# A Novel Dual-Loop Disturbance Observer-Based Attitude Control for a Space-Based Observation Microsatellite

Authors: Jian Sun; Jian Yu; Haijun Rong
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

Aiming at strong coupling and nonlinear dynamic equations for a space-based observation satellite, a sliding mode controller with feed-forward compensation is proposed in this paper. The theorem of moment of momentum is applied to formulate the exact nonlinear dynamic equations of a multibody satellite. On this basis, sliding-mode control-based, dual-loop forward-feed compensation control is used to control the attitude of the space-based observation microsatellite. By comparing with the conventional control method, simulation results demonstrate that the proposed control method has superior performance in terms of suppression from external disturbances and vibration. Better dynamic and static performance indices than the conventional control method are achieved.


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## Body

## 1. Introduction

Space observation systems consist of two parts, that is, the space- and ground-based space observation systems. The latter is subject to atmospheric propagation jitter, astronomical refraction, ionosphere scintillation, and limits of national boundaries. Space-based observation techniques can effectively solve such problems and have the advantages of flexible maneuverability, wide monitoring ranges, long-distance observation, and so forth. Space-based observation is an important direction of future space target detection.However, space-based observation satellites contain rotating rigid parts and flexible appendages. During the observation process, the rotation of rigid parts, elastic vibration of flexible appendages, and motion of satellite attitude influence each other, comprising a strongly coupling nonlinear system. Highly accurate and highly stable attitude control is difficult to realize, especially for micro-satellites used in space target observation, because of the small mass and moment inertias of micro-satellites, as well as the limited quantity and capacity of actuators. Therefore, some special control methods should be applied to the attitude control system of space-based observation micro-satellites.Numerous efforts have been exerted to achieve high-accuracy satellite attitude control. Cheng et al. [1] have used the feedback linearization method to eliminate the coupling terms between the attitude angular velocity and reaction wheel angular momentum, as well as reduce the influences of the vibration of panels and turning of rigid parts on the satellite attitude. Li et al. [2] have proposed a method combining variable structure control with neural networks based on the Lyapunov stability theory, which can ensure that the attitude angular velocity exponentially converges to the given limited range in the case of uncertain disturbance. Chen et al. [3] has proposed an adaptive mixed H2/H∞ attitude control for nonlinear spacecraft systems with unknown or uncertain inertia matrices and external disturbances. Experimental simulation results based on the Republic of China Satellite-1 spacecraft system are presented to demonstrate the effectiveness of the proposed design methods. Azadi et al. [4] have applied an adaptive-robust control scheme to control the three-axes maneuver of a flexible satellite. The controller is robust against disturbances and uncertainties, and adapted to the bounds of the unknown parameters of the system. Jan and Chiou [5] have applied sliding mode control to perform a rigid spacecraft large-angle maneuver in minimal time.Modeling of nonlinear dynamic factors, parameter changes during movement, and external disturbances can be called disturbance terms. Disturbance observer (DOB) can suppress such disturbances. The design of observers has been investigated exhaustively in the theoretical framework and also in many related applications [6–10]. De La Sen and Luo  [7] designs linear observers for a class of linear hybrid systems. Such systems are composed of continuous-time and digital substates and  possess. Santhakumar [9] investigates the performance of a robust tracking control on the basis of proportional-derivative observer-based backstepping control applied on a three degrees of freedom underwater spatial manipulator.  Yamashita et al. [6] have applied the disturbance observation-based PID control method to the Japanese satellite HALCA, whose mission is very long baseline interferometry observations in orbit.However, all the above mentioned methods involve the linearization process or some simplification of the satellite dynamic model, which to some extent affects the final precision. In this paper, the nonlinear dynamic equations of multi-body satellites with flexible appendages and rigid turntables were explored based on the theorem of moment of momentum. On this basis, satellite attitude control with variable structure based on dual-loop feed-forward compensation of disturbance estimator was proposed. The mathematical simulation results showed that the proposed method is far superior to the traditional method of attitude control. The Lyapunov analysis and its treatment on stability and robustness of the proposed scheme under parameter uncertainties are not explicitly dealt with, and is left in this work for the scope of future work. The dynamic model of a space-based observation satellite derived in this paper has a high reference value for the design of attitude control systems of space-based observation satellites.


## 2. Dynamic Modeling of a Multi-Body Satellite

### 2.1. Kinematic Equation of a Space-Based Observation Satellite

The angular velocity of a camera coordinate system relevant to the satellite body frame is denoted asωB1B so that its projection under a camera coordinate system is written as
(2.1)[ωB1B]B1=[ωx1ωy1ωz1]T.
As shown in Figure 1, the motion of the turntable of the camera along the z-axes is constrained relative to the satellite body. Therefore, the rotation angle turntable of the camera along the x2- and y2-axes in the camera coordination system is set as α and β. The corresponding angular velocities are α˙ and β˙, respectively.Figure 1
Simplified structure of a space-based observation satellite.From Euler’s theorem, the rotation of a rigid body around a fixed point can be decomposed into a number of limited rotations around such point. As shown in Figure2, the coordinate transformation matrix  TB1B  from the satellite body coordinate system to the camera coordinate system with a rotation sequence of 2-3-1 is as follows:
(2.2)TB1B(α,β,0)=Tx(α)Tz(0)Ty(β)=[cβ0-sβsαsβcαsαcβcαsβ-sαcαcβ].
The angular velocity of the camera ωB1B is connected to the transformation matrix TB1B by the following relationship:
(2.3)ΩB1B=-T·B1B(α,β,0)TB1BT(α,β,0),
where ΩB1B is the antisymmetric matrix of  ωB1B.Figure 2
Transformation between the satellite body frame and camera frame.Based on the above equation, the following matrix can be obtained:(2.4)[ωx1ωy1ωz1]=[α˙cα·β˙-sα·β˙].


### 2.2. Dynamic Equations of a Space-Based Observation Satellite Attitude

As shown in Figure3, a space-based observation satellite can be simplified into three parts: camera turntable, satellite body, and solar panels. The camera turntable and satellite body can be approximated as rigid bodies, and the solar panels as flexible appendages. The following subsections describe the attitude dynamic modeling of a space-based observation satellite.Figure 3
A space-based observation satellite.


