# Attitude Maneuvering and Vibration Reducing Control of Flexible Spacecraft Subject to Actuator Saturation and Misalignment

Authors: Jiawei Tao; Tao Zhang; Yongfang Nie
Journal: Shock and Vibration
 (2018)

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## Abstract

A robust adaptive constrained control scheme is proposed for flexible spacecraft attitude maneuver and vibration suppression, in which multiple constraints are simultaneously considered, such as uncertain inertia parameters, external disturbance, unmeasured elastic vibration, actuator saturation, and even actuator misalignment. More specifically, a novel path planning scheme based on quintic polynomial transition is firstly developed to realize smooth acceleration variate and therefore decrease the vibration of flexible appendages. Secondly, an elastic modal estimator is employed to estimate the unmeasured variables, such as the modal position and velocity. Thirdly, an adaptive updating technique is used to spare the extra knowledge of system parameters and the bound of the external disturbance. In addition, an auxiliary design system is constructed to address the actuator saturation problem, and a compensation term is synthesized and integrated into the controller to handle the actuator misalignment. Finally, overall system stabilization is proved within the framework of Lyapunov theory, and numerical simulation results are presented to illustrate the effectiveness of the proposed scheme.


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## Body

## 1. Introduction

Flexible spacecraft with large flexible structures is usually expected to achieve high pointing and fast attitude maneuvering in future space missions. However, the attitude maneuvering operation will introduce certain levels of vibration to flexible appendages due to the rigid-flexible coupling effect, which will deteriorate its pointing performance. For flexible spacecrafts, the governing differential equations for attitude kinematics and dynamics are strongly nonlinear in nature. The attitude maneuvering problem is further complicated by the uncertainty of spacecraft inertia parameters due to onboard payload motion and fuel consumption. Furthermore, it is also affected by various external disturbances that influence the mission objectives significantly. Additionally, the actuator misalignment during installation and actuator saturation increases the complexity and difficulty further.All these factors in a realistic environment cause a considerable difficulty in the design of attitude control system for meeting high-precision pointing requirement and desired control performance during the attitude maneuver process, especially when all these issues are treated simultaneously.Over the last few decades, considerable works have been found for vibration suppression and attitude control system design. A nonlinear state feedback attitude control law [1] in combination with path planning is proposed. Specifically, the planned attitude maneuver path is smooth, and thus, appendages’ vibration excited by attitude maneuver can be attenuated greatly. Specially, variable structure control (VSC) is known as an efficient way to deal with system uncertainty and external disturbance and has been applied to attitude control problem of flexible spacecrafts in [2, 3].However, these design methods require the information on the bounds on the uncertainties or disturbances for the computation of control gains. Recently, to overcome the drawbacks of each method, a combination of these techniques with adaptation mechanism to tune the controller gains are also studied for flexible spacecrafts with parameter uncertainties and disturbances [4–6]. A new adaptive system [4] for rotational maneuver and vibration suppression of an orbiting spacecraft with flexible appendages has been designed. Then adaptive output regulation of the closed-loop system is accomplished in spite of parameter uncertainties and disturbances. A new variable structure control approach [5] has been proposed for attitude control and vibration suppression of flexible spacecrafts during attitude maneuvering, and the adaptive version of the proposed controller is achieved through releasing the limitation of knowing the bounds of the uncertainties and perturbations in advance.Relevant drawback of these control strategies is either the extra necessity to measure the modal variables or to treat the rigid-flexible coupling effect as an additional disturbance acting on a rigid structure. With regard to the latter situation, an extended state observer [7, 8] is designed to estimate and thus to attenuate the total disturbance in finite time, including an external disturbance torque and the coupling effect. As a result, the prior knowledge of the total disturbance is not required.Unfortunately, in some cases, the availability of the measured modal variables is an unrealistic hypothesis due to the impracticability of using appropriate sensors or the economical requisite. For removing this disadvantage, reconstructing the unmeasured modal position and velocity by means of appropriate dynamics is an alternative way. A class of nonlinear controllers incorporating modal state estimator [9–13] has been derived for spacecraft with flexible appendages. It does not ask for measures of the modal variables, but only uses the parameters describing the attitude and the spacecraft angular velocity. The controller derived then uses estimates of the modal variables and its rate to avoid direct measurement.However, a typical feature in all of the mentioned attitude-control schemes and methods is that the control device is assumed to be able to produce big enough control torque without taking actuator saturation into account. Extensive results pertaining to spacecraft attitude control systems containing actuator saturation nonlinearities have been presented in [14–22]. A robust variable structure controller in [14, 15] has been skillfully designed to control the spacecraft attitude under input saturation. However, these control schemes lose the generality to nonlinear flexible spacecraft system. A modified adaptive backstepping attitude controller [17, 18] is developed considering external disturbances and input saturation. But the way to include the effects of the flexible dynamics in the lumped disturbance for the rigid dynamics deprives the controller of a direct compensation of the dynamic terms caused by the flexibility.A typical feature in the previous approaches is that they do not consider actuator misalignments. However, whether due to finite manufacturing tolerances or warping of the spacecraft structure during launch, some actuator alignment error will definitely exist in practice. This problem may cause mission performance to degrade and thus pose significant risk to the successful operation of the spacecraft. Therefore, it is desirable to design a control scheme to handle the actuator misalignments. Unfortunately, there has been insufficient research on attitude control in the presence of actuator misalignments. An adaptive control law [23] is developed to accomplish attitude maneuver in the presence of relatively small gimbals’ alignment error of variable speed control moment gyros.More specially, an adaptive control approach [24] was proposed for satellite formation flying. The backstepping technique is used to synthesize the controller, and the thrust misalignment is successfully handled. In another related work, a nonlinear model reference adaptive control scheme [25] is tested in the presence of misalignment errors up to fifteen degrees. Although an extended Kalman filter is used in another approach to develop methods for on-orbit actuator alignment calibration, uncertain inertia properties have not been taken into account.With a view to handle the above challenges and potential problems simultaneously, a new constrained robust adaptive control scheme is proposed in this paper. The main contributions are summarized as follows:(1)
A novel path planning scheme based on quantic polynomial transition is applied to smooth acceleration variate and therefore decrease the vibration of flexible appendages.(2)
This paper investigates the feasibility of attitude maneuver and vibration suppression in the presence of uncertain inertia parameters, external disturbances, unmeasured elastic vibrations, actuator saturation, and even actuator misalignment simultaneously, which has not been previously examined. As compared with the controller in [16] which only considers attitude stabilization, that is, rest-to-rest maneuver of a flexible spacecraft, the control scheme in the present paper can be applied to handle attitude tracking control problem. The controller in [6] investigates the attitude tracking problem for flexible spacecrafts, while it does not take the actuator saturation and actuator misalignment into consideration explicitly.(3)
The proposed controller is designed without requiring prior knowledge of the measurement of vibration variables, of the parameter uncertainties, and the upper bound of external disturbance. Robust control terms are synthesized to compensate all uncertainties including parametric uncertainties, unmeasured elastic vibration, external disturbances, actuator saturation, and actuator misalignment.The rest of this paper is organized as follows: Section2 states flexible spacecraft modelling with actuator misalignment and control problem formulation. A constrained robust adaptive-control scheme is proposed in Section 3. Numerical simulation results are presented in Section 4 to demonstrate the effectiveness and superiority of the proposed control scheme. Finally, conclusion is given in Section 5.


