# Reaction Wheel Installation Deviation Compensation for Overactuated Spacecraft with Finite-Time Attitude Control

Authors: Aihua Zhang; Jianfei Ni; Hamid Reza Karimi
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

A novel attitude tracking control scheme is presented for overactuated spacecraft to address the attitude stabilization problem in presence of reaction wheel installation deviation, external disturbance and uncertain mass of moment inertia. An adaptive sliding mode control technique is proposed to track the uncertainty. A Lyapunov-based analysis shows that the compensation control law can guarantee that the desired attitude trajectories are followed in finite-time. The key feature of the proposed control strategy is that it globally asymptotically stabilizes the system, even in the presence of reaction wheel installation deviation, external disturbances, and uncertain mass of moment inertia. The attitude track performance using the proposed finite-time compensation control is evaluated through a numerical example.


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## Body

## 1. Introduction

In present, nearly all of the highly accurate slewing maneuvers necessitate the use of nonlinear differential equations for the kinematics and dynamics during the control system design [1]. However, the attitude tracking problem is further complicated by the external disturbance and uncertain mass of moment inertia. To address these issues, there have been several important developments in the design of feedback control laws for spacecraft maneuvering. A number of control design approaches using adaptive control [2, 3], sliding mode control [4–7], H∞ [8, 9], optimal control [10–13], and data driven control [14–16] have been proposed. However, few of them focus on the reaction wheel installation deviation that are of great theoretical and practical interest. In fact, the installation deviation is a widespread phenomenon, such as the actuator misalignment which is limited by the installation technique or generated by materials deforms the vehicle violent vibration during the launching process. In the area of actuator misalignment compensation, there currently exist few unified frameworks for the design of simple control structures.Several solutions to actuator installation deviation have been presented in the literature [17–20]. In [17], the authors presented a general adaptive tracking attitude controller design framework for a spacecraft subject to the actuator installation minor angle deviation. In [18], an adaptive attitude tracking method is proposed to compensate the actuator misalignment of nearly 15 degree. And in [19], a novel algorithm is employed precisely to estimate the information, such as installation angle of wheel and CMG alignments. And then the controller design can be on for the estimation information. Moreover, another recent paper in [20] proposed an adaptive control approach for satellite formation flying, in which backstepping technique is used to synthesize a controller to handle thrust magnitude error and misalignment. However, the torque is different between thruster and reaction wheel, one is literal and the other is time-variable. That is to say that this control strategy is not suitable for reaction wheel installation deviation compensation for overactuated spacecraft attitude control.Treating the uncertain mass of moment inertia caused by the reaction wheel misalignment is another impossibly avoided problem. In practice, in order to ensure the reliability of on-orbit spacecraft operation, especially under high altitude sever external environment, overactuatation is widely employed to guarantee the control system reliability service. And finite-time is meanwhile necessary for time critical missions. As a result, more and more investigations also have focused on attitude control design with finite-time convergence. In [21–23], the finite-time control technique was applied to design an attitude controller. Feng et al. [24] proposed a terminal sliding mode controller to solve the singular problem for a second-order nonlinear dynamic system. A terminal sliding mode and the Chebyshev neural network were used in [25] to guarantee that the attitude manoeuvre was accomplished in finite time, even in the presence of an unknown inertia matrix, external disturbances, and control input constraints. Furthermore, two robust sliding mode controllers were proposed in [26] to realize attitude tracking in finite-time. Similar finite-time fault tolerant controllers for spacecrafts were investigated in [27–29].This work focuses on developing a control scheme to perform attitude compensation for an overactuated spacecraft with reaction wheel installation deviation, external disturbances, and uncertain inertia parameters. More specifically, the attitude tracking error is required to be zero in finite time. The proposed approach is illustrated in Figure1. The compensation control module is added to the output of the nominal controller to compensate for the reaction wheel misalignment, disturbances, and uncertain moment of inertia. The proposed scheme solves a difficult problem of reliable and high accuracy attitude tracking control in finite time that rejects external disturbances and, at the same time, compensates for actuator misalignment and system uncertainties so that the control objective is met.Figure 1
Structure of the attitude compensation controller.The remainder of this paper is organized as follows. In Section2, we summarize the mathematical model for the rigid spacecraft attitude and control problem. A compensation control solution with the misalignment, disturbance, and mass moment of inertia is presented in Section 3. Simulation results are presented in Section 4. Some conclusions are given in Section 5.


## 2. Mathematical Model and Problem Formulation

The notation adopted throughout this paper is introduced as follows. The symbol∥·∥ denotes the standard Euclidean norm or its induced norm; the symbol ∥·∥∞ denotes the infinite norm of a vector or matrix. For any given matrix A∈Rp×q with full row rank, A† denotes its pseudoinverse.


### 2.1. Dynamic Model of Rigid Spacecraft

Consider a rigid space system described by the following attitude kinematics and dynamics equations [30]:
(1)q˙=12(q×+q0I3)ω,(2)q˙0=-12qTω,(3)Jω˙+ω×Jω=u+d,whereω∈R3 is the angular velocity of a body-fixed reference frame expressed in the body-fixed reference frame, J∈R3×3 (positive and definite) is the total inertia matrix of the spacecraft, u=[u1u2u3]T∈R3 denotes the combined control torque produced by the actuators, and d(t)=[d1d2d3]T∈R3 denotes the external disturbance torque from the environment, which is assumed to be unknown but bounded; q0, q are the scalar and vector components of the unit quaternion, respectively, with q=[q1q2q3]T∈R3, satisfying the constraint q02+qTq=1; I3 represents the identity matrix with proper dimensions, and for ∀a=[a1a2a3]T, a× denotes a skew-symmetric matrix, more precisely,
(4)a×=[0-a3a2a30-a1-a2a10].