#### 2.2.1. Attitude Dynamic Modeling of a Space-Based Observation Satellite

A space-based observation satellite can be simplified into a camera turntable and a satellite body when not considering the reaction wheel and flexible appendages. In this case, the attitude dynamic equation can be derived from the theorem of moment of momentum as follows:(2.5)DILBBI=MB,
where LBBI is the total moment of momentum, and MB is the total outside torque the satellite experiences relative to the centroid B.The total moment of momentum of satellite consists of two parts, which can be written as(2.6)LBBI=LBB1I+LBB2I,
where LBB1I,LBB2I can be expressed as
(2.7)LBB1I=LB1B1I+mB1rB1B×DIrB1B=IB1B1ωB1I+mB1RB1BDIrB1B,LBB2I=IB2B2ωB2I+mB2RB2BDIrB2B,
where RB1B  represents the antisymmetric matrix of rB1B.Equations (2.5)–(2.7) can yield
(2.8)DI(IB1B1ωB1I+mB1RB1BDIrB1B+IB2B2ωB2I+mB2RB2BDIrB2B)=MB.Given thatmB1rB1B+mB2rB2B=0, (2.8) can yield
(2.9)DI(IB1B1ωB1I)+DI(IB2B2ωB2I)+DI(mB2RB2BDIrB2B1)=MB.
The transforms DI(IB1B1ωB1I), DI(IB2B2ωB2I), and DI(mB2RB2BDIrB2B1) are discussed below.ForDI(IB1B1ωB1I), based on the equation DI(s)=DB1(s)+ωB1I×s, we can obtain
(2.10)DI(IB1B1ωB1I)=DB1(IB1B1ωB1I)+ωB1I×IB1B1ωB1I=IB1B1DB1(ωB1I)+ΩB1IIB1B1ωB1I,
where ΩB1I is the antisymmetric matrix of ωB1I.Given thatωB1I=ωB1B+ωBI, substitution into (2.10) yields
(2.11)DI(IB1B1ωB1I)=IB1B1DB1ωB1B+IB1B1DBωBI+(ΩB1BIB1B1+ΩBIIB1B1-IB1B1ΩB1B)ωBI+(ΩB1B+ΩBI)IB1B1ωB1B.ForDI(IB2B2ωB2I),
(2.12)DI(IB2B2ωB1I)=IB2B2DB2ωB2B+IB2B2DBωBI+(ΩB2BIB2B2+ΩBIIB2B2-IB2B2ΩB2B)ωBI+(ΩB2B+ΩBI)IB2B2ωB2B.The following conversion relations exist between the angular velocities:(2.13)ωB2I=ωB2B+ωBI,ωB2B=0,
where ωB2B=0 because B2 is fixed in the satellite body coordinate system.Substituting (2.13) into (2.12) yields
(2.14)DI(IB2B2ωB2I)=IB2B2DBωBI+ΩBIIB2B2ωBI.ForDI(mB2RB2BDIrB2B1),
(2.15)DI(mB2RB2BDIrB2B1)=DI(mB2rB2B×DIrB2B1)=mB2DIrB2B×DIrB2B1+mB2RB2BDIDIrB2B1,
where
(2.16)DIrB2B×DIrB2B=0,DIrB2B1=ΩBIrB2B1+DBrB2B1,DBrB2B1=0 because the position vector rB2B1 remains unchanged in the body coordinate system.Substituting (2.16) into (2.15) yields
(2.17)DI(mB2RB2BDIrB1B)=mB2RB2BDI(ΩBIrB2B1).Based onDI(s)=DB1(s)+ωB1I×s, (2.17) becomes
(2.18)mB2RB2BDI(ΩBIrB2B1)=mB2RB2BΩBIΩBIrB2B1+mB2RB2BDB(ΩBIrB2B1).Given that(2.19)DB(ΩBIrB2B1)=DB(ωBI×rB2B1)=-DB(rB2B1×ωBI)=-RB2B1DBωBI,
then
(2.20)DI(mB2RB2BDIrB1B)=mB2RB2BΩBIΩBIrB2B1-mB2RB2BRB2B1DBωBI.
Based on (2.9), (2.11), (2.14), and (2.20), the attitude dynamic equation of a space-based observation satellite is as follows:
(2.21)IB1B1DB1ωB1B+IB1B1DBωBI+(ΩB1BIB1B1+ΩBIIB1B1-IB1B1ΩB1B)ωBI+(ΩB1B+ΩBI)IB1B1ωB1B+IB2B2DBωBI+ΩBIIB2B2ωBI+mB2RB2BΩBIΩBIrB2B1-mB2RB2BRB2B1DBωBI=MB,
where
(2.22)[IB1B1]B=[TBB1][IB1B1]B1[TBB1]T,[ΩB1B]B=[TBB1][ΩB1B]B1[TBB1]T,[ωB1B]B=[TBB1][ωB1B]B1.
Considering the orthogonality of [TBB1], the dynamic equation of attitude in the body coordinate system is as follows:
(2.23)[TBB1][IB1B1]B1[DB1ωB1B]B1+[TBB1][IB1B1]B1[TBB1]TDBωBI+([TBB1][ΩB1B]B1[IB1B1]B1[TBB1]T+ΩBI[TBB1][IB1B1]B1[TBB1]T-[TBB1][IB1B1]B1[ΩB1B]B1[TBB1]TωBI)+([TBB1][ΩB1B]B1[TBB1]T+[ΩBI]B)[TBB1][IB1B1]B1[ωB1B]B1+IB2B2DBωBI+ΩBIIB2B2ωBI+mB2RB2BΩBIΩBIrB2B1-mB2RB2BRB2B1DBωBI=MB.
Assuming the disturbance torque introduced by the camera turntable is Mrotate, then
(2.24)Mrotate=[TBB1][IB1B1]B1[DB1ωB1B]B1+[TBB1][IB1B1]B1[TBB1]TDBωBI+([TBB1][ΩB1B]B1[IB1B1]B1[TBB1]T+ΩBI[TBB1][IB1B1]B1[TBB1]T-[TBB1][IB1B1]B1[ΩB1B]B1[TBB1]T)ωBI+([TBB1][ΩB1B]B1[TBB1]T+[ΩBI]B)[TBB1][IB1B1]B1[ωB1B]B1.The satellite attitude dynamic equation can then be simplified as(2.25)IB2B2DBωBI+ΩBIIB2B2ωBI+Mrotate=MB.


#### 2.2.2. Dynamic Model of a Space-Based Observation Satellite with Reaction Wheels

The reaction wheel, as the actuator of a satellite, is used to adjust the satellite attitude. When equipped with reaction wheels, the total moment of momentum of a satellite becomes(2.26)DILBBI=DI(LBB1I+LBB2I+LBWB)=DI(LBB1I+LBB2I)+DBLBWB+ΩBILBWB.
From (2.25) and (2.26), the attitude dynamic equation of a space-based observation satellite with reaction wheels is as follows:
(2.27)IB2B2DBωBI+ΩBIIB2B2ωBI+ΩBILBWB+Mrotate=MB+Mf,
where Mf=-DBLBWB is the control torque produced by a reaction wheel.


#### 2.2.3. Dynamic Model of a Space-Based Observation Satellite with Flexible Appendages

The motion equation of solar panels is [6]
(2.28)q¨+2ξq˙+Λ2q+CTω˙=0,
where q is the flexible modal coordinates, CT is the coupling matrix between the solar panel and satellite body,  ξ is the structural damping of the solar panel, and  Λ  is the vibration frequency of the solar panel.Then, the attitude dynamic model of the satellite with a reaction wheel and flexible appendages is(2.29)IB2B2DBωBI+ΩBIIB2B2ωBI+ΩBILBWB+Cq¨+Mrotate=MB+Mf.