## 2. Flexible Spacecraft Modelling and Problem Formulation

### 2.1. Kinematics and Dynamics Equation

This section briefly reviews the quaternion mathematical description of the attitude motion of a flexible spacecraft.The attitude kinematic equation of the spacecraft can be written in terms of unit quaternion for global representation without singularities as follows [26]:(1)q˙=q˙0q˙v=12−qvTq0I3+qv×ω,where q=q0,qvTT represents the attitude quaternion, and I is an identity matrix of the dimension specified by its subscript.The dynamic model of a flexible spacecraft is governed by the following differential equations:(2)Jω˙+δTη¨=−ω×Jω+δTη˙+u+d,(3)η¨+Cη˙+Kη=−δω˙,where ω is the spacecraft angular velocity in the body-fixed frame, ω× is the skew-symmetric matrix of ω, η is the modal coordinate vector of the flexible appendages, the matrix J denotes the total inertia of spacecraft moment, δ is the coupling matrix between the flexible appendages and the rigid spacecraft, u is the control torque, d is the external disturbance torque vector, and C and K denote the damping and stiffness matrices, respectively, and they are defined as(4)C=diag2ξiωni,i=1,…,N,K=diagωni2,i=1,…,N,where N is the number of elastic modes considered, ωni is the natural frequency, and ξi is the corresponding damping ratio.For simplicity of the development, let us first introduce the following variableψ=η˙+δω representing the total angular velocity expressed in modal variables, so the dynamics of the flexible spacecraft from (2) to (3) can be further expressed as(5)η˙ψ˙=0I−K−Cηψ+−ICδω,(6)Jmω˙=−ω×Jmbω+δTψ+δTCψ+Kη−Cδω+u+d,where Jm=J−δTδ, with δTδ as the contribution of the flexible parts to the total inertia matrix.


### 2.2. Attitude Tracking Model

Letqd=qd0,qdvTT be the unit quaternion representing the desired attitude, ωd be the desired angular velocity.Then, the quaternion errorqe and angular velocity error ωe are given by(7)qe=qd−1∘q=q0qd0+qvTqdvqd0qv−q0qdv+Sqvqdv,(8)ωe=ω−ωr,where qd−1 is the inverse of the desired attitude unit quaternion, the symbol ∘ is the operator for quaternion multiplication; with ωr=Rqeωd, we have(9)ω˙r=−ωe×Rqeωd+Rqeω˙d,where Rqe is the transformation matrix from the desired coordinate frame to the body coordinate frame with(10)Rqe=qe02−qevTqevI3+2qevqevT−2qe0Sqev.From (1) to (10), tracking error dynamic equation can be obtained as follows:(11)q˙e=q˙e0q˙ev=12−qevTqe0I+qev×ωe,(12)Jmω˙e=−ω×Jmω−Jmω˙r+δTKη+δTCψ−ω×δTδω−δTCδωe+u+d,(13)η˙ψ˙=Aηψ−ABδωe−Bδω˙r,where A and B are given by(14)A=0I−K−C,B=0I.


### 2.3. Actuator Uncertainty Model

In practical aerospace engineering, redundant actuators are fixed to improve the reliability of the attitude control system. The considered spacecraft is controlled by using four reaction wheels. The configuration structure of four actuators in [27, 28] is adopted. Three reaction wheels are fixed orthogonally aligned with the axes of the body-fixed frame. The fourth, redundant, actuator is mounted skewed at α4 and β4.With this configuration, the total control torqueu can be calculated as(15)u=Dτ,where τ=τ1,τ2,τ3,τ4T denotes the nominal torque generated by the four reaction wheels, and D is the reaction wheel configuration matrix, representing the influence of each wheel on the angular acceleration of the spacecraft.However, in practice, the knowledge of orthogonal configuration of actuator will never be perfect. Due to finite-manufacturing tolerances or warping of the spacecraft structure during launch, actuator misalignments may exist. The reaction wheel mounted on theX-axis is offset from the nominal direction by constant angles, Δα1 and Δβ1. The reaction wheels mounted on Y-axis and Z-axis are assumed to be tilted away from their nominal directions by Δα2, Δβ2, Δα3, and Δβ3, while the redundant reaction wheel is tilted from its nominal direction by Δα4 and Δβ4.Then, the real control torque acting on spacecrafts with misalignment is expressed as(16)τxτyτz=cosΔα1sinΔα1cosΔβ1sinΔα1sinΔβ1τ1+sinΔα2sinΔβ2cosΔα2sinΔα2cosΔβ2τ2+sinΔα3cosΔβ3sinΔα3sinΔβ3cosΔα3τ3+cosα4+Δα4cosβ4+Δβ4cosα4+Δα4sinβ4+Δβ4sinα4+Δα4τ4.The definition of the misalignment angles suggests thatΔαi,Δβi,i∈1,2,3,4 are small angles. Hence, the following relationships are used to approximate (16):(17)cosΔαi≈cosΔβi≈1,sinΔαi≈Δαi,sinΔβi≈Δβi.Then for the considered actuator configuration, the configuration matrixD can be represented as(18)D=D0+ΔD,where D0 denotes the nominal configuration matrix, and ΔD denotes the actuator misalignment. They can be written as(19)D0=100cosα4cosβ4010cosα4sinβ4001sinα4,(20)ΔD=00Δα3ΔD14Δα100ΔD240Δα20Δα4cosα4,ΔD14=−Δα4sinα4cosβ4−Δβ4cosα4sinβ4,ΔD24=−Δα4sinα4sinβ4+Δβ4cosα4cosβ4.In view of (15) and (18), tracking error dynamics in (12) can be rewritten as(21)Jmω˙e=−ω×Jmω−Jmω˙r+δTKη+δTCψ−ω×δTδω−δTCδωe+D0+ΔDτ+d.To facilitate control system design, the following assumptions and lemmas are presented and will be used in the subsequent developments.Assumption 1.
The components of external disturbanced in (21) are assumed to be bounded by a set of unknown bounded constants, that is(22)di≤ρii=1,2,3.Assumption 2.
Due to physical limitations on the reaction wheels considered, the maximum output torque of each actuator output torque has the same limit valueτ¯,τ¯, that is [29](23)τ∈Ω=τiτ¯≤τi≤τ¯,i=1,2,3,4,where τ¯ and τ¯ represent the known saturation levels of reaction wheels.Assumption 3.
The uncertaintyΔD due to misalignment is an unknown but bounded matrix satisfying(24)ΔDF≤Δm,where Δm is a known positive constant.Remark 1.
According to (20), every element of actuator misalignment matrix ΔD is a function of misalignment angle errors which are small in practice, then the Frobenius norm of ΔD is bounded by a known quantity Δm. It is easy to find a feasible Δm in practice.Lemma 1.
For arbitrary positive constantε and variable η, the following inequality holds [30, 31](25)0≤η−ηtanhηε≤κε,with κ=e−κ+1 and κ=0.2785.