### 2.2. Reaction Wheel Configuration with Installation Deviation

For orbiting spacecraft, loosely speaking, they have more than three reaction wheels aligned with the spacecraft body axes. However, in practice, the configuration of actuators will never be perfect; that is, to say, whether due to finite manufacturing tolerances or warping of the spacecraft structure during launch, some alignment errors can always exist. Thus, in this section, the reaction wheels misalignment is taken into consideration; the faulty dynamics can be described by(5)Jω=-ω×Jω+(D+ΔD)τ+d,whereD∈R3×N denotes the actuator distribution matrix, ΔD denotes the actuator distribution matrix induced by misalignment, and τ=[τ1,…,τN]T∈RN denotes the actual output torque of the N reaction wheels.Due to the rotation of the payload or the existence of the flywheel installation deviation, the moment of inertiaJ is uncertain but positive definite symmetric matrices and record J=J0+ΔJ, where J0 denotes the nominal rotational inertia and ΔJ denotes the uncertain rotational inertia. Here set  0<∥ΔJ∥≤∥J∥≤Jmax<∞ and Jmax is a positive constant.


### 2.3. Attitude Tracking Model

Assume that the desired attitude to be followed is described with a desired frame𝒯 with respect to ℐ. It is specified by the desired unit quaternion Qd=(qd0,qdT)∈R×R3. The desired angular velocity is denoted by ωd∈R3. Let the error quaternion Qe=(e0,eT)∈R×R3 denote the attitude between ℬ and 𝒯, and let ωe∈R3 represent the corresponding error angular velocity. One has
(6)ωe=ω-R~ωd,whereR~∈R3×3 denote the corresponding rotation matrix that brings 𝒯 onto ℬ, and
(7)R~=(e02-eTe)I3+2eeT-2e0e×.With (1)–(5), the attitude tracking error dynamics is given by:
(8)Jω˙e+(ω˙e+R~ωd)×J(ωe+R~ωd)-J(ωe×R~ωd-R~ωd)=(D+ΔD)E(t)τ(t)+d(t),(9)e˙0=-12eTωe,(10)e˙=12(e×+e0I3)ωe.


### 2.4. Control Objective

The control objective of this work can be stated as considering the uncertain attitude tracking system (8)–(10) and design a control law to guarantee that the attitude tracking error converges to zero in finite-time, even in the presence of actuator misalignment, uncertain inertia matrix, and external disturbance d(t).We present now the main results of this study.


## 2.1. Dynamic Model of Rigid Spacecraft

Consider a rigid space system described by the following attitude kinematics and dynamics equations [30]:
(1)q˙=12(q×+q0I3)ω,(2)q˙0=-12qTω,(3)Jω˙+ω×Jω=u+d,whereω∈R3 is the angular velocity of a body-fixed reference frame expressed in the body-fixed reference frame, J∈R3×3 (positive and definite) is the total inertia matrix of the spacecraft, u=[u1u2u3]T∈R3 denotes the combined control torque produced by the actuators, and d(t)=[d1d2d3]T∈R3 denotes the external disturbance torque from the environment, which is assumed to be unknown but bounded; q0, q are the scalar and vector components of the unit quaternion, respectively, with q=[q1q2q3]T∈R3, satisfying the constraint q02+qTq=1; I3 represents the identity matrix with proper dimensions, and for ∀a=[a1a2a3]T, a× denotes a skew-symmetric matrix, more precisely,
(4)a×=[0-a3a2a30-a1-a2a10].


## 2.2. Reaction Wheel Configuration with Installation Deviation

For orbiting spacecraft, loosely speaking, they have more than three reaction wheels aligned with the spacecraft body axes. However, in practice, the configuration of actuators will never be perfect; that is, to say, whether due to finite manufacturing tolerances or warping of the spacecraft structure during launch, some alignment errors can always exist. Thus, in this section, the reaction wheels misalignment is taken into consideration; the faulty dynamics can be described by(5)Jω=-ω×Jω+(D+ΔD)τ+d,whereD∈R3×N denotes the actuator distribution matrix, ΔD denotes the actuator distribution matrix induced by misalignment, and τ=[τ1,…,τN]T∈RN denotes the actual output torque of the N reaction wheels.Due to the rotation of the payload or the existence of the flywheel installation deviation, the moment of inertiaJ is uncertain but positive definite symmetric matrices and record J=J0+ΔJ, where J0 denotes the nominal rotational inertia and ΔJ denotes the uncertain rotational inertia. Here set  0<∥ΔJ∥≤∥J∥≤Jmax<∞ and Jmax is a positive constant.