## 2.1. Kinematic Equation of a Space-Based Observation Satellite

The angular velocity of a camera coordinate system relevant to the satellite body frame is denoted asωB1B so that its projection under a camera coordinate system is written as
(2.1)[ωB1B]B1=[ωx1ωy1ωz1]T.
As shown in Figure 1, the motion of the turntable of the camera along the z-axes is constrained relative to the satellite body. Therefore, the rotation angle turntable of the camera along the x2- and y2-axes in the camera coordination system is set as α and β. The corresponding angular velocities are α˙ and β˙, respectively.Figure 1
Simplified structure of a space-based observation satellite.From Euler’s theorem, the rotation of a rigid body around a fixed point can be decomposed into a number of limited rotations around such point. As shown in Figure2, the coordinate transformation matrix  TB1B  from the satellite body coordinate system to the camera coordinate system with a rotation sequence of 2-3-1 is as follows:
(2.2)TB1B(α,β,0)=Tx(α)Tz(0)Ty(β)=[cβ0-sβsαsβcαsαcβcαsβ-sαcαcβ].
The angular velocity of the camera ωB1B is connected to the transformation matrix TB1B by the following relationship:
(2.3)ΩB1B=-T·B1B(α,β,0)TB1BT(α,β,0),
where ΩB1B is the antisymmetric matrix of  ωB1B.Figure 2
Transformation between the satellite body frame and camera frame.Based on the above equation, the following matrix can be obtained:(2.4)[ωx1ωy1ωz1]=[α˙cα·β˙-sα·β˙].


## 2.2. Dynamic Equations of a Space-Based Observation Satellite Attitude

As shown in Figure3, a space-based observation satellite can be simplified into three parts: camera turntable, satellite body, and solar panels. The camera turntable and satellite body can be approximated as rigid bodies, and the solar panels as flexible appendages. The following subsections describe the attitude dynamic modeling of a space-based observation satellite.Figure 3
A space-based observation satellite.


### 2.2.1. Attitude Dynamic Modeling of a Space-Based Observation Satellite

A space-based observation satellite can be simplified into a camera turntable and a satellite body when not considering the reaction wheel and flexible appendages. In this case, the attitude dynamic equation can be derived from the theorem of moment of momentum as follows:(2.5)DILBBI=MB,
where LBBI is the total moment of momentum, and MB is the total outside torque the satellite experiences relative to the centroid B.The total moment of momentum of satellite consists of two parts, which can be written as(2.6)LBBI=LBB1I+LBB2I,
where LBB1I,LBB2I can be expressed as
(2.7)LBB1I=LB1B1I+mB1rB1B×DIrB1B=IB1B1ωB1I+mB1RB1BDIrB1B,LBB2I=IB2B2ωB2I+mB2RB2BDIrB2B,
where RB1B  represents the antisymmetric matrix of rB1B.Equations (2.5)–(2.7) can yield
(2.8)DI(IB1B1ωB1I+mB1RB1BDIrB1B+IB2B2ωB2I+mB2RB2BDIrB2B)=MB.Given thatmB1rB1B+mB2rB2B=0, (2.8) can yield
(2.9)DI(IB1B1ωB1I)+DI(IB2B2ωB2I)+DI(mB2RB2BDIrB2B1)=MB.
The transforms DI(IB1B1ωB1I), DI(IB2B2ωB2I), and DI(mB2RB2BDIrB2B1) are discussed below.ForDI(IB1B1ωB1I), based on the equation DI(s)=DB1(s)+ωB1I×s, we can obtain
(2.10)DI(IB1B1ωB1I)=DB1(IB1B1ωB1I)+ωB1I×IB1B1ωB1I=IB1B1DB1(ωB1I)+ΩB1IIB1B1ωB1I,
where ΩB1I is the antisymmetric matrix of ωB1I.Given thatωB1I=ωB1B+ωBI, substitution into (2.10) yields
(2.11)DI(IB1B1ωB1I)=IB1B1DB1ωB1B+IB1B1DBωBI+(ΩB1BIB1B1+ΩBIIB1B1-IB1B1ΩB1B)ωBI+(ΩB1B+ΩBI)IB1B1ωB1B.ForDI(IB2B2ωB2I),
(2.12)DI(IB2B2ωB1I)=IB2B2DB2ωB2B+IB2B2DBωBI+(ΩB2BIB2B2+ΩBIIB2B2-IB2B2ΩB2B)ωBI+(ΩB2B+ΩBI)IB2B2ωB2B.The following conversion relations exist between the angular velocities:(2.13)ωB2I=ωB2B+ωBI,ωB2B=0,
where ωB2B=0 because B2 is fixed in the satellite body coordinate system.Substituting (2.13) into (2.12) yields
(2.14)DI(IB2B2ωB2I)=IB2B2DBωBI+ΩBIIB2B2ωBI.ForDI(mB2RB2BDIrB2B1),
(2.15)DI(mB2RB2BDIrB2B1)=DI(mB2rB2B×DIrB2B1)=mB2DIrB2B×DIrB2B1+mB2RB2BDIDIrB2B1,
where
(2.16)DIrB2B×DIrB2B=0,DIrB2B1=ΩBIrB2B1+DBrB2B1,DBrB2B1=0 because the position vector rB2B1 remains unchanged in the body coordinate system.Substituting (2.16) into (2.15) yields
(2.17)DI(mB2RB2BDIrB1B)=mB2RB2BDI(ΩBIrB2B1).Based onDI(s)=DB1(s)+ωB1I×s, (2.17) becomes
(2.18)mB2RB2BDI(ΩBIrB2B1)=mB2RB2BΩBIΩBIrB2B1+mB2RB2BDB(ΩBIrB2B1).Given that(2.19)DB(ΩBIrB2B1)=DB(ωBI×rB2B1)=-DB(rB2B1×ωBI)=-RB2B1DBωBI,
then
(2.20)DI(mB2RB2BDIrB1B)=mB2RB2BΩBIΩBIrB2B1-mB2RB2BRB2B1DBωBI.
Based on (2.9), (2.11), (2.14), and (2.20), the attitude dynamic equation of a space-based observation satellite is as follows:
(2.21)IB1B1DB1ωB1B+IB1B1DBωBI+(ΩB1BIB1B1+ΩBIIB1B1-IB1B1ΩB1B)ωBI+(ΩB1B+ΩBI)IB1B1ωB1B+IB2B2DBωBI+ΩBIIB2B2ωBI+mB2RB2BΩBIΩBIrB2B1-mB2RB2BRB2B1DBωBI=MB,
where
(2.22)[IB1B1]B=[TBB1][IB1B1]B1[TBB1]T,[ΩB1B]B=[TBB1][ΩB1B]B1[TBB1]T,[ωB1B]B=[TBB1][ωB1B]B1.
Considering the orthogonality of [TBB1], the dynamic equation of attitude in the body coordinate system is as follows:
(2.23)[TBB1][IB1B1]B1[DB1ωB1B]B1+[TBB1][IB1B1]B1[TBB1]TDBωBI+([TBB1][ΩB1B]B1[IB1B1]B1[TBB1]T+ΩBI[TBB1][IB1B1]B1[TBB1]T-[TBB1][IB1B1]B1[ΩB1B]B1[TBB1]TωBI)+([TBB1][ΩB1B]B1[TBB1]T+[ΩBI]B)[TBB1][IB1B1]B1[ωB1B]B1+IB2B2DBωBI+ΩBIIB2B2ωBI+mB2RB2BΩBIΩBIrB2B1-mB2RB2BRB2B1DBωBI=MB.
Assuming the disturbance torque introduced by the camera turntable is Mrotate, then
(2.24)Mrotate=[TBB1][IB1B1]B1[DB1ωB1B]B1+[TBB1][IB1B1]B1[TBB1]TDBωBI+([TBB1][ΩB1B]B1[IB1B1]B1[TBB1]T+ΩBI[TBB1][IB1B1]B1[TBB1]T-[TBB1][IB1B1]B1[ΩB1B]B1[TBB1]T)ωBI+([TBB1][ΩB1B]B1[TBB1]T+[ΩBI]B)[TBB1][IB1B1]B1[ωB1B]B1.The satellite attitude dynamic equation can then be simplified as(2.25)IB2B2DBωBI+ΩBIIB2B2ωBI+Mrotate=MB.