### 2.4. Control Problem Formulation

Given any initial attitude and angular velocity, the control objective can be stated as considering the flexible spacecraft attitude system described by (1)–(3), design a torque command τ such that the following goals are met in the presence of uncertain inertia parameters, external disturbance, unmeasured elastic vibration, actuator saturation (23), and even actuator misalignment:(1)
The attitude orientation and angular velocity tracking errors are driven to zero, or a small set containing the origin.(2)
The vibration induced by the maneuver rotation should be attenuated as soon as possible in the presence of parametric uncertainties, external disturbance, actuator saturation, and even actuator misalignment.


## 2.1. Kinematics and Dynamics Equation

This section briefly reviews the quaternion mathematical description of the attitude motion of a flexible spacecraft.The attitude kinematic equation of the spacecraft can be written in terms of unit quaternion for global representation without singularities as follows [26]:(1)q˙=q˙0q˙v=12−qvTq0I3+qv×ω,where q=q0,qvTT represents the attitude quaternion, and I is an identity matrix of the dimension specified by its subscript.The dynamic model of a flexible spacecraft is governed by the following differential equations:(2)Jω˙+δTη¨=−ω×Jω+δTη˙+u+d,(3)η¨+Cη˙+Kη=−δω˙,where ω is the spacecraft angular velocity in the body-fixed frame, ω× is the skew-symmetric matrix of ω, η is the modal coordinate vector of the flexible appendages, the matrix J denotes the total inertia of spacecraft moment, δ is the coupling matrix between the flexible appendages and the rigid spacecraft, u is the control torque, d is the external disturbance torque vector, and C and K denote the damping and stiffness matrices, respectively, and they are defined as(4)C=diag2ξiωni,i=1,…,N,K=diagωni2,i=1,…,N,where N is the number of elastic modes considered, ωni is the natural frequency, and ξi is the corresponding damping ratio.For simplicity of the development, let us first introduce the following variableψ=η˙+δω representing the total angular velocity expressed in modal variables, so the dynamics of the flexible spacecraft from (2) to (3) can be further expressed as(5)η˙ψ˙=0I−K−Cηψ+−ICδω,(6)Jmω˙=−ω×Jmbω+δTψ+δTCψ+Kη−Cδω+u+d,where Jm=J−δTδ, with δTδ as the contribution of the flexible parts to the total inertia matrix.


## 2.2. Attitude Tracking Model

Letqd=qd0,qdvTT be the unit quaternion representing the desired attitude, ωd be the desired angular velocity.Then, the quaternion errorqe and angular velocity error ωe are given by(7)qe=qd−1∘q=q0qd0+qvTqdvqd0qv−q0qdv+Sqvqdv,(8)ωe=ω−ωr,where qd−1 is the inverse of the desired attitude unit quaternion, the symbol ∘ is the operator for quaternion multiplication; with ωr=Rqeωd, we have(9)ω˙r=−ωe×Rqeωd+Rqeω˙d,where Rqe is the transformation matrix from the desired coordinate frame to the body coordinate frame with(10)Rqe=qe02−qevTqevI3+2qevqevT−2qe0Sqev.From (1) to (10), tracking error dynamic equation can be obtained as follows:(11)q˙e=q˙e0q˙ev=12−qevTqe0I+qev×ωe,(12)Jmω˙e=−ω×Jmω−Jmω˙r+δTKη+δTCψ−ω×δTδω−δTCδωe+u+d,(13)η˙ψ˙=Aηψ−ABδωe−Bδω˙r,where A and B are given by(14)A=0I−K−C,B=0I.


## 2.3. Actuator Uncertainty Model

In practical aerospace engineering, redundant actuators are fixed to improve the reliability of the attitude control system. The considered spacecraft is controlled by using four reaction wheels. The configuration structure of four actuators in [27, 28] is adopted. Three reaction wheels are fixed orthogonally aligned with the axes of the body-fixed frame. The fourth, redundant, actuator is mounted skewed at α4 and β4.With this configuration, the total control torqueu can be calculated as(15)u=Dτ,where τ=τ1,τ2,τ3,τ4T denotes the nominal torque generated by the four reaction wheels, and D is the reaction wheel configuration matrix, representing the influence of each wheel on the angular acceleration of the spacecraft.However, in practice, the knowledge of orthogonal configuration of actuator will never be perfect. Due to finite-manufacturing tolerances or warping of the spacecraft structure during launch, actuator misalignments may exist. The reaction wheel mounted on theX-axis is offset from the nominal direction by constant angles, Δα1 and Δβ1. The reaction wheels mounted on Y-axis and Z-axis are assumed to be tilted away from their nominal directions by Δα2, Δβ2, Δα3, and Δβ3, while the redundant reaction wheel is tilted from its nominal direction by Δα4 and Δβ4.Then, the real control torque acting on spacecrafts with misalignment is expressed as(16)τxτyτz=cosΔα1sinΔα1cosΔβ1sinΔα1sinΔβ1τ1+sinΔα2sinΔβ2cosΔα2sinΔα2cosΔβ2τ2+sinΔα3cosΔβ3sinΔα3sinΔβ3cosΔα3τ3+cosα4+Δα4cosβ4+Δβ4cosα4+Δα4sinβ4+Δβ4sinα4+Δα4τ4.The definition of the misalignment angles suggests thatΔαi,Δβi,i∈1,2,3,4 are small angles. Hence, the following relationships are used to approximate (16):(17)cosΔαi≈cosΔβi≈1,sinΔαi≈Δαi,sinΔβi≈Δβi.Then for the considered actuator configuration, the configuration matrixD can be represented as(18)D=D0+ΔD,where D0 denotes the nominal configuration matrix, and ΔD denotes the actuator misalignment. They can be written as(19)D0=100cosα4cosβ4010cosα4sinβ4001sinα4,(20)ΔD=00Δα3ΔD14Δα100ΔD240Δα20Δα4cosα4,ΔD14=−Δα4sinα4cosβ4−Δβ4cosα4sinβ4,ΔD24=−Δα4sinα4sinβ4+Δβ4cosα4cosβ4.In view of (15) and (18), tracking error dynamics in (12) can be rewritten as(21)Jmω˙e=−ω×Jmω−Jmω˙r+δTKη+δTCψ−ω×δTδω−δTCδωe+D0+ΔDτ+d.To facilitate control system design, the following assumptions and lemmas are presented and will be used in the subsequent developments.Assumption 1.
The components of external disturbanced in (21) are assumed to be bounded by a set of unknown bounded constants, that is(22)di≤ρii=1,2,3.Assumption 2.
Due to physical limitations on the reaction wheels considered, the maximum output torque of each actuator output torque has the same limit valueτ¯,τ¯, that is [29](23)τ∈Ω=τiτ¯≤τi≤τ¯,i=1,2,3,4,where τ¯ and τ¯ represent the known saturation levels of reaction wheels.Assumption 3.
The uncertaintyΔD due to misalignment is an unknown but bounded matrix satisfying(24)ΔDF≤Δm,where Δm is a known positive constant.Remark 1.
According to (20), every element of actuator misalignment matrix ΔD is a function of misalignment angle errors which are small in practice, then the Frobenius norm of ΔD is bounded by a known quantity Δm. It is easy to find a feasible Δm in practice.Lemma 1.
For arbitrary positive constantε and variable η, the following inequality holds [30, 31](25)0≤η−ηtanhηε≤κε,with κ=e−κ+1 and κ=0.2785.