## 2.3. Attitude Tracking Model

Assume that the desired attitude to be followed is described with a desired frame𝒯 with respect to ℐ. It is specified by the desired unit quaternion Qd=(qd0,qdT)∈R×R3. The desired angular velocity is denoted by ωd∈R3. Let the error quaternion Qe=(e0,eT)∈R×R3 denote the attitude between ℬ and 𝒯, and let ωe∈R3 represent the corresponding error angular velocity. One has
(6)ωe=ω-R~ωd,whereR~∈R3×3 denote the corresponding rotation matrix that brings 𝒯 onto ℬ, and
(7)R~=(e02-eTe)I3+2eeT-2e0e×.With (1)–(5), the attitude tracking error dynamics is given by:
(8)Jω˙e+(ω˙e+R~ωd)×J(ωe+R~ωd)-J(ωe×R~ωd-R~ωd)=(D+ΔD)E(t)τ(t)+d(t),(9)e˙0=-12eTωe,(10)e˙=12(e×+e0I3)ωe.


## 2.4. Control Objective

The control objective of this work can be stated as considering the uncertain attitude tracking system (8)–(10) and design a control law to guarantee that the attitude tracking error converges to zero in finite-time, even in the presence of actuator misalignment, uncertain inertia matrix, and external disturbance d(t).We present now the main results of this study.


## 3. Finite-Time Attitude Compensation Control

For the proposed control approach shown in Figure1, the nominal control power and the compensation control effort are presented in this section. First, a finite-time sliding mode surface is proposed. Then, based on the finite-time sliding mode surface, a compensation controller is synthesized and added to the nominal controller to guarantee the global asymptotic stability of the resulting closed-loop attitude tracking system with finite-time convergence.


### 3.1. Finite-Time Sliding Mode Surface Design

We first introduce some lemmas which will be utilized in the subsequent control development and analysis.Lemma 1 (see [31]).
Ifp∈(0,1), then the following inequality holds for any vector x=(x1,x2,…,xn)T∈Rn:
(11)∑i=1n|xi|1+p≥(∑i=1n|xi|2)(1+p)/2.Lemma 2 (see [32]).
Suppose thatV(x) is a C1 smooth positive-definite function such that
(12)V˙(x)+λ1V(x)+λ2Vβ(x)≤0,
whereλ1∈R+, λ2∈R+, β∈R+ and 0<β<1. Then for any initial value x(0)=x0, it follows that V(x(t))=0 for all the time t≥tF1,
(13)tF1≤1λ1(1-β)lnλ1V1-β(x0)+λ2λ2.
To this end, in this work, a sliding mode surface is introduced as(14)s=ωe+μ1e+μ2sgn(e)r,
wheres=[s1s2s3]T∈R3, μ1>0, μ2>0, and 0<γ<1 are the design parameters and sgn(e) is the sign function defined by
(15)sgn(e)r=(|e1|rsgn(e1),|e2|rsgn(e2),|e3|rsgn(e3))T.Theorem 3.
If an controlleru(t) is appropriately designed to let the states reach the sliding surface s, then it has e(t)≡0, e0(t)≡1, and ωe(t)≡0 for all the t≥tF1.Proof.
From the sliding mode theory [33], it is known that once the state trajectories of the attitude tracking system reach the sliding surface, that is, s=0, it follows that
(16)ωe=-μ1e-μ2sgn(e)r.
Consider a candidate Lyapunov function as(17)V1=(1-e0)2+eTe.
Because the inequality∑i=03|ei|r+1≥(∑i=03|ei|2)(r+1)/2 holds for 0<r<1,  it is obtained from (15) that
(18)V˙1=-2e˙0=eTωe=-μ1eTe-μ2∑i=13|ei|r+1
which implies thatV˙1=0 if and only if e=0. Thus, V1 is really a Lyapunov function such that the signal e will converge to zero and, accordingly, e0 tends to ±1 as t→∞ by using the constraint in (18). Note that the equilibrium point (e0,e)=(-1,0) is not a stable equilibrium point [34]. Then, by Lemma 1, we obtain
(19)V˙1≤-μ1eTe-μ2(eTe)(r+1)/2.
Becausee=(-1,0)T is not the stable equilibrium point, the signal e will converge to zero. Thus, limt→∞e0(t)=1 can be obtained from the constraint eTe+e02=1. There exists a finite time t-≥0 such that e0(t)>0 for t≥t-. Then, for t≥t-, one has
(20)(1-e0)2=2e02+eTe-2e0=2e0(e0-1)+eTe-2≤eTe.
Then,(21)V1≤2eTe.
Using (21), (19) can be further bounded by
(22)V˙1≤-12μ1V1-(12)(r+1)/2μ2V1(r+1)/2.
Using0.5<(r+1)/2<1 and Lemma 2, one has V1(t)≡0 for all t≥tF1. According to definition of V1(t) in (17), e0(t)≡1, e(t)≡0, and ωe(t)≡0 for all t≥tF1 are concluded. Thereby, the proof is completed here.