### 2.2.2. Dynamic Model of a Space-Based Observation Satellite with Reaction Wheels

The reaction wheel, as the actuator of a satellite, is used to adjust the satellite attitude. When equipped with reaction wheels, the total moment of momentum of a satellite becomes(2.26)DILBBI=DI(LBB1I+LBB2I+LBWB)=DI(LBB1I+LBB2I)+DBLBWB+ΩBILBWB.
From (2.25) and (2.26), the attitude dynamic equation of a space-based observation satellite with reaction wheels is as follows:
(2.27)IB2B2DBωBI+ΩBIIB2B2ωBI+ΩBILBWB+Mrotate=MB+Mf,
where Mf=-DBLBWB is the control torque produced by a reaction wheel.


### 2.2.3. Dynamic Model of a Space-Based Observation Satellite with Flexible Appendages

The motion equation of solar panels is [6]
(2.28)q¨+2ξq˙+Λ2q+CTω˙=0,
where q is the flexible modal coordinates, CT is the coupling matrix between the solar panel and satellite body,  ξ is the structural damping of the solar panel, and  Λ  is the vibration frequency of the solar panel.Then, the attitude dynamic model of the satellite with a reaction wheel and flexible appendages is(2.29)IB2B2DBωBI+ΩBIIB2B2ωBI+ΩBILBWB+Cq¨+Mrotate=MB+Mf.


## 2.2.1. Attitude Dynamic Modeling of a Space-Based Observation Satellite

A space-based observation satellite can be simplified into a camera turntable and a satellite body when not considering the reaction wheel and flexible appendages. In this case, the attitude dynamic equation can be derived from the theorem of moment of momentum as follows:(2.5)DILBBI=MB,
where LBBI is the total moment of momentum, and MB is the total outside torque the satellite experiences relative to the centroid B.The total moment of momentum of satellite consists of two parts, which can be written as(2.6)LBBI=LBB1I+LBB2I,
where LBB1I,LBB2I can be expressed as
(2.7)LBB1I=LB1B1I+mB1rB1B×DIrB1B=IB1B1ωB1I+mB1RB1BDIrB1B,LBB2I=IB2B2ωB2I+mB2RB2BDIrB2B,
where RB1B  represents the antisymmetric matrix of rB1B.Equations (2.5)–(2.7) can yield
(2.8)DI(IB1B1ωB1I+mB1RB1BDIrB1B+IB2B2ωB2I+mB2RB2BDIrB2B)=MB.Given thatmB1rB1B+mB2rB2B=0, (2.8) can yield
(2.9)DI(IB1B1ωB1I)+DI(IB2B2ωB2I)+DI(mB2RB2BDIrB2B1)=MB.
The transforms DI(IB1B1ωB1I), DI(IB2B2ωB2I), and DI(mB2RB2BDIrB2B1) are discussed below.ForDI(IB1B1ωB1I), based on the equation DI(s)=DB1(s)+ωB1I×s, we can obtain
(2.10)DI(IB1B1ωB1I)=DB1(IB1B1ωB1I)+ωB1I×IB1B1ωB1I=IB1B1DB1(ωB1I)+ΩB1IIB1B1ωB1I,
where ΩB1I is the antisymmetric matrix of ωB1I.Given thatωB1I=ωB1B+ωBI, substitution into (2.10) yields
(2.11)DI(IB1B1ωB1I)=IB1B1DB1ωB1B+IB1B1DBωBI+(ΩB1BIB1B1+ΩBIIB1B1-IB1B1ΩB1B)ωBI+(ΩB1B+ΩBI)IB1B1ωB1B.ForDI(IB2B2ωB2I),
(2.12)DI(IB2B2ωB1I)=IB2B2DB2ωB2B+IB2B2DBωBI+(ΩB2BIB2B2+ΩBIIB2B2-IB2B2ΩB2B)ωBI+(ΩB2B+ΩBI)IB2B2ωB2B.The following conversion relations exist between the angular velocities:(2.13)ωB2I=ωB2B+ωBI,ωB2B=0,
where ωB2B=0 because B2 is fixed in the satellite body coordinate system.Substituting (2.13) into (2.12) yields
(2.14)DI(IB2B2ωB2I)=IB2B2DBωBI+ΩBIIB2B2ωBI.ForDI(mB2RB2BDIrB2B1),
(2.15)DI(mB2RB2BDIrB2B1)=DI(mB2rB2B×DIrB2B1)=mB2DIrB2B×DIrB2B1+mB2RB2BDIDIrB2B1,
where
(2.16)DIrB2B×DIrB2B=0,DIrB2B1=ΩBIrB2B1+DBrB2B1,DBrB2B1=0 because the position vector rB2B1 remains unchanged in the body coordinate system.Substituting (2.16) into (2.15) yields
(2.17)DI(mB2RB2BDIrB1B)=mB2RB2BDI(ΩBIrB2B1).Based onDI(s)=DB1(s)+ωB1I×s, (2.17) becomes
(2.18)mB2RB2BDI(ΩBIrB2B1)=mB2RB2BΩBIΩBIrB2B1+mB2RB2BDB(ΩBIrB2B1).Given that(2.19)DB(ΩBIrB2B1)=DB(ωBI×rB2B1)=-DB(rB2B1×ωBI)=-RB2B1DBωBI,
then
(2.20)DI(mB2RB2BDIrB1B)=mB2RB2BΩBIΩBIrB2B1-mB2RB2BRB2B1DBωBI.
Based on (2.9), (2.11), (2.14), and (2.20), the attitude dynamic equation of a space-based observation satellite is as follows:
(2.21)IB1B1DB1ωB1B+IB1B1DBωBI+(ΩB1BIB1B1+ΩBIIB1B1-IB1B1ΩB1B)ωBI+(ΩB1B+ΩBI)IB1B1ωB1B+IB2B2DBωBI+ΩBIIB2B2ωBI+mB2RB2BΩBIΩBIrB2B1-mB2RB2BRB2B1DBωBI=MB,
where
(2.22)[IB1B1]B=[TBB1][IB1B1]B1[TBB1]T,[ΩB1B]B=[TBB1][ΩB1B]B1[TBB1]T,[ωB1B]B=[TBB1][ωB1B]B1.
Considering the orthogonality of [TBB1], the dynamic equation of attitude in the body coordinate system is as follows:
(2.23)[TBB1][IB1B1]B1[DB1ωB1B]B1+[TBB1][IB1B1]B1[TBB1]TDBωBI+([TBB1][ΩB1B]B1[IB1B1]B1[TBB1]T+ΩBI[TBB1][IB1B1]B1[TBB1]T-[TBB1][IB1B1]B1[ΩB1B]B1[TBB1]TωBI)+([TBB1][ΩB1B]B1[TBB1]T+[ΩBI]B)[TBB1][IB1B1]B1[ωB1B]B1+IB2B2DBωBI+ΩBIIB2B2ωBI+mB2RB2BΩBIΩBIrB2B1-mB2RB2BRB2B1DBωBI=MB.
Assuming the disturbance torque introduced by the camera turntable is Mrotate, then
(2.24)Mrotate=[TBB1][IB1B1]B1[DB1ωB1B]B1+[TBB1][IB1B1]B1[TBB1]TDBωBI+([TBB1][ΩB1B]B1[IB1B1]B1[TBB1]T+ΩBI[TBB1][IB1B1]B1[TBB1]T-[TBB1][IB1B1]B1[ΩB1B]B1[TBB1]T)ωBI+([TBB1][ΩB1B]B1[TBB1]T+[ΩBI]B)[TBB1][IB1B1]B1[ωB1B]B1.The satellite attitude dynamic equation can then be simplified as(2.25)IB2B2DBωBI+ΩBIIB2B2ωBI+Mrotate=MB.