## 2.4. Control Problem Formulation

Given any initial attitude and angular velocity, the control objective can be stated as considering the flexible spacecraft attitude system described by (1)–(3), design a torque command τ such that the following goals are met in the presence of uncertain inertia parameters, external disturbance, unmeasured elastic vibration, actuator saturation (23), and even actuator misalignment:(1)
The attitude orientation and angular velocity tracking errors are driven to zero, or a small set containing the origin.(2)
The vibration induced by the maneuver rotation should be attenuated as soon as possible in the presence of parametric uncertainties, external disturbance, actuator saturation, and even actuator misalignment.


## 3. Robust Adaptive Constrained Controller Design

The proposed attitude control scheme for the flexible spacecraft is shown in Figure1, in which the controller is composed of an auxiliary design system to compensate the effect of actuator saturation and modal estimator to estimate unmeasured modal position and velocity, as well as path planning, which will be given in the following subsections.Figure 1
Structure of the closed-loop system.


### 3.1. Path Planning

From (2) and (3), it is known that attitude maneuvering would excite appendages’ vibration [8], which is closely related to the attitude angular acceleration. A sudden change in attitude signal, especially in the form of step signal, may cause a serious appendages’ vibration. Here, in order to achieve high performance of attitude control and attenuate the residual vibration, a novel path planning scheme based on quintic polynomial transition is developed, which is motivated by the previous study of robot trajectory planning in joint space [32]. With regard to the limit of the angular velocity and the angular acceleration, the maneuvering path is designed as follows:The angular acceleration signal of the path is planned as a quintic polynomial curve.(26)φt=a5t5+a4t4+a3t3+a2t2+a1t+a0.Then, parametersa5, a4, a3, a2, a1, and a0 should be selected appropriately to satisfy the following equality and inequality constraints.(27)φ0=φ0,φ˙0=0,φ¨0=0φtf=φf,φ˙tf=0,φ¨tf=0,(28)φ˙t≤φ˙max,φ¨≤φ¨max,where φ˙max and φ¨max are the permitted maximum angular velocity and acceleration, respectively.Using (27) yields(29)a0=φ0,a1=a2=0,a3=5a5tf23,a4=−5a5tf2.Angular velocity and acceleration with regard to (26) and (29) are obtained as(30)φt=a5t5−52tft4+53tf2t3+φ0,φ˙t=a55t4−10tft3+5tf2t2,φ¨t=a520t3−30tft2+10tf2t.With the notationτ=t/tf and a^5=a5tf5, (30) can be rewritten as(31)φt=a^5τ5−52τ4+53τ3+φ0,φ^˙t=a^55τ4−10τ3+5τ2=tfφ˙t,φ^¨t=a^520τ3−30τ2+10τ=tf2φ¨t.For notational convenience, we definefτ as(32)fτ=τ5−52τ4+53τ3,τ∈0,1.It can be easily verified that(33)0≤fτ≤f1=16,0≤f′τ≤f′12=516,f″12+36≤f″τ≤f″12−36,where(34)f″12+36=−0.9623,f″12−36=0.9623.Then, in view of (27) and (28), a^5 and tf satisfy(35)a^5=6φf−φ0,(36)tf≥φ^˙maxφ˙max=5a^516φ˙max,tf2≥φ^¨maxφ¨max=0.9623a^5φ¨max.Hence, angular path planning procedure can be described as follows:(1)
Determinea^5 from (35)(2)
Determinetf from (36)(3)
Determinea5, a4, a3, a2, a1, and a0 from (29)(4)
Obtain the angular velocity and angular acceleration according to (30).