### 3.2. Attitude Compensation Controller Design

Considering the reaction wheel installation deviation and external disturbance, it is obtained from the sliding surface (14) that
(23)Js˙=Dτ+ΔDτ+L-βqe-0.5J˙s.BecauseJ is unknown but bounded, then J˙=0 is established. Then, L can be represented to be bounded by ∥L∥≤α0+α1∥ω∥+α2∥ω∥2, where αi, i=1,2,3 are positive constants [35].In order to facilitate analysis and proof, firstly, defineκ=λmin(DDT), 3∥ΔD∥∞∥D†∥∞=ε<1, and D† is the pseudoinverse of D. Now, we are ready to present the main result in Theorem 4.Theorem 4.
Considering the uncertainty attitude tracking dynamics described by (5) with actuator misalignment ΔD and external disturbance torque d(t), design an attitude compensation control law as
(24)τ=τnom(t)+τadp(t)+τmis(t),
where(25)τnom(t)=-k1βDT∥qe∥s∥s∥-KDTs∥s∥2,(26)τadp(t)=DT(-k^3-k^4∥ω∥-k^5∥ω∥2)s∥s∥,(27)τmis(t)=-(π^1-1)φ(t)D†s∥s∥∞,
whereK∈R+ is control parameter and k1 is carefully chosen such that k1κ-1>0; π^1 is the estimate of π1=1/(1-δ); k^3 is the estimate of k3=α0/κ; k^4 is the estimate of k4=α1/κ; k^5 is the estimate of k5=α2/κ. Moreover, π^˙1, k^i, i=3,4,5 are adaptively updated by k^˙2=(∥sT∥/ℓ4)k^i, k^˙3=∥ω∥∥s∥/ℓ5, k^˙4=∥ω∥2∥s∥/ℓ6, and π^˙1=l1φ(t)∥s∥∞, respectively. Then the system states reach the sliding mode surface s(t)=0 in finite-time for any initial state Q(0) and ω(0).Proof.
Whens≠0, consider a candidate Lyapunov function:
(28)V2=12sTJs+(1-ε)2l1π~12+12l3k~32+12l4k~42+12l5k~52,
whereπ~1=π1-π^1, k~i=ki-k^i, i=3,4,5, and l1, l3, l4, l5 are the positive constants.
Calculating the time-derivative ofV2, it yields
(29)V˙2≤sTJs˙-(1-ε)l1π~π^˙-l3k~3k^˙3-l4k~4k^˙4-l5k^˙3=s(Dτnom-βqe)+s(Dτadp+L)+s(Dτmis+ΔDτ)-(1-ε)l1π~π^˙-l3k~3k^˙3-l4k~4k^˙4-l5k^˙3.
With (25), it follows that
(30)s(Dτnom-βqe)≤sDτnom+β∥qe∥∥s∥≤-κk1β∥s∥∥qe∥+β∥qe∥∥s∥-Kκ=-(k1κβ-β)∥qe∥-κK≤-κK.
In the same way, with (26), the following equality is yielded:
(31)s(Dτadp+L)≤sDτadp+∥s∥∥L∥≤-κ(k^3-α0κ)∥s∥-κ(k^4-α1κ)∥s∥∥ω∥-κ(k^5-α2κ)∥s∥∥ω∥2=κk~3∥s∥+κk~4∥s∥∥ω∥+κk~5∥s∥∥ω∥2.
Defineφ(t)=∥D†(τnom+τadp)∥∞. If the choice of the control gains is such that 3∥ΔD∥∞∥D-1∥∞=ε<1, using the inequality xy≤3∥x∥∞∥y∥∞>0 for any vector x,y∈Rn and applying (29) lead to
(32)s(Dτmis+ΔDτ)=sΔD(τnom+τadp+τmis)+sDτmis≤3∥ΔD∥∞∥D†∥∞φ(t)∥s∥∞+sΔDτmis+sDτmis≤εφ(t)∥s∥∞-(π^1-1)φ(t)∥s∥∞+sΔD[-(π^1-1)φ(t)D†s∥s∥∞]=(1-ε)π~1φ(t)∥s∥∞.
Substituting (30)–(32) into (29), consequently, it follows that
(33)V˙≤-κK.
And then,(34)∫0tV˙2(μ)dμ≤-κK∫0tdμ;thatis,V2(t)-V2(0)≤-κKt.
Due toV2(t)≥0, solving (34) leads to V2(t)≡0, for t≥tF2,
(35)tF2≤-V2(t)-V2(0)κK.
Then, it can be concluded that the system states reach the surfaces(t)=0 in finite time. Thereby, the proof is completed here.Remark 5.
The controller (24) includes three parts: τadp(t) is used to compensate for system uncertainty caused by the external disturbance and moment inertia, τmis(t) is used to accommodate actuator misalignment, and τnom(t) is the nominal control.Theorem 6.
Consider the attitude tracking system given by (5), (9), and (10). If the control scheme (24) is implemented, then the attitude tracking maneuver can be accomplished in a finite time tF=tF1+tF2; that is, ωe(t)≡0 and e(t)≡0 are guaranteed for all the time t≥tF.Proof.
It is obtained from Theorem4 that all the states of the attitude tracking system reach the sliding mode surface s(t)=0 in finite-time tF2 and maintain the motion state on the slide mode surface. Furthermore, from Theorem 3, it is obtained that once the system state reaches the slide mode surface (14) the system state can reach the equilibrium point (e0,e)=(1,0) in finite time tF1. Therefore, for any initial state Q(0) and ω(0), the desired attitude trajectory can be followed in a finite time tF;that is,e(t)≡0, e0(t)≡1, and ωe(t)≡0 are achieved for all the time t≥tF. Thereby, the proof is completed here.