## 2.2.2. Dynamic Model of a Space-Based Observation Satellite with Reaction Wheels

The reaction wheel, as the actuator of a satellite, is used to adjust the satellite attitude. When equipped with reaction wheels, the total moment of momentum of a satellite becomes(2.26)DILBBI=DI(LBB1I+LBB2I+LBWB)=DI(LBB1I+LBB2I)+DBLBWB+ΩBILBWB.
From (2.25) and (2.26), the attitude dynamic equation of a space-based observation satellite with reaction wheels is as follows:
(2.27)IB2B2DBωBI+ΩBIIB2B2ωBI+ΩBILBWB+Mrotate=MB+Mf,
where Mf=-DBLBWB is the control torque produced by a reaction wheel.


## 2.2.3. Dynamic Model of a Space-Based Observation Satellite with Flexible Appendages

The motion equation of solar panels is [6]
(2.28)q¨+2ξq˙+Λ2q+CTω˙=0,
where q is the flexible modal coordinates, CT is the coupling matrix between the solar panel and satellite body,  ξ is the structural damping of the solar panel, and  Λ  is the vibration frequency of the solar panel.Then, the attitude dynamic model of the satellite with a reaction wheel and flexible appendages is(2.29)IB2B2DBωBI+ΩBIIB2B2ωBI+ΩBILBWB+Cq¨+Mrotate=MB+Mf.


## 3. Dual-Loop Compensation-Based High-Precision Attitude Control

A matter of primary concern in designing high-accuracy  attitude control is how to eliminate the disturbance introduced by rotation of rigid parts and flexible solar panels. [6] have applied the attitude control system as shown in Figure 4(a) to the Japanese satellite HALCA for the very long baseline interferometry observations in orbit. F(s) is a third-order low-pass filter, called the Chebyshev-type filter. Therefore,
(3.1)F(s)=10.686s3+0.736s2+2.061τs+1,
where the cut-off frequency of F(s) is 0.286 Hz.Structure of the attitude control system. (a) The conventional method used in Japanese HALCA satellite. (b) The method proposed in this paper.
(a)
(b)In this paper, a novel dual-loop DOB is proposed as shown in Figure4(b), where the DOB includes two parts: loop 1 is used to compensate for the disturbance introduced by the motion of the turntable, while loop 2 is used to compensate for the disturbance made by solar panels. The numerical simulation results show that the proposed method is far superior to the conventional method.


### 3.1. Design of Loop 1

Based on (2.24), the estimation of disturbance introduced by the turntable is as follows:
(3.2)M^rotate=[TBB1][IB1B1]B1[DB1ωB1B]B1+[TBB1][IB1B1]B1[TBB1]TDBωBI+(ΩBI[TBB1][IB1B1]B1[TBB1]T)ωBI+[TBB1][ΩB1B]B1[IB1B1]B1[ωB1B]B1+mB2RB2BΩBIΩBIrB2B1-mB2RB2BRB2B1DBωBI+([TBB1][ΩB1B]B1[IB1B1]B1[TBB1]T-[TBB1][IB1B1]B1[ΩB1B]B1[TBB1]T)ωBI+[ΩBI]B[TBB1][IB1B1]B1[ωB1B]B1,M^rotate=[M^BxM^ByM^Bz].
Then the estimation of disturbance in x-axis is
(3.3)M^Bx=IB1x(ω˙x1cβ+ω˙y1sαsβ+ω˙z1cαsβ+ω˙x+ωyωz1cαcβ-ωyωx1sβ+ωyωy1sαcβ-ωzωy1cα+ωzωz1sα)+IB1y(ωy1ωzs2αcβ-ωyωz1cαcβ+ωyωx1sβ-ωyωy1sαcβ-ωyωz+IB1y-ωy1ωz1c2αcβ+ωx1ωy1cαsβ-ωy12sαcαcβ+ωz12sαcαcβ+IB1y-ωx1ωz1sαsβ+ωy1ωzs2αcβ-ωzωy1cα+ωzωz1sα)+IB1z(ωzωy1cα-ωzωz1sα+ωyωz-ωx1ωy1cαsβ+ωx1ωz1sαsβ+IB1y+ωy12sαcαcβ-ωy1ωz1s2αcβ+ωy1ωz1c2αcβ-ωz12sαcαcβ+IB1y-ωyωx1sβ+ωyωy1sαcβ+ωyωz1cαcβ)+IB2xω˙x-ωyωzIB2y+ωyωzIB2z,
where α, β, ωx, ωy, and ωz can be measured by sensors on the satellite. ωx1, ωy1, and ωz1 can be yielded by (2.4).The model of the reaction wheel is described as(3.4)A(s)=KmTms+1s.Therefore,Z(s) is given by
(3.5)Z(s)=Tms+1Kms.The compensation of loop1 inx-axis is then described as
(3.6)U2=Z(s)MBx.