### 3.2. Controller Design

To remove the hypothesis of the measurability of the modal position and velocity, an elastic modal estimator to supply their estimates is constructed as follows:(37)η^ψ^˙=Aη^ψ^−ABδωe−Bδω˙r,where η^, ψ^ are the estimates of modal variables, and eη=η−η^, eψ=ψ−ψ^ are their estimation errors.From (13) and (37), the response of eη, eψ can be algebraically arranged as(38)e˙ηe˙ψ=Aeηeψ.Since matrixA is a Hurwitz matrix, the estimation errors eη, eψ will converge to zero asymptotically.Considering the actuator saturation in (23), the actual control input τ=τ1,τ2,τ3,τ4T in (21) can be further defined as(39)τi=satτci=τ¯,τci>τ¯,τci,τ¯≤τci≤τ¯,τ¯,τci<τ¯,where τc=τc1,τc2,τc3,τc4T is the desired control input that will be designed in the subsequent developments.Then, (21) can be rewritten as(40)Jmω˙e=−ω×Jmω−Jmω˙r+δTKη+δTCψ−ω×δTδω−δTCδωe+D0+ΔDsatτc+d.Step 1. We start with (11) and (13) by considering ωe as the virtual control variable. Define the tracking error as(41)z=ωe−αc−ξ=ωe−zα−α−ξ,where αc is the output of the following first-order filter utilized to approximate the derivative of the virtual control α˙.(42)Γαα˙c+αc=α,where Γα=diagτα1,τα2,τα3 is the filter parameter matrix and the components τα1,i=1,2,3 are positive scalars to be selected. zα=αc−α denotes the estimation error of the filter in (42).The auxiliary signalξ in (41) is designed to compensate the effect of actuator saturation and can be produced by the following system:(43)J^mξ˙=−Kξξ+D0Δτ,where Kξ is a positive matrix to be chosen, Δτ denotes the difference between the applied control and the designed control input(44)Δτ=τ−τc=satτc−τc.The first Lyapunov candidate function is chosen as(45)V1=1−qe02+qevTqev+12η^Tψ^TK12K+C2CC2Iη^ψ^+12eηTeψTK22K+C2CC2Ieηeψ,where the positive definite matrices K1 and K2 are partitioned as(46)K1=k11I00k12I,K2=k21I00k22I.Using (11), (37), (38) and (41), the time derivative of (45) is given by(47)V˙1=xTωe−y−η^Tψ^Tk11CK−2k11K2k12Kk12Cη^ψ^−eηTeψTk21CK−2k21K2k22Kk22Ceηeψ=xTz+zα+α+ξ−y−η^Tψ^Tk11CK−2k11K2k12Kk12Cη^ψ^−eηTeψTk21CK−2k21K2k22Kk22Ceηeψ,where x and y are defined as(48)xT=qevT+k12ψ^TC−2k11η^TKδ−k11η^TC+2k12ψ^TδSRqeωd,y=k11η^TC+2k12ψ^TδRqeω˙d.Define the virtual controlα as(49)α=−K3x−ξ+sgnxx1y.Based on what is mentioned above, (47) be rewritten as(50)V˙1=xTz+xTzα−xTK3x−η^Tψ^Tk11CK−2k11K2k12Kk12Cη^ψ^−eηTeψTk21CK−2k21K2k22Kk22Ceηeψ.Step 2. Take the derivate of z left-multiplied by inertia matrix with respect to (40), then we have(51)Jmz˙=−ω×Jmω−Jmω˙r+δTKη+δTCψ−ω×δTδω−δTCδωe+D0satτc+ΔDsatτc+d−Jmα˙c−Jmξ˙.Although the inertia matrix is unknown for the system design, it can be observed that the inertia parameters appear linearly in (51). To isolate these parameters, a linear operator L⋅:R3→R3×6 acting on a=a1a2a3T is introduced as follows(52)La=a100a2a300a20a10a300a30a1a2.LetθmT=Jm,11,Jm,22,Jm,33,Jm,12,Jm,13,Jm,23, it follows that Jma=Laθm and then (51) can be rewritten as(53)Jmz˙=F1θm+Lξ˙θ˜m+δTKη+δTCψ−ω×δTδω−δTCδωe+ΔDsatτc+D0τc+d+Kξξ,with F1=−ω×Lω−Lω˙r−Lα˙c. θ˜m denotes the estimate error of θm and is defined as θ˜m=θ^m−θm.The design procedure can be summarized in the following theorem.Theorem 1.
Consider the flexible spacecraft system involving uncertain inertia parameters, external disturbance, unmeasured elastic vibration, actuator saturation, and even actuator misalignment. If the control law is designed by(54)τc=−D0+x+Kξξ+K4z−D0+F1θ^mb−D0+δTKη^+δTCψ^−ω×δTδω−δTCδωe−12D0+KδTKδz+CδTCδz−2D0+Δmτmtanh2Δmτmzε−D0+tanhzρ^,where tanhz=diagtanhzj/εdj, the adaptive control is selected as(55)θ^˙m=ProjΓ1F2Tz,ρ^˙=Γ2tanhzz−kρρ^,where F2=F1−Lξ˙, the projection operator Proj⋅ is defined in [33] to avoid the parameter drift problem, then the attitude orientation and angular velocity tracking errors are uniformly ultimately bounded.Proof.
Consider the composite Lyapunov functionV2 as(56)V2=V1+12zTJmz+12θ˜mTΓ1−1θ˜m+12ρ˜TΓ2−1ρ˜.
Substituting the aforementioned control torque (54), taking the derivative of the above Lyapunov function with regard to (50) and (53), it follows(57)V˙2=−xTK3x−zTK4z+xTzα−η^Tψ^Tk11CK−2k11K2k12Kk12Cη^ψ^−eηTeψTk21CK−2k21K2k22Kk22Ceηeψ+θ˜mTΓ1−1θ^˙mb−zTF2θ˜m+ρ˜TΓ2−1ρ^˙+zTd−zTtanhzρ^+zTΔDsatτc−2ΔmτmzTtanh2Δmτmzε+zTδTKeη−12KδzTKδz+zTδTCeψ−12CδzTCδz.
In view of the conclusion in [34], we have zα≤μ, then(58)−xTK3x+xTzα≤−xTK3−12Ix+12μ2.
The last two terms in (57) can be expanded as(59)zTδTKeη−12KδzTKδz≤12eηTeη,zTδTCeψ−12CδzTCδz≤12eψTeψ.
According to Lemma 1, we have(60)∑j=13zjρj≤∑j=13κdjεdj+zjtanhzjεdjρj=φTρ+zTtanhzρ≤φ22+ρ22+zTtanhzρ,where(61)φ=κεd1,κεd2,κεd3T,ρ=ρ1,ρ2,ρ3T.
Then, substituting the aforementioned adaptive law and considering the property of projection operator as well as the following inequality(62)−ρ˜Tρ^≤−12ρ˜2+12ρ2,gives(63)θ˜mTΓ1−1θ^˙m−zTF2θ˜m≤0,ρ˜TΓ2−1ρ^˙+zTd−zTtanhzρ^≤−kρ2ρ˜2+φ22+1+kρ2ρ2.
Assumption 2 dictates thatsatτc≤2τm, τm is the permitted maximum control torque. Combining Assumption 2 and Lemma 1 yields to(64)zTΔDsatτc−2ΔmτmzTtanh2Δmτmzε≤3κε.
From the above development, (57) can be further derived as(65)V˙2=−zTK4z−xTK3−12Ix−kρ2ρ˜2−η^Tψ^Tk11CK−2k11K2k12Kk12Cη^ψ^−eηTeψTk21CK−2k21K2k22Kk22Ceηeψ+φ22+1+kρ2ρ2+12μ2+3κε.
If parametersK1, K2, and K3 are selected such that(66)K¯1=k11CK−2k11K2k12Kk12C>0,K¯2=k21CK−2k21K2k22Kk22C>0,K¯3=K3−12I>0,then V˙2 can be further upper bounded by(67)V˙2≤−kV2+εt,where εt is given by(68)εt=φ22+1+kρ2ρ2+12μ2+3κε,which readily concludes that(69)0≤V2t≤εtk+V20−εtke−kt.
Hence,V2 is bounded. It follows the definition of V2 that the quaternion tracking error qev, angular velocity control error z, modal variables estimates η^T,ψ^TT, and parameter estimation error θ˜m are all bounded. Therefore, the controlled closed-loop system is uniformly ultimately bounded.
In addition, following the definition ofV2 also yields(70)maxqevTqev,12λminJzTz≤V2t≤εtk+V20−εtke−kt,such that quaternion tracking error qev and angular velocity control error z, respectively, converge to the following compact sets:(71)Ω1=qevqev≤εtk,Ω2=zz≤2εtkλminJ.
According to (41), the angular velocity tracking error ωe is also uniformly ultimately bounded. The convergence domain Ω1 and Ω2 can be adjusted by an explicit choice of design parameters.