## 3.1. Finite-Time Sliding Mode Surface Design

We first introduce some lemmas which will be utilized in the subsequent control development and analysis.Lemma 1 (see [31]).
Ifp∈(0,1), then the following inequality holds for any vector x=(x1,x2,…,xn)T∈Rn:
(11)∑i=1n|xi|1+p≥(∑i=1n|xi|2)(1+p)/2.Lemma 2 (see [32]).
Suppose thatV(x) is a C1 smooth positive-definite function such that
(12)V˙(x)+λ1V(x)+λ2Vβ(x)≤0,
whereλ1∈R+, λ2∈R+, β∈R+ and 0<β<1. Then for any initial value x(0)=x0, it follows that V(x(t))=0 for all the time t≥tF1,
(13)tF1≤1λ1(1-β)lnλ1V1-β(x0)+λ2λ2.
To this end, in this work, a sliding mode surface is introduced as(14)s=ωe+μ1e+μ2sgn(e)r,
wheres=[s1s2s3]T∈R3, μ1>0, μ2>0, and 0<γ<1 are the design parameters and sgn(e) is the sign function defined by
(15)sgn(e)r=(|e1|rsgn(e1),|e2|rsgn(e2),|e3|rsgn(e3))T.Theorem 3.
If an controlleru(t) is appropriately designed to let the states reach the sliding surface s, then it has e(t)≡0, e0(t)≡1, and ωe(t)≡0 for all the t≥tF1.Proof.
From the sliding mode theory [33], it is known that once the state trajectories of the attitude tracking system reach the sliding surface, that is, s=0, it follows that
(16)ωe=-μ1e-μ2sgn(e)r.
Consider a candidate Lyapunov function as(17)V1=(1-e0)2+eTe.
Because the inequality∑i=03|ei|r+1≥(∑i=03|ei|2)(r+1)/2 holds for 0<r<1,  it is obtained from (15) that
(18)V˙1=-2e˙0=eTωe=-μ1eTe-μ2∑i=13|ei|r+1
which implies thatV˙1=0 if and only if e=0. Thus, V1 is really a Lyapunov function such that the signal e will converge to zero and, accordingly, e0 tends to ±1 as t→∞ by using the constraint in (18). Note that the equilibrium point (e0,e)=(-1,0) is not a stable equilibrium point [34]. Then, by Lemma 1, we obtain
(19)V˙1≤-μ1eTe-μ2(eTe)(r+1)/2.
Becausee=(-1,0)T is not the stable equilibrium point, the signal e will converge to zero. Thus, limt→∞e0(t)=1 can be obtained from the constraint eTe+e02=1. There exists a finite time t-≥0 such that e0(t)>0 for t≥t-. Then, for t≥t-, one has
(20)(1-e0)2=2e02+eTe-2e0=2e0(e0-1)+eTe-2≤eTe.
Then,(21)V1≤2eTe.
Using (21), (19) can be further bounded by
(22)V˙1≤-12μ1V1-(12)(r+1)/2μ2V1(r+1)/2.
Using0.5<(r+1)/2<1 and Lemma 2, one has V1(t)≡0 for all t≥tF1. According to definition of V1(t) in (17), e0(t)≡1, e(t)≡0, and ωe(t)≡0 for all t≥tF1 are concluded. Thereby, the proof is completed here.