### 3.2. Design of Loop 2

A description of the basic structure of disturbance observer is shown in Figure5. The system output can be expressed in terms of the reference control input, the external disturbance, and the measurement noise. Thus, the behavior of the actual system is to be the same as the given nominal model.Figure 5
Block diagram of the disturbance observer.From Figure5 we can find
(3.7)u(s)=c(s)-d^(s)+d(s),
where c(s) is the controller output and d^(s) is the disturbance estimation.And we can also find(3.8)GCY(s)=Y(s)U(s)=Gp(s)Gn(s)Gn(s)+Q(s)(Gp(s)-Gn(s)),GDY(s)=Y(s)D(s)=Gp(s)Gn(s)(1-Q(s))Gn(s)+Q(s)(Gp(s)-Gn(s)),GNY(s)=Y(s)N(s)=Gp(s)Q(s)Gn(s)+Q(s)(Gp(s)-Gn(s)).Assumingfq is the cut-off frequency of low-pass filter Q(s) and f is the signal frequency. When f≤fq, Q≈1, GCY(s)≈Gn(s), GDY≈0, GNY≈1.This shows that in low-frequency range the disturbance observer could make system characteristics approximately equal to the nominal model characteristics, and the disturbance observer can overcome the various disturbances effectively.In this section, the nonlinear terms and small quantities in the satellite dynamics are treated as the external disturbances so that linear controller design tools (Laplace transform, disturbance observer) can be utilized. We take thex-axes as examples to design loop 2. Taking the ΩBILBWB,Cq¨,Mrotate  in the satellite dynamics equation (2.29) as external disturbances, then the x-axis nominal modal could be written as
(3.9)Gn(s)=1Ixωxs+(Iz-Iy)ωzωy.Then the inverse model is(3.10)Gn-1(s)=Ixωxs+(Iz-Iy)ωzωy.TheQ(s) is a key part in loop 2, which has the form [11]:
(3.11)Q(s)=[1+∑k=1N-rak(τs)k][1+∑k=1Nak(τs)k]-1,
where N is the order of Q(s), τ is the filter time constant, and r is the relative degree of Q(s).The main issue in the design ofQ(s) is the tradeoff between making |Q(jw)| small and |1-Q(jw)| small [12]. One of the typical solutions is to choose Q(s) so that the right-hand-side slope of |Q(jw)| is the same as the left-hand-side slope of |1-Q(jw)|. In the frequency domain, the slope of |Q(jw)| in low frequency and the slope of |1-Q(jw)| in high frequency are, respectively, approximated as [12]
(3.12)Lm|1-Q(jw)|Lm(w)≈N-r+1,Lm|Q(jw)|Lm(w)≈-r,
where Lm is log magnitude. Since the optimal Q(s) is obtained by letting the slopes be equivalent, the following relation is achieved:
(3.13)r=(N+1)2.It is notable that theQ31-filter is obtained as the minimum-order Q-filter for the second-order system [12]
(3.14)Q31(s)=3τs+1s3+3τs2+3τs+1,
where τ is the time constant.The compensation of loop 2 as shown in Figure4(b) in x-axis is then described as
(3.15)U3=Z(s)Q(s)[UM-Ixωxs-(Iz-Iy)ωyωz].Other axes employ the same compensation structure as (3.15).


### 3.3. Design of the External Loop

There are many uncertain factors in the parameters and structure of an observation satellite. For example, the moment of inertiaJ of the system is not quite accurate, and the space environmental disturbance torques, such as gravity gradient torque, aerodynamic torque and, solar radiation torque, are uncertainties. For a satellite in 1000 km orbit, a typical environmental disturbance torque equation is written as
(3.16)Mx=10-5(3cosω0t+1),My=10-5(1.5sinω0t+3cosω0t),Mz=10-5(3sinω0t+1).Consequently, the control scheme must robustly restrain all uncertainties, and the property of the actuators must be considered. The control law should be suitable for limited and discontinuous control torque and gas consumption.The SMC method has some advantages, such as robustness to parameter uncertainty, insensitivity to bounded disturbances, fast dynamic response, a remarkable computational simplicity with respect to other robust control approaches, and easy implementation of the controller.Variable structure control (VSC) is recognized as a powerful theoretical technique for robust control, even under variations in system parameters or in the presence of external disturbances. A well-known property of a VSC system is that its sliding modes are invariant with respect to a class of disturbances and parameter perturbations. If the disturbances and perturbations are bounded, the so-called reaching process is also robust against them with the controller appropriately designed. Thus, in this paper, VSC was employed as the external loop controller, which is written as(3.17)U={-T·sgn(s)|s|>δ1-Tf·sgn(s)δ<|s|<δ1-Tf·sδ|s|<δ,s=ce+e˙,
where s is the sliding mode, δ1 is a bound value for adjusting the thrusters and wheels, δ is the cutting gene to avoid vibrations of the VSC on the switching surface, Tp is the thruster maximal output torque, Tf is the wheel maximal output torque, e is the angle error (Figure 6), and e˙ is the angular velocity error.Figure 6
Track of the system in the phase plane.Considering the effects produced by the motion of the two-dimensional turntable and elastic vibration of flexible panels, the  disturbance estimation can be used as the forward to compensate for the disturbance from the camera turntable.Then, the final control law is rewritten as follows:(3.18)U={-T·sgn(s)|s|>δ1-Tf·sgn(s)δ<|s|<δ1-Tf·sδ+U2+U3|s|<δ,
where U2, U3 are the compensations of disturbance by the DOB.FilterFH(s) is to prevent excitation of the ignored higher flexible-mode’s vibration, where we chose a common filter structure as
(3.19)FH=ωhs+ωh,
where ωh is the cut-off frequency. It is preferred that the cut-off frequency ωh of FH is designed equal to or lower than the bandwidth of the Q(s). In this paper, the cut-off frequency of Q(s) is 0.6, so ωh is set as 0.6.