## 3.1. Path Planning

From (2) and (3), it is known that attitude maneuvering would excite appendages’ vibration [8], which is closely related to the attitude angular acceleration. A sudden change in attitude signal, especially in the form of step signal, may cause a serious appendages’ vibration. Here, in order to achieve high performance of attitude control and attenuate the residual vibration, a novel path planning scheme based on quintic polynomial transition is developed, which is motivated by the previous study of robot trajectory planning in joint space [32]. With regard to the limit of the angular velocity and the angular acceleration, the maneuvering path is designed as follows:The angular acceleration signal of the path is planned as a quintic polynomial curve.(26)φt=a5t5+a4t4+a3t3+a2t2+a1t+a0.Then, parametersa5, a4, a3, a2, a1, and a0 should be selected appropriately to satisfy the following equality and inequality constraints.(27)φ0=φ0,φ˙0=0,φ¨0=0φtf=φf,φ˙tf=0,φ¨tf=0,(28)φ˙t≤φ˙max,φ¨≤φ¨max,where φ˙max and φ¨max are the permitted maximum angular velocity and acceleration, respectively.Using (27) yields(29)a0=φ0,a1=a2=0,a3=5a5tf23,a4=−5a5tf2.Angular velocity and acceleration with regard to (26) and (29) are obtained as(30)φt=a5t5−52tft4+53tf2t3+φ0,φ˙t=a55t4−10tft3+5tf2t2,φ¨t=a520t3−30tft2+10tf2t.With the notationτ=t/tf and a^5=a5tf5, (30) can be rewritten as(31)φt=a^5τ5−52τ4+53τ3+φ0,φ^˙t=a^55τ4−10τ3+5τ2=tfφ˙t,φ^¨t=a^520τ3−30τ2+10τ=tf2φ¨t.For notational convenience, we definefτ as(32)fτ=τ5−52τ4+53τ3,τ∈0,1.It can be easily verified that(33)0≤fτ≤f1=16,0≤f′τ≤f′12=516,f″12+36≤f″τ≤f″12−36,where(34)f″12+36=−0.9623,f″12−36=0.9623.Then, in view of (27) and (28), a^5 and tf satisfy(35)a^5=6φf−φ0,(36)tf≥φ^˙maxφ˙max=5a^516φ˙max,tf2≥φ^¨maxφ¨max=0.9623a^5φ¨max.Hence, angular path planning procedure can be described as follows:(1)
Determinea^5 from (35)(2)
Determinetf from (36)(3)
Determinea5, a4, a3, a2, a1, and a0 from (29)(4)
Obtain the angular velocity and angular acceleration according to (30).


## 3.2. Controller Design

To remove the hypothesis of the measurability of the modal position and velocity, an elastic modal estimator to supply their estimates is constructed as follows:(37)η^ψ^˙=Aη^ψ^−ABδωe−Bδω˙r,where η^, ψ^ are the estimates of modal variables, and eη=η−η^, eψ=ψ−ψ^ are their estimation errors.From (13) and (37), the response of eη, eψ can be algebraically arranged as(38)e˙ηe˙ψ=Aeηeψ.Since matrixA is a Hurwitz matrix, the estimation errors eη, eψ will converge to zero asymptotically.Considering the actuator saturation in (23), the actual control input τ=τ1,τ2,τ3,τ4T in (21) can be further defined as(39)τi=satτci=τ¯,τci>τ¯,τci,τ¯≤τci≤τ¯,τ¯,τci<τ¯,where τc=τc1,τc2,τc3,τc4T is the desired control input that will be designed in the subsequent developments.Then, (21) can be rewritten as(40)Jmω˙e=−ω×Jmω−Jmω˙r+δTKη+δTCψ−ω×δTδω−δTCδωe+D0+ΔDsatτc+d.Step 1. We start with (11) and (13) by considering ωe as the virtual control variable. Define the tracking error as(41)z=ωe−αc−ξ=ωe−zα−α−ξ,where αc is the output of the following first-order filter utilized to approximate the derivative of the virtual control α˙.(42)Γαα˙c+αc=α,where Γα=diagτα1,τα2,τα3 is the filter parameter matrix and the components τα1,i=1,2,3 are positive scalars to be selected. zα=αc−α denotes the estimation error of the filter in (42).The auxiliary signalξ in (41) is designed to compensate the effect of actuator saturation and can be produced by the following system:(43)J^mξ˙=−Kξξ+D0Δτ,where Kξ is a positive matrix to be chosen, Δτ denotes the difference between the applied control and the designed control input(44)Δτ=τ−τc=satτc−τc.The first Lyapunov candidate function is chosen as(45)V1=1−qe02+qevTqev+12η^Tψ^TK12K+C2CC2Iη^ψ^+12eηTeψTK22K+C2CC2Ieηeψ,where the positive definite matrices K1 and K2 are partitioned as(46)K1=k11I00k12I,K2=k21I00k22I.Using (11), (37), (38) and (41), the time derivative of (45) is given by(47)V˙1=xTωe−y−η^Tψ^Tk11CK−2k11K2k12Kk12Cη^ψ^−eηTeψTk21CK−2k21K2k22Kk22Ceηeψ=xTz+zα+α+ξ−y−η^Tψ^Tk11CK−2k11K2k12Kk12Cη^ψ^−eηTeψTk21CK−2k21K2k22Kk22Ceηeψ,where