## 3.2. Attitude Compensation Controller Design

Considering the reaction wheel installation deviation and external disturbance, it is obtained from the sliding surface (14) that
(23)Js˙=Dτ+ΔDτ+L-βqe-0.5J˙s.BecauseJ is unknown but bounded, then J˙=0 is established. Then, L can be represented to be bounded by ∥L∥≤α0+α1∥ω∥+α2∥ω∥2, where αi, i=1,2,3 are positive constants [35].In order to facilitate analysis and proof, firstly, defineκ=λmin(DDT), 3∥ΔD∥∞∥D†∥∞=ε<1, and D† is the pseudoinverse of D. Now, we are ready to present the main result in Theorem 4.Theorem 4.
Considering the uncertainty attitude tracking dynamics described by (5) with actuator misalignment ΔD and external disturbance torque d(t), design an attitude compensation control law as
(24)τ=τnom(t)+τadp(t)+τmis(t),
where(25)τnom(t)=-k1βDT∥qe∥s∥s∥-KDTs∥s∥2,(26)τadp(t)=DT(-k^3-k^4∥ω∥-k^5∥ω∥2)s∥s∥,(27)τmis(t)=-(π^1-1)φ(t)D†s∥s∥∞,
whereK∈R+ is control parameter and k1 is carefully chosen such that k1κ-1>0; π^1 is the estimate of π1=1/(1-δ); k^3 is the estimate of k3=α0/κ; k^4 is the estimate of k4=α1/κ; k^5 is the estimate of k5=α2/κ. Moreover, π^˙1, k^i, i=3,4,5 are adaptively updated by k^˙2=(∥sT∥/ℓ4)k^i, k^˙3=∥ω∥∥s∥/ℓ5, k^˙4=∥ω∥2∥s∥/ℓ6, and π^˙1=l1φ(t)∥s∥∞, respectively. Then the system states reach the sliding mode surface s(t)=0 in finite-time for any initial state Q(0) and ω(0).Proof.
Whens≠0, consider a candidate Lyapunov function:
(28)V2=12sTJs+(1-ε)2l1π~12+12l3k~32+12l4k~42+12l5k~52,
whereπ~1=π1-π^1, k~i=ki-k^i, i=3,4,5, and l1, l3, l4, l5 are the positive constants.
Calculating the time-derivative ofV2, it yields
(29)V˙2≤sTJs˙-(1-ε)l1π~π^˙-l3k~3k^˙3-l4k~4k^˙4-l5k^˙3=s(Dτnom-βqe)+s(Dτadp+L)+s(Dτmis+ΔDτ)-(1-ε)l1π~π^˙-l3k~3k^˙3-l4k~4k^˙4-l5k^˙3.
With (25), it follows that
(30)s(Dτnom-βqe)≤sDτnom+β∥qe∥∥s∥≤-κk1β∥s∥∥qe∥+β∥qe∥∥s∥-Kκ=-(k1κβ-β)∥qe∥-κK≤-κK.
In the same way, with (26), the following equality is yielded:
(31)s(Dτadp+L)≤sDτadp+∥s∥∥L∥≤-κ(k^3-α0κ)∥s∥-κ(k^4-α1κ)∥s∥∥ω∥-κ(k^5-α2κ)∥s∥∥ω∥2=κk~3∥s∥+κk~4∥s∥∥ω∥+κk~5∥s∥∥ω∥2.
Defineφ(t)=∥D†(τnom+τadp)∥∞. If the choice of the control gains is such that 3∥ΔD∥∞∥D-1∥∞=ε<1, using the inequality xy≤3∥x∥∞∥y∥∞>0 for any vector x,y∈Rn and applying (29) lead to
(32)s(Dτmis+ΔDτ)=sΔD(τnom+τadp+τmis)+sDτmis≤3∥ΔD∥∞∥D†∥∞φ(t)∥s∥∞+sΔDτmis+sDτmis≤εφ(t)∥s∥∞-(π^1-1)φ(t)∥s∥∞+sΔD[-(π^1-1)φ(t)D†s∥s∥∞]=(1-ε)π~1φ(t)∥s∥∞.
Substituting (30)–(32) into (29), consequently, it follows that
(33)V˙≤-κK.
And then,(34)∫0tV˙2(μ)dμ≤-κK∫0tdμ;thatis,V2(t)-V2(0)≤-κKt.
Due toV2(t)≥0, solving (34) leads to V2(t)≡0, for t≥tF2,
(35)tF2≤-V2(t)-V2(0)κK.
Then, it can be concluded that the system states reach the surfaces(t)=0 in finite time. Thereby, the proof is completed here.Remark 5.
The controller (24) includes three parts: τadp(t) is used to compensate for system uncertainty caused by the external disturbance and moment inertia, τmis(t) is used to accommodate actuator misalignment, and τnom(t) is the nominal control.Theorem 6.
Consider the attitude tracking system given by (5), (9), and (10). If the control scheme (24) is implemented, then the attitude tracking maneuver can be accomplished in a finite time tF=tF1+tF2; that is, ωe(t)≡0 and e(t)≡0 are guaranteed for all the time t≥tF.Proof.
It is obtained from Theorem4 that all the states of the attitude tracking system reach the sliding mode surface s(t)=0 in finite-time tF2 and maintain the motion state on the slide mode surface. Furthermore, from Theorem 3, it is obtained that once the system state reaches the slide mode surface (14) the system state can reach the equilibrium point (e0,e)=(1,0) in finite time tF1. Therefore, for any initial state Q(0) and ω(0), the desired attitude trajectory can be followed in a finite time tF;that is,e(t)≡0, e0(t)≡1, and ωe(t)≡0 are achieved for all the time t≥tF. Thereby, the proof is completed here.


