=35πsinπ5t.


### 4.2. Test Results with PD Controller

In order to validate the feasibility of the proposed compound controller, the PD controller is carried out firstly in this paper. The control results are shown in Figures5–8.Figure 5
Attitude angle of payload module with PD controller.Figure 6
Angular velocity of payload module with PD controller.Figure 7
Relative attitude angle between two modules with PD controller.Figure 8
Relative angular velocity between two modules with PD controller.Figures5 and 6 show the attitude angle and angular velocity curves of the payload module while the PD controller is adopted. It can be seen that the attitude angle and angular velocity curves are deformed and the whole attitude maneuver process is not completed.Figures7 and 8 show the attitude angle and angular velocity curves of the service module relative to the payload module. It can be seen that the relative attitude angle of the two modules has exceeded the air clearance constraint of the noncontact Lorentz actuator and the relative attitude angular velocity curve of the two modules is greater than 0.5°/s, indicating that the service module could not track the payload module well, and the two modules is colliding in the attitude maneuver process.


### 4.3. Test Results with Compound Controller

The control results of the proposed compound controller are shown in Figures9–12. Figures 9 and 10 show the attitude angle and angular velocity curves of the payload module while the compound attitude maneuver and collision avoiding control proposed in this paper are adopted. It can be seen that the X-axis can realize the maneuver control with 30°/10 s. Meanwhile, the angular runout is less than 0.15°, and the angular velocity runout is less than 0.1°/s in the attitude maneuver process.Figure 9
Attitude angle of payload module with compound controller.Figure 10
Angular velocity of payload module with compound controller.Figure 11
Relative attitude between two modules with compound controller.Figure 12
Relative angular velocity between two modules with compound controller.As shown in Figures11 and 12, the relative attitude angle between the two modules is less than 0.3°, which is in the range of the small air clearance constraint of the noncontact Lorentz actuator. Meanwhile, the relative attitude angular velocity is less than 0.25°/s. Obviously, the cooperative control of the service module is convergent, suggesting that the service module tracked the payload module very well in the attitude maneuver process without collision.The above-mentioned experimental results clearly demonstrate that the compound controller with the variable-parameter sliding model control for payload module and disturbance observer-based feedforward compensation for service module can well realize attitude maneuver without collision within the small air clearance constraint of the noncontact Lorentz actuator. The comparison of the control performance of the two methods is shown in Table2.Table 2
Controller performance comparison.
IndexesPD methodProposed methodAttitude maneuverabilityFailed30°/10sCollision avoidance abilityFailedSuccessConvergence rateAttitude angle of payload module25 s10sAngular velocity of payload module25 s10sRelative attitude≥35 s30sRelative angular velocity≥35 s30sRunout of the other two axesAttitude angle of payload module≥0.7°≤0.15°Angular velocity of payload module≥1°≤0.1°/sRelative attitude≥0.5°≤0.4°Relative angular velocity≥0.5°/s≤0.35°/s


## 4.1. Experimental System

In order to verify the proposed compound attitude maneuver and collision avoiding control of the noncontact close-proximity formation satellite, the schematic diagram of the physical air-floating platform with 3R-DOF and 2T-DOF is shown in Figure3. The system is composed of two air-floating platforms, in which the left air-floating platform is used to simulate the service module, the right air-floating platform is used to simulate the payload module, and eight noncontact Lorentz actuators are installed between the two modules.Figure 3
Schematic diagram of test system.The payload module is equipped with the batteries, DC-DC, industrial computer, fiber optic gyros, camera, etc., and the service module is equipped with the batteries, DC-DC, industrial control computer, eddy current sensors, magnetic suspension control moment gyroscopes, and flexible solar panel.The control system is adopted as the MATLAB/Simulink XPC system. The wide-range angle measurement system and fiber optic gyros are adopted to obtain the current attitude information of the payload module. The relative attitude and position between the service module and payload module are measured and calculated by eight eddy current sensors fixed with the noncontact Lorentz actuators. The attitude information of the experimental system is sent to the industrial computer through RS422 communication. The industrial computer is used to calculate the command input through the attitude control algorithm according to the difference between the current attitude information and the expected attitude and to drive the noncontact Lorentz actuators and control moment gyroscopes to output the command input. The under-stage computer and the on-stage industrial computer are used to upload the control program and to download the telemetry information through the wireless network.According to the above-mentioned design, the physical air-floating platform of the noncontact close-proximity formation satellite is shown in Figure4, in which the related parameters are shown in Table 1. Combining with the placement position in the laboratory, the reference coordinate system is established in the east-north-sky direction. Taking the center of mass of the service module as the coordinate origin, the body coordinate system OsXsYsZs of the service module is established along the three directions of the inertia axis. In the same way, the payload module body coordinate system OpXpYpZp of the payload module is established along the three directions of the inertia axis with the center of mass as the coordinate origin.Figure 4
Physical system of the noncontact close-proximity formation satellite.Table 1
Related parameters of physical system platform.
ParameterValueCapacity of air-floating platform1000-2000 kgRotation range of air-floating platform±25° (X-axis)Centroid fine adjustment error of air-floating platform≤0.2 kgmCentroid coarse adjustment error of air-floating platform≤5 gcmSystem total interference torque of air-floating platform≤0.1 NmDynamic range of wide-range angle measurement±30°Accuracy of wide-range angle measurement1″ (3σ)Angular momentum of control moment gyroscope≤210 NmsOutput torque of control moment gyroscope≤55 NmFrame angular velocity stability of control moment gyroscope3.9×10−3°/sOutput force of noncontact Lorentz actuator≤15 NAir-gap constraint of noncontact Lorentz actuator±0.9°Due to the gravity constraint, the physical system of the noncontact close-proximity formation satellite is carried out alongX-axis to verify the attitude maneuver control of the payload module and cooperative control of the service module. Setting the initial attitude angle of the two modules as -15°, the desired attitude angle as +15°, and attitude maneuver control as 30°/10s in a sinusoidal manner; therefore, the maneuver angle, angular velocity, and angular acceleration planning command curve can be obtained as follows:
(23)φ=3t−15πsinπ5t,ωx=3−3cosπ5t,t∈t0,t0+10,ω̇x=35πsinπ5t.