## 3.1. Design of Loop 1

Based on (2.24), the estimation of disturbance introduced by the turntable is as follows:
(3.2)M^rotate=[TBB1][IB1B1]B1[DB1ωB1B]B1+[TBB1][IB1B1]B1[TBB1]TDBωBI+(ΩBI[TBB1][IB1B1]B1[TBB1]T)ωBI+[TBB1][ΩB1B]B1[IB1B1]B1[ωB1B]B1+mB2RB2BΩBIΩBIrB2B1-mB2RB2BRB2B1DBωBI+([TBB1][ΩB1B]B1[IB1B1]B1[TBB1]T-[TBB1][IB1B1]B1[ΩB1B]B1[TBB1]T)ωBI+[ΩBI]B[TBB1][IB1B1]B1[ωB1B]B1,M^rotate=[M^BxM^ByM^Bz].
Then the estimation of disturbance in x-axis is
(3.3)M^Bx=IB1x(ω˙x1cβ+ω˙y1sαsβ+ω˙z1cαsβ+ω˙x+ωyωz1cαcβ-ωyωx1sβ+ωyωy1sαcβ-ωzωy1cα+ωzωz1sα)+IB1y(ωy1ωzs2αcβ-ωyωz1cαcβ+ωyωx1sβ-ωyωy1sαcβ-ωyωz+IB1y-ωy1ωz1c2αcβ+ωx1ωy1cαsβ-ωy12sαcαcβ+ωz12sαcαcβ+IB1y-ωx1ωz1sαsβ+ωy1ωzs2αcβ-ωzωy1cα+ωzωz1sα)+IB1z(ωzωy1cα-ωzωz1sα+ωyωz-ωx1ωy1cαsβ+ωx1ωz1sαsβ+IB1y+ωy12sαcαcβ-ωy1ωz1s2αcβ+ωy1ωz1c2αcβ-ωz12sαcαcβ+IB1y-ωyωx1sβ+ωyωy1sαcβ+ωyωz1cαcβ)+IB2xω˙x-ωyωzIB2y+ωyωzIB2z,
where α, β, ωx, ωy, and ωz can be measured by sensors on the satellite. ωx1, ωy1, and ωz1 can be yielded by (2.4).The model of the reaction wheel is described as(3.4)A(s)=KmTms+1s.Therefore,Z(s) is given by
(3.5)Z(s)=Tms+1Kms.The compensation of loop1 inx-axis is then described as
(3.6)U2=Z(s)MBx.


## 3.2. Design of Loop 2

A description of the basic structure of disturbance observer is shown in Figure5. The system output can be expressed in terms of the reference control input, the external disturbance, and the measurement noise. Thus, the behavior of the actual system is to be the same as the given nominal model.Figure 5
Block diagram of the disturbance observer.From Figure5 we can find
(3.7)u(s)=c(s)-d^(s)+d(s),
where c(s) is the controller output and d^(s) is the disturbance estimation.And we can also find(3.8)GCY(s)=Y(s)U(s)=Gp(s)Gn(s)Gn(s)+Q(s)(Gp(s)-Gn(s)),GDY(s)=Y(s)D(s)=Gp(s)Gn(s)(1-Q(s))Gn(s)+Q(s)(Gp(s)-Gn(s)),GNY(s)=Y(s)N(s)=Gp(s)Q(s)Gn(s)+Q(s)(Gp(s)-Gn(s)).Assumingfq is the cut-off frequency of low-pass filter Q(s) and f is the signal frequency. When f≤fq, Q≈1, GCY(s)≈Gn(s), GDY≈0, GNY≈1.This shows that in low-frequency range the disturbance observer could make system characteristics approximately equal to the nominal model characteristics, and the disturbance observer can overcome the various disturbances effectively.In this section, the nonlinear terms and small quantities in the satellite dynamics are treated as the external disturbances so that linear controller design tools (Laplace transform, disturbance observer) can be utilized. We take thex-axes as examples to design loop 2. Taking the ΩBILBWB,Cq¨,Mrotate  in the satellite dynamics equation (2.29) as external disturbances, then the x-axis nominal modal could be written as
(3.9)Gn(s)=1Ixωxs+(Iz-Iy)ωzωy.Then the inverse model is(3.10)Gn-1(s)=Ixωxs+(Iz-Iy)ωzωy.TheQ(s) is a key part in loop 2, which has the form [11]:
(3.11)Q(s)=[1+∑k=1N-rak(τs)k][1+∑k=1Nak(τs)k]-1,
where N is the order of Q(s), τ is the filter time constant, and r is the relative degree of Q(s).The main issue in the design ofQ(s) is the tradeoff between making |Q(jw)| small and |1-Q(jw)| small [12]. One of the typical solutions is to choose Q(s) so that the right-hand-side slope of |Q(jw)| is the same as the left-hand-side slope of |1-Q(jw)|. In the frequency domain, the slope of |Q(jw)| in low frequency and the slope of |1-Q(jw)| in high frequency are, respectively, approximated as [12]
(3.12)Lm|1-Q(jw)|Lm(w)≈N-r+1,Lm|Q(jw)|Lm(w)≈-r,
where Lm is log magnitude. Since the optimal Q(s) is obtained by letting the slopes be equivalent, the following relation is achieved:
(3.13)r=(N+1)2.It is notable that theQ31-filter is obtained as the minimum-order Q-filter for the second-order system [12]
(3.14)Q31(s)=3τs+1s3+3τs2+3τs+1,
where τ is the time constant.The compensation of loop 2 as shown in Figure4(b) in x-axis is then described as
(3.15)U3=Z(s)Q(s)[UM-Ixωxs-(Iz-Iy)ωyωz].Other axes employ the same compensation structure as (3.15).


## 3.3. Design of the External Loop

There are many uncertain factors in the parameters and structure of an observation satellite. For example, the moment of inertiaJ of the system is not quite accurate, and the space environmental disturbance torques, such as gravity gradient torque, aerodynamic torque and, solar radiation torque, are uncertainties. For a satellite in 1000 km orbit, a typical environmental disturbance torque equation is written as
(3.16)Mx=10-5(3cosω0t+1),My=10-5(1.5sinω0t+3cosω0t),Mz=10-5(3sinω0t+1).Consequently, the control scheme must robustly restrain all uncertainties, and the property of the actuators must be considered. The control law should be suitable for limited and discontinuous control torque and gas consumption.The SMC method has some advantages, such as robustness to parameter uncertainty, insensitivity to bounded disturbances, fast dynamic response, a remarkable computational simplicity with respect to other robust control approaches, and easy implementation of the controller.Variable structure control (VSC) is recognized as a powerful theoretical technique for robust control, even under variations in system parameters or in the presence of external disturbances. A well-known property of a VSC system is that its sliding modes are invariant with respect to a class of disturbances and parameter perturbations. If the disturbances and perturbations are bounded, the so-called reaching process is also robust against them with the controller appropriately designed. Thus, in this paper, VSC was employed as the external loop controller, which is written as(3.17)U={-T·sgn(s)|s|>δ1-Tf·sgn(s)δ<|s|<δ1-Tf·sδ|s|<δ,s=ce+e˙,
where s is the sliding mode, δ1 is a bound value for adjusting the thrusters and wheels, δ is the cutting gene to avoid vibrations of the VSC on the switching surface, Tp is the thruster maximal output torque, Tf is the wheel maximal output torque, e is the angle error (Figure 6), and e˙ is the angular velocity error.Figure 6
Track of the system in the phase plane.Considering the effects produced by the motion of the two-dimensional turntable and elastic vibration of flexible panels, the  disturbance estimation can be used as the forward to compensate for the disturbance from the camera turntable.Then, the final control law is rewritten as follows:(3.18)U={-T·sgn(s)|s|>δ1-Tf·sgn(s)δ<|s|<δ1-Tf·sδ+U2+U3|s|<δ,
where U2, U3 are the compensations of disturbance by the DOB.FilterFH(s) is to prevent excitation of the ignored higher flexible-mode’s vibration, where we chose a common filter structure as
(3.19)FH=ωhs+ωh,
where ωh is the cut-off frequency. It is preferred that the cut-off frequency ωh of FH is designed equal to or lower than the bandwidth of the Q(s). In this paper, the cut-off frequency of Q(s) is 0.6, so ωh is set as 0.6.