## 4. Numerical Simulation Results

### 4.1. Reaction Wheel Configuration

To demonstrate the effectiveness and performance of the proposed compensation control scheme, numerical simulations have been carried out using the rigid spacecraft system (3) and (6) in conjunction with the developed compensation control law (24). The spacecraft is activated by four reaction wheels with a limited control torque umax=0.1 N·m. The configuration of those four actuators is shown in Figure 2. αi=35.26° and βi=45° are the nominal alignment angles, i=1,2,3,4. Δαi and Δβi are the misalignment angles.Figure 2
Configuration of four reaction wheels.With the configuration shown in Figure2, the relation between the actual output torque of reaction wheel and the total torque acting on the spacecraft is to be calculated as
(36)u(t)=τ1(cos(α1+Δα1)sin(β1+Δβ1)-sin(α1+Δα1)cos(α1+Δα1)cos(β1+Δβ1))+τ2(-cos(α2+Δα2)cos(β2+Δβ2)-sin(α2+Δα2)cos(α2+Δα2)sin(β2+Δβ2))+τ3(-cos(α3+Δα3)sin(β3+Δβ3)-sin(α3+Δα3)-cos(α3+Δα3)cos(β3+Δβ3))+τ4(cos(α4+Δα4)cos(β4+Δβ4)-sin(α4+Δα4)-cos(α4+Δα4)sin(β4+Δβ4)).Although the misalignment angles exist due to finite-manufacture technique and vehicle vibration, those anglesΔαi, Δβi  (i=1,2,3,4) are small values. They can be approximated by
(37)cosΔαi≈1,sinΔαi≈Δαi,cosΔβi≈1,sinΔβi≈Δβi,sinΔαisinΔβi≈0(i=1,2,3,4).Hence, (36) can be re written as
(38)u(t)=Dτ(t)+ΔDτ(t),whereD and ΔD=(ΔD1,ΔD2,ΔD3,ΔD4) are calculated as
(39)D=(cosα1sinβ1-cosα2sinβ2-cosα3sinβ3cosα4sinβ4-sinα1-sinα2-sinα3-sinα4cosα1cosβ1cosα2cosβ2-cosα3cosβ3-cosα4cosβ4)=(33-33-3333-33-33-33-333333-33-33),ΔD1=(Δβ1cosα1cosβ1-Δα1sinα1sinβ1-Δα1cosα1-Δβ1cosα1sinβ1-Δβ1sinα1cosβ1)ΔD2=(-Δβ2cosα2cosβ2+Δα2sinα2sinβ2-Δα2cosα2-Δβ2cosα2sinβ2-Δβ2sinα2cosβ2)ΔD3=(-Δβ3cosα3cosβ3+Δα3sinα3sinβ3-Δα3cosα3Δβ3cosα3sinβ3+Δβ3sinα3cosβ3)ΔD4=(Δβ4cosα4cosβ4-Δα4sinα4sinβ4-Δα4cosα4Δβ4cosα4sinβ4+Δα4sinα4cosβ4).Remark 7.
Theorem4 gives out the sufficient condition 3∥ΔD∥∞∥D†∥∞=ε<1 of efficacious processes on reaction wheel installation deviation ΔD for guaranteeing the attitude controller (24). Particulary, according to the definition ΔD and matrix norm, then ∥ΔD∥∞≤maxi=1,2,3θi, where
(40)θ1=∑i=14|Δαisinαisinβi|+∑i=14|Δβicosαicosβi|=66∑i=14|Δαi|+33∑i=14|Δβi|,θ2=∑i=14|Δαicosαi|=63∑i=14|Δαi|,θ3=∑i=14|Δαisinαicosβi|+∑i=14|Δβicosαisinβi|=66∑i=14|Δαi|+33∑i=14|Δβi|.
On the other hand, inequality3∥ΔD∥∞∥D†∥∞=ε<1 means that  ∥ΔD∥∞=ε/(3∥D†∥∞)<1/(3∥D†∥∞). Therefore, the establishment conditions maxi=1,2,3,4θi<1/3∥D†∥∞=0.2566 rad of Theorem 4 from (40) can be obtained, that is to say, the installation deviation angle of any two reaction wheels is not larger than 0.2566 rad. Relying on this, this largest installation deviation angle, that is maxi=1,2,3,4θi = 14.7021°, of the reaction wheel installation structure is considered in this paper.


### 4.2. Simulation Results

The nominal inertia matrix of the considered spacecraft is specified by [36]
(41)J0=(353-1.53282-1.5230)kg·m2,(42)ΔJ=(1+e-0.1t+2ϑ(t-10)-4ϑ(t-20))diag(3,2,1)which incorporated into the model, whereϑ(·) is defined as ϑ(t≥0)=1 and ϑ(t<0)=0. External disturbance d(t) is chosen as [35]
(43)d(t)=(∥ω∥2+0.05)(sin0.8t,cos0.5t,cos0.3t)T.The reaction wheel misalignment angleΔαi  (i=1,2,3,4) can be selected randomly   between -4.5°~+4.5°, and Δβi  (i=1,2,3,4) can be selected randomly  between -5.5°~+5.5°.In this simulation, spacecraft initial parameter is set as follows: initial angular velocityωi(0)=[0.1-0.1-0.05]°/s, i=1,2,3; initial attitude quaternary qi(0)=[0.181-0.2870.792-0.524]T, i=0,1,2,3; the corresponding initial roll angle, pitch angle, and yaw angle are set 0.2°, −0.4° and −0.3° respectively. The control gains are selected by the following list: β=0.7, K=0.05, κ=1.05, ki=1.5  (i=1,3,4,5), as li=1.5  (i=1,3,4,5); moreover, the initial of the adaptive update laws are π^˙1(0)=1.25, k^˙3(0)=0.68, k^˙4(0)=0.42, k^˙5(0)=0.22.To demonstrate the effectiveness of the proposed misalignment compensation and disturbance rejection scheme, a spacecraft is numerically simulated using the proposed control compensation strategy (24).We see in Figures3–8 the controller managed to stabilize the origin equilibrium point in 30 seconds with great pointing accuracy. Indeed, since the knowledge of spacecraft inertia parameters was not required and an implicit integral item was incorporated in the control law design, external disturbance effect on the attitude control performance can be compensated efficiently, and also great robustness to system uncertainties, such as misalignment, can be guaranteed.Figure 3
Time responses of angular velocity.Figure 4
Time response of satellite attitude angle.Figure 5
Time response of control torque.Figure 6
Time response of control torque for reaction wheels.Figure 7
Time responses of quaternion.Figure 8
Time response of sliding mode surface.We can see in Figures3 and 4 the time responses of angle velocity and attitude angle; the proposed control scheme surely realized the high precision stable control in the presence of external disturbance, uncertain moment of inertia, and reaction wheel misalignment, and the pointing accuracy is superior to 0.01°; the attitude stable precision is superior to 0.001°/s. Meanwhile, from Figures 5 and 6, we can see that for compensating the misalignment and other uncertainties, the designed control command of control redundancy configuration for 4 reaction wheels τ is allocated to the three-actual-output torque u, and then the purpose of compensation, attitude high precision control is realized. In addition, the finite-time control validity is shown in Figure 8. And from Figure 8, we can see that the spacecraft attitude control system status has realized the tracking control at tF=30.5. Thereinto, the spacecraft attitude has arrived at slide mode surface s at tF1=25.9; afterwards, the statuses converge to equilibrium point at tF2=5.4 under the normal control τnom(t). The same validity of finite-time attitude compensation control strategy proposed in this paper can be further proved from the time response of the quaternion as shown in Figure 7.From the above illustrated simulation results, it is shown that the proposed scheme can accomplish the attitude stabilization in finite-time in presence of time-varying external disturbances, uncertain inertial parameters, and even reaction wheel installation deviation.