x and y are defined as(48)xT=qevT+k12ψ^TC−2k11η^TKδ−k11η^TC+2k12ψ^TδSRqeωd,y=k11η^TC+2k12ψ^TδRqeω˙d.Define the virtual controlα as(49)α=−K3x−ξ+sgnxx1y.Based on what is mentioned above, (47) be rewritten as(50)V˙1=xTz+xTzα−xTK3x−η^Tψ^Tk11CK−2k11K2k12Kk12Cη^ψ^−eηTeψTk21CK−2k21K2k22Kk22Ceηeψ.Step 2. Take the derivate of z left-multiplied by inertia matrix with respect to (40), then we have(51)Jmz˙=−ω×Jmω−Jmω˙r+δTKη+δTCψ−ω×δTδω−δTCδωe+D0satτc+ΔDsatτc+d−Jmα˙c−Jmξ˙.Although the inertia matrix is unknown for the system design, it can be observed that the inertia parameters appear linearly in (51). To isolate these parameters, a linear operator L⋅:R3→R3×6 acting on a=a1a2a3T is introduced as follows(52)La=a100a2a300a20a10a300a30a1a2.LetθmT=Jm,11,Jm,22,Jm,33,Jm,12,Jm,13,Jm,23, it follows that Jma=Laθm and then (51) can be rewritten as(53)Jmz˙=F1θm+Lξ˙θ˜m+δTKη+δTCψ−ω×δTδω−δTCδωe+ΔDsatτc+D0τc+d+Kξξ,with F1=−ω×Lω−Lω˙r−Lα˙c. θ˜m denotes the estimate error of θm and is defined as θ˜m=θ^m−θm.The design procedure can be summarized in the following theorem.Theorem 1.
Consider the flexible spacecraft system involving uncertain inertia parameters, external disturbance, unmeasured elastic vibration, actuator saturation, and even actuator misalignment. If the control law is designed by(54)τc=−D0+x+Kξξ+K4z−D0+F1θ^mb−D0+δTKη^+δTCψ^−ω×δTδω−δTCδωe−12D0+KδTKδz+CδTCδz−2D0+Δmτmtanh2Δmτmzε−D0+tanhzρ^,where tanhz=diagtanhzj/εdj, the adaptive control is selected as(55)θ^˙m=ProjΓ1F2Tz,ρ^˙=Γ2tanhzz−kρρ^,where F2=F1−Lξ˙, the projection operator Proj⋅ is defined in [33] to avoid the parameter drift problem, then the attitude orientation and angular velocity tracking errors are uniformly ultimately bounded.Proof.
Consider the composite Lyapunov functionV2 as(56)V2=V1+12zTJmz+12θ˜mTΓ1−1θ˜m+12ρ˜TΓ2−1ρ˜.
Substituting the aforementioned control torque (54), taking the derivative of the above Lyapunov function with regard to (50) and (53), it follows(57)V˙2=−xTK3x−zTK4z+xTzα−η^Tψ^Tk11CK−2k11K2k12Kk12Cη^ψ^−eηTeψTk21CK−2k21K2k22Kk22Ceηeψ+θ˜mTΓ1−1θ^˙mb−zTF2θ˜m+ρ˜TΓ2−1ρ^˙+zTd−zTtanhzρ^+zTΔDsatτc−2ΔmτmzTtanh2Δmτmzε+zTδTKeη−12KδzTKδz+zTδTCeψ−12CδzTCδz.
In view of the conclusion in [34], we have zα≤μ, then(58)−xTK3x+xTzα≤−xTK3−12Ix+12μ2.
The last two terms in (57) can be expanded as(59)zTδTKeη−12KδzTKδz≤12eηTeη,zTδTCeψ−12CδzTCδz≤12eψTeψ.
According to Lemma 1, we have(60)∑j=13zjρj≤∑j=13κdjεdj+zjtanhzjεdjρj=φTρ+zTtanhzρ≤φ22+ρ22+zTtanhzρ,where(61)φ=κεd1,κεd2,κεd3T,ρ=ρ1,ρ2,ρ3T.
Then, substituting the aforementioned adaptive law and considering the property of projection operator as well as the following inequality(62)−ρ˜Tρ^≤−12ρ˜2+12ρ2,gives(63)θ˜mTΓ1−1θ^˙m−zTF2θ˜m≤0,ρ˜TΓ2−1ρ^˙+zTd−zTtanhzρ^≤−kρ2ρ˜2+φ22+1+kρ2ρ2.
Assumption 2 dictates thatsatτc≤2τm, τm is the permitted maximum control torque. Combining Assumption 2 and Lemma 1 yields to(64)zTΔDsatτc−2ΔmτmzTtanh2Δmτmzε≤3κε.
From the above development, (57) can be further derived as(65)V˙2=−zTK4z−xTK3−12Ix−kρ2ρ˜2−η^Tψ^Tk11CK−2k11K2k12Kk12Cη^ψ^−eηTeψTk21CK−2k21K2k22Kk22Ceηeψ+φ22+1+kρ2ρ2+12μ2+3κε.
If parametersK1, K2, and K3 are selected such that(66)K¯1=k11CK−2k11K2k12Kk12C>0,K¯2=k21CK−2k21K2k22Kk22C>0,K¯3=K3−12I>0,then V˙2 can be further upper bounded by(67)V˙2≤−kV2+εt,where εt is given by(68)εt=φ22+1+kρ2ρ2+12μ2+3κε,which readily concludes that(69)0≤V2t≤εtk+V20−εtke−kt.
Hence,V2 is bounded. It follows the definition of V2 that the quaternion tracking error qev, angular velocity control error z, modal variables estimates η^T,ψ^TT, and parameter estimation error θ˜m are all bounded. Therefore, the controlled closed-loop system is uniformly ultimately bounded.
In addition, following the definition ofV2 also yields(70)maxqevTqev,12λminJzTz≤V2t≤εtk+V20−εtke−kt,such that quaternion tracking error qev and angular velocity control error z, respectively, converge to the following compact sets:(71)Ω1=qevqev≤εtk,Ω2=zz≤2εtkλminJ.
According to (41), the angular velocity tracking error ωe is also uniformly ultimately bounded. The convergence domain Ω1 and Ω2 can be adjusted by an explicit choice of design parameters.