## 4.2. Test Results with PD Controller

In order to validate the feasibility of the proposed compound controller, the PD controller is carried out firstly in this paper. The control results are shown in Figures5–8.Figure 5
Attitude angle of payload module with PD controller.Figure 6
Angular velocity of payload module with PD controller.Figure 7
Relative attitude angle between two modules with PD controller.Figure 8
Relative angular velocity between two modules with PD controller.Figures5 and 6 show the attitude angle and angular velocity curves of the payload module while the PD controller is adopted. It can be seen that the attitude angle and angular velocity curves are deformed and the whole attitude maneuver process is not completed.Figures7 and 8 show the attitude angle and angular velocity curves of the service module relative to the payload module. It can be seen that the relative attitude angle of the two modules has exceeded the air clearance constraint of the noncontact Lorentz actuator and the relative attitude angular velocity curve of the two modules is greater than 0.5°/s, indicating that the service module could not track the payload module well, and the two modules is colliding in the attitude maneuver process.


## 4.3. Test Results with Compound Controller

The control results of the proposed compound controller are shown in Figures9–12. Figures 9 and 10 show the attitude angle and angular velocity curves of the payload module while the compound attitude maneuver and collision avoiding control proposed in this paper are adopted. It can be seen that the X-axis can realize the maneuver control with 30°/10 s. Meanwhile, the angular runout is less than 0.15°, and the angular velocity runout is less than 0.1°/s in the attitude maneuver process.Figure 9
Attitude angle of payload module with compound controller.Figure 10
Angular velocity of payload module with compound controller.Figure 11
Relative attitude between two modules with compound controller.Figure 12
Relative angular velocity between two modules with compound controller.As shown in Figures11 and 12, the relative attitude angle between the two modules is less than 0.3°, which is in the range of the small air clearance constraint of the noncontact Lorentz actuator. Meanwhile, the relative attitude angular velocity is less than 0.25°/s. Obviously, the cooperative control of the service module is convergent, suggesting that the service module tracked the payload module very well in the attitude maneuver process without collision.The above-mentioned experimental results clearly demonstrate that the compound controller with the variable-parameter sliding model control for payload module and disturbance observer-based feedforward compensation for service module can well realize attitude maneuver without collision within the small air clearance constraint of the noncontact Lorentz actuator. The comparison of the control performance of the two methods is shown in Table2.Table 2
Controller performance comparison.
IndexesPD methodProposed methodAttitude maneuverabilityFailed30°/10sCollision avoidance abilityFailedSuccessConvergence rateAttitude angle of payload module25 s10sAngular velocity of payload module25 s10sRelative attitude≥35 s30sRelative angular velocity≥35 s30sRunout of the other two axesAttitude angle of payload module≥0.7°≤0.15°Angular velocity of payload module≥1°≤0.1°/sRelative attitude≥0.5°≤0.4°Relative angular velocity≥0.5°/s≤0.35°/s


## 5. Conclusion

The noncontact close-proximity formation satellite has been successfully demonstrated for stability control. This paper extends its advantage into the attitude maneuver operation by using a compound control strategy with variable-parameter sliding mode control and disturbance observer-based feedforward compensation. The experimental verification results with the established physical air-floating platform show that the PD controller cannot be used to deform the attitude angle and angular velocity and the collision of the two modules would be occurred. The proposed compound control not only can be used to guarantee the attitude maneuver performance of the payload module but also can be used to guarantee the synchronization of the two separated modules within the small air clearance constraint of the noncontact Lorentz actuator.



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*Source: 2606233-2022-01-31.xml*