## 4. Numerical Simulation and Results

In this section, the results and analysis from the algorithm developed to control the space-based observation micro-satellite are presented. For some space observation tasks, the objective of the control system is to keep the satellite attitude angle error < 0.02° and the angular velocity error < 0.001°/s.


### 4.1. Initial Assumptions

See Tables1 and 2.Table 1
Initial parameters of satellite
Value
Initial attitude angle of satellite
θ
x
=
1
°
,  θy=1°,  θz=1°
Initial attitude angular velocity of satellite
ω
x
=
0
.
1
°
/
s,  ωy=0.1°/s,  ωz=0.1°/s
Angular velocity of camera turntable
α
=
(
1.2
sin
ω
t
)
°
/
s, β=(1.2sinωt)°/s
ω
=
0.03
rad
/
s
The uncertainties of the matrix of moment of inertia
Δ
I
=
[
3
0
0
0
4
0
0
0
2
]
(kg
·
m
2
)
Space environmental disturbance torques
M
x
=
1
0
-
5
(
3
cos
ω
0
t
+
1
)
M
y
=
1
0
-
5
(
1.5
sin
ω
0
t
+
3
cos
ω
0
t
)
M
z
=
1
0
-
5
(
3
sin
ω
0
t
+
1
)Table 2
Satellite parameters.
Satellite parameters
Value
Mass of platformm2
m
2
=
250
kg
Mass of camera turntablem1
m
1
=
30
kg
The matrix of moment of inertia of platformIB2
I
B
2
=
[
90
0
0
0
150
0
0
0
85
]
(kg · m2)
The matrix of moment of inertia of camera turntableIB1
I
B
1
=
[
9
0
0
0
15
0
0
0
8.5
]
(
kg
·
m
2
)
r
B
2
B
1
|
r
B
2
B
1
|
=
0.8
m
r
B
2
B
|
r
B
2
B
|
=
0.08
m
Maximal output torque of reaction wheel
0.1 Nm
The matrix coupling coefficient of solar panel
C
=
[
-
2.8137
0
0
0
3.142
0
-
0.0441
0
1.826
]
Vibration frequency of solar panelsΛ
Λ
=
[
0.11
0
0
0
0.33
0
0
0
0.73
]
Hz
Structural damping of solar panels
ξ = 0.002


### 4.2. Results

See Figures7, 8, and 9, and Table 3.Table 3
Comparison of the results of the two cases.
Angle errors
Angular velocity errors
The conventional method
0.03°
(
1
.
2
×
10
-
3
)
°
/
s
The proposed method
0.004°
(
0
.
2
×
10
-
3
)
°
/
sAngle error results: (a)x-, (b) y-, and (c) z-axes. The red line represents proposed method. The blue line represents conventional method.
(a)
(b)
(c)Angular velocity error results: (a)x-, (b) y-, and (c) z-axes. The red line represents proposed method. The blue line represents conventional method.
(a)
(b)
(c)Estimation of the disturbance torques.
(a)
(b)
(c)


### 4.3. Conclusion

From Figures7–9 and Table 3, we can conclude that by the dual-loop DOB, the control method proposed in this paper can significantly reduce the errors of attitude angle and angular velocity (as shown in Table 3), and the proposed control scheme can meet the requirements of the attitude control task. In addition, the derived-dynamic model of a space-based observation satellite has a high reference value to the design of attitude control systems for space-based observation satellites.


## 4.1. Initial Assumptions

See Tables1 and 2.Table 1
Initial parameters of satellite
Value
Initial attitude angle of satellite
θ
x
=
1
°
,  θy=1°,  θz=1°
Initial attitude angular velocity of satellite
ω
x
=
0
.
1
°
/
s,  ωy=0.1°/s,  ωz=0.1°/s
Angular velocity of camera turntable
α
=
(
1.2
sin
ω
t
)
°
/
s, β=(1.2sinωt)°/s
ω
=
0.03
rad
/
s
The uncertainties of the matrix of moment of inertia
Δ
I
=
[
3
0
0
0
4
0
0
0
2
]
(kg
·
m
2
)
Space environmental disturbance torques
M
x
=
1
0
-
5
(
3
cos
ω
0
t
+
1
)
M
y
=
1
0
-
5
(
1.5
sin
ω
0
t
+
3
cos
ω
0
t
)
M
z
=
1
0
-
5
(
3
sin
ω
0
t
+
1
)Table 2
Satellite parameters.
Satellite parameters
Value
Mass of platformm2
m
2
=
250
kg
Mass of camera turntablem1
m
1
=
30
kg
The matrix of moment of inertia of platformIB2
I
B
2
=
[
90
0
0
0
150
0
0
0
85
]
(kg · m2)
The matrix of moment of inertia of camera turntableIB1
I
B
1
=
[
9
0
0
0
15
0
0
0
8.5
]
(
kg
·
m
2
)
r
B
2
B
1
|
r
B
2
B
1
|
=
0.8
m
r
B
2
B
|
r
B
2
B
|
=
0.08
m
Maximal output torque of reaction wheel
0.1 Nm
The matrix coupling coefficient of solar panel
C
=
[
-
2.8137
0
0
0
3.142
0
-
0.0441
0
1.826
]
Vibration frequency of solar panelsΛ
Λ
=
[
0.11
0
0
0
0.33
0
0
0
0.73
]
Hz
Structural damping of solar panels
ξ = 0.002


## 4.2. Results

See Figures7, 8, and 9, and Table 3.Table 3
Comparison of the results of the two cases.
Angle errors
Angular velocity errors
The conventional method
0.03°
(
1
.
2
×
10
-
3
)
°
/
s
The proposed method
0.004°
(
0
.
2
×
10
-
3
)
°
/
sAngle error results: (a)x-, (b) y-, and (c) z-axes. The red line represents proposed method. The blue line represents conventional method.
(a)
(b)
(c)Angular velocity error results: (a)x-, (b) y-, and (c) z-axes. The red line represents proposed method. The blue line represents conventional method.
(a)
(b)
(c)Estimation of the disturbance torques.
(a)
(b)
(c)


## 4.3. Conclusion

From Figures7–9 and Table 3, we can conclude that by the dual-loop DOB, the control method proposed in this paper can significantly reduce the errors of attitude angle and angular velocity (as shown in Table 3), and the proposed control scheme can meet the requirements of the attitude control task. In addition, the derived-dynamic model of a space-based observation satellite has a high reference value to the design of attitude control systems for space-based observation satellites.



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*Source: 987256-2012-09-27.xml*