## 4. Numerical Simulation

Numerical simulations have been carried out in this section to illustrate and verify the effectiveness of the proposed control scheme. The spacecraft is characterized by a nominal main body inertia matrix(72)J=35034328010410190kg⋅m2,and by the coupling matrix(73)δ=6.456371.278142.15629−1.256190.91756−1.672641.116872.48901−0.836741.23637−2.6581−1.12503kg1/2⋅m,respectively. Then matrix Jm is given by(74)Jm=303.9613−3.5930−9.6975−3.5930264.26387.8709−9.69757.8709180.5869kg⋅m2The first four elastic modes have been taken into account for the implemented spacecraft model resulting from the modal analysis of the structure, with natural frequency and damping presented in Table1.Table 1
Parameters of the flexible dynamics.
Natural frequency (rad/s)
Damping
Mode 1
1.0973
0.05
Mode 2
1.2761
0.06
Mode 3
1.6538
0.08
Mode 4
2.2893
0.025The maximum angular velocity and acceleration of the flexible spacecraft arevmax=0.0343rad/s and amax=0.0286rad/s2, respectively. Herein, we consider a three-axis large angle maneuver. The initial attitude in Euler angles is 0°30°45°T, and the corresponding initial attitude in quaternion is qT0=0.8924,−0.0990,0.2391,0.3696. The desired attitude is 30°45°60°T. The initial and desired angular velocity are chosen to be ωT0=0,0,0. Sample time is ts=0.01s, and the span of simulation time is from 0 to 200 s. In addition, the initial modal variables and its time derivative are given by(75)ηT0=0.01242,0.01584,−0.01749,0.01125,and η˙T0=0,0,0,0, respectively.To examine the robustness to external disturbance, simulation is done corresponding to the following periodic disturbance torque(76)dt=0.03cos0.01t+0.010.015sin0.02t+0.03cos0.025t0.03sin0.01t+0.01Nm.For the purpose of comparison, two different sets of simulation are conducted to demonstrate the effectiveness of the proposed control approach as follows:Case 1. Attitude control without considering the actuator saturation and even actuator misalignment. The robust adaptive control (RAC) law and corresponding adaptive law are derived in steps identical to those employed in Section 3.2 as(77)τc=−D0+x+K4z−D0+F1θ^mb−D0+δTKη^+δTCψ^−ω×δTδω−δTCδωe−12D0+KδTKδz+CδTCδz−D0+tanhzρ^,(78)θ^˙m=ProjΓ1F1Tz,ρ^˙=Γ2tanhzz−kρρ^.Case 2. Attitude control using constrained robust adaptive control (CRAC) law (54) and adaptive law (55), taking actuator saturation and even actuator misalignment into account explicitly.In the following simulations, the control and adaptation gains are provided in Table2. The maximum output torque of reaction wheel is assumed to be 10 Nm. For the overactuated reaction wheel with configuration misalignments, the normal assembling angles are α4=35.26deg and β4=45deg, with the misalignment angles of reaction wheel are, respectively, Δα=2,3,4,5Tdeg and Δβ=5,4,3,2Tdeg.Table 2
Design parameters for the different controllers.
Control schemes
Parameters and values
Proposed controller in (77) and (78)
k
11
=
k
12
=
0.1
,
k
21
=
4
,
k
22
=
5
,
K
3
=
0.55
I
3
,
K
4
=
0.1
I
3
,
Γ
α
=
0.01
I
3
,
Γ
1
=
0.01
I
6
,
Γ
2
=
0.01
I
3
,
k
ρ
=
0.001
,
ε
d
=
0.01
,
Proposed controller in (54) and (55)
k
11
=
k
12
=
0.1
,
k
21
=
4
,
k
22
=
5
,
K
3
=
0.55
I
3
,
K
4
=
0.1
I
3
,
K
ξ
=
0.3
I
3
,
Γ
α
=
0.01
I
3
,
Γ
1
=
0.01
I
6
,
Γ
2
=
0.01
I
3
,
k
ρ
=
0.001
,
ε
d
=
0.01
,
ε
=
0.01
,
τ
m
=
10
,The planned maneuvering path is illustrated in Figures2–4. For the comparison, the conventional path planning method based on bang-coast-bang (BCB) is also performed at the same simulation conditions. It is illustrated that the maneuver path utilizing planning scheme based on quantic polynomial transition is much smoother than the one by applying conventional method. Smooth angular acceleration variate will introduce the least level of vibration to flexible appendages.Figure 2
Time response of planned attitude path.Figure 3
Time response of planned angular velocity.Figure 4
Time response of planned angular acceleration.With the application of the proposed CRAC scheme with path planning in Case 2, the simulation results are summarized in Figures5–21. Since the attitude variables in terms of Euler angles are easier to understand than quaternion, the attitude tracking error in Euler angles denoted as eφ, eθ, and eϕ are also displayed in Figures 5 and 6.Figure 5
Time response of attitude tracking error in Euler angle.Figure 6
Enlarged version of Figure5.Figure 7
Time response of attitude tracking error in quaternion.Figure 8
Time response of angular velocity tracking error.Figure 9
Time response of estimated parameters of inertia in Case 1.Figure 10
Time response of estimated parameters of the product of inertia in Case 1.Figure 11
Time response of estimated parameters of inertia in Case 2.Figure 12
Time response of estimated parameters of the product of inertia in Case 2.Figure 13
Time response of vibration displacements and their estimates.Figure 14
Time response of vibration estimate errors.Figure 15
Time response of required control torqueux.Figure 16
Time response of required control torqueuy.Figure 17
Time response of required control torqueuz.Figure 18
Time response of output torqueτ1.Figure 19
Time response of output control torqueτ2.Figure 20
Time response of output control torqueτ3.Figure 21
Time response of output control torqueτ4.It can been seen from Figures5, 7, and 8 that it takes about 100 s to drive the tracking errors of attitude Euler angles, attitude quaternion, and angular velocity to their stable points. Through enlarged version of attitude tracking errors illustrated in Figure 6, it can be observed that steady errors of Euler angles are maintained within a range of less than 0.0006 rad. Figure 8 shows the response of angular velocity tracking error, and the steady errors are no more than 2.9e − 6 rad/s. Thus, the high-precision attitude control performance can be achieved despite the presence of parameters uncertainties, external disturbance, actuator saturation, and even actuator misalignments.The responses of estimated inertial parameters corresponding to update law of (55) and (78) are illustrated in Figures 9–12. It is clear that the convergence of these estimated parameters can be achieved, but not to the true values. That is because sufficient frequency components in the tracking error states are not guaranteed. In other words, the persistent excitation (PE) condition is not satisfied.The behavior of the modal displacements and their estimates are given in Figure13. It is noted that all the elastic vibrations and their rates approach zero at time 70 s. It can be observed that not only the vibrations excited by the attitude maneuver are effectively suppressed but also the modal displacements can be well estimated by the modal observer whose performance is explicitly demonstrated in Figure 14. The steady observation errors of modal observer in (37) are tabulated in Table 3.Table 3
Steady observation errors of modal observer.
Steady observation errors
Mode 1η1−η^1
7.381e − 6
Mode 2η2−η^2
1.61e − 7
Mode 3η3−η^3
0
Mode 4η4−η^4
3.92e − 7Furthermore, the time responses of total required control torques and reaction wheel output torques are shown in Figures15–17 and Figures 18–21, respectively. The output torque of each reaction wheel is within the saturation limitation, that is, ±10Nm, which is because the proposed control scheme addresses the control saturation constraints explicitly.For comparison, the system is also controlled by RAC scheme in Case 1. When it is applied under the same initial conditions and control parameters, the control performance is also illustrated in Figures5–21 where it is denoted as “RAC.”The steady errors of Euler angles depicted in Figure6 are 0.005 rad. As demonstrated in Figure 8, it can be observed that severe oscillations of the angular velocity tracking error occur in transient phase and its steady values are no more than 4.1e − 5 rad/s, which are larger than those of the proposed CRAC scheme in Case 2. Thus, although the attitude rotational maneuver using RAC scheme is successfully performed, the control performances are much poorer than those using the proposed CRAC scheme owing to not considering the actuator misalignment.Further, as is shown in Figures15–21, the saturation phenomenon of the reaction wheel torques using RAC scheme is not addressed.Thus, it can be concluded that desired performance of the system can be achieved by the proposed control scheme subject to multiple undesired constraints including uncertain inertia parameters, external disturbance, unmeasured elastic vibration, actuator saturation, and even actuator misalignment.


## 5. Conclusion

Taking uncertain inertia parameters, external disturbance, unmeasured elastic vibration, actuator saturation, and even actuator misalignment into account simultaneously, this paper investigates the problem of attitude maneuver and vibration suppression of flexible spacecrafts. Firstly, a new way of path planning, based on quintic polynomial transition, is designed to suppress the flexible appendages’ vibration effectively. Then, by utilizing the inherent physical properties of flexible appendages, a modal observer is constructed to supply elastic modal estimates. Plus, an adaptive law is derived so that the requirements to know the prior knowledge of parameter uncertainties and the upper bound of external disturbance is eliminated. In addition, an auxiliary design system is introduced to compensate the actuator saturation effect, and a compensation term is synthesized and integrated into the controller to handle the actuator misalignment. Finally, the stability is rigorously proved and the simulation results demonstrated the effectiveness and robustness of proposed control scheme.



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*Source: 3129834-2018-09-05.xml*