## 4.1. Reaction Wheel Configuration

To demonstrate the effectiveness and performance of the proposed compensation control scheme, numerical simulations have been carried out using the rigid spacecraft system (3) and (6) in conjunction with the developed compensation control law (24). The spacecraft is activated by four reaction wheels with a limited control torque umax=0.1 N·m. The configuration of those four actuators is shown in Figure 2. αi=35.26° and βi=45° are the nominal alignment angles, i=1,2,3,4. Δαi and Δβi are the misalignment angles.Figure 2
Configuration of four reaction wheels.With the configuration shown in Figure2, the relation between the actual output torque of reaction wheel and the total torque acting on the spacecraft is to be calculated as
(36)u(t)=τ1(cos(α1+Δα1)sin(β1+Δβ1)-sin(α1+Δα1)cos(α1+Δα1)cos(β1+Δβ1))+τ2(-cos(α2+Δα2)cos(β2+Δβ2)-sin(α2+Δα2)cos(α2+Δα2)sin(β2+Δβ2))+τ3(-cos(α3+Δα3)sin(β3+Δβ3)-sin(α3+Δα3)-cos(α3+Δα3)cos(β3+Δβ3))+τ4(cos(α4+Δα4)cos(β4+Δβ4)-sin(α4+Δα4)-cos(α4+Δα4)sin(β4+Δβ4)).Although the misalignment angles exist due to finite-manufacture technique and vehicle vibration, those anglesΔαi, Δβi  (i=1,2,3,4) are small values. They can be approximated by
(37)cosΔαi≈1,sinΔαi≈Δαi,cosΔβi≈1,sinΔβi≈Δβi,sinΔαisinΔβi≈0(i=1,2,3,4).Hence, (36) can be re written as
(38)u(t)=Dτ(t)+ΔDτ(t),whereD and ΔD=(ΔD1,ΔD2,ΔD3,ΔD4) are calculated as
(39)D=(cosα1sinβ1-cosα2sinβ2-cosα3sinβ3cosα4sinβ4-sinα1-sinα2-sinα3-sinα4cosα1cosβ1cosα2cosβ2-cosα3cosβ3-cosα4cosβ4)=(33-33-3333-33-33-33-333333-33-33),ΔD1=(Δβ1cosα1cosβ1-Δα1sinα1sinβ1-Δα1cosα1-Δβ1cosα1sinβ1-Δβ1sinα1cosβ1)ΔD2=(-Δβ2cosα2cosβ2+Δα2sinα2sinβ2-Δα2cosα2-Δβ2cosα2sinβ2-Δβ2sinα2cosβ2)ΔD3=(-Δβ3cosα3cosβ3+Δα3sinα3sinβ3-Δα3cosα3Δβ3cosα3sinβ3+Δβ3sinα3cosβ3)ΔD4=(Δβ4cosα4cosβ4-Δα4sinα4sinβ4-Δα4cosα4Δβ4cosα4sinβ4+Δα4sinα4cosβ4).Remark 7.
Theorem4 gives out the sufficient condition 3∥ΔD∥∞∥D†∥∞=ε<1 of efficacious processes on reaction wheel installation deviation ΔD for guaranteeing the attitude controller (24). Particulary, according to the definition ΔD and matrix norm, then ∥ΔD∥∞≤maxi=1,2,3θi, where
(40)θ1=∑i=14|Δαisinαisinβi|+∑i=14|Δβicosαicosβi|=66∑i=14|Δαi|+33∑i=14|Δβi|,θ2=∑i=14|Δαicosαi|=63∑i=14|Δαi|,θ3=∑i=14|Δαisinαicosβi|+∑i=14|Δβicosαisinβi|=66∑i=14|Δαi|+33∑i=14|Δβi|.
On the other hand, inequality3∥ΔD∥∞∥D†∥∞=ε<1 means that  ∥ΔD∥∞=ε/(3∥D†∥∞)<1/(3∥D†∥∞). Therefore, the establishment conditions maxi=1,2,3,4θi<1/3∥D†∥∞=0.2566 rad of Theorem 4 from (40) can be obtained, that is to say, the installation deviation angle of any two reaction wheels is not larger than 0.2566 rad. Relying on this, this largest installation deviation angle, that is maxi=1,2,3,4θi = 14.7021°, of the reaction wheel installation structure is considered in this paper.


## 4.2. Simulation Results

The nominal inertia matrix of the considered spacecraft is specified by [36]
(41)J0=(353-1.53282-1.5230)kg·m2,(42)ΔJ=(1+e-0.1t+2ϑ(t-10)-4ϑ(t-20))diag(3,2,1)which incorporated into the model, whereϑ(·) is defined as ϑ(t≥0)=1 and ϑ(t<0)=0. External disturbance d(t) is chosen as [35]
(43)d(t)=(∥ω∥2+0.05)(sin0.8t,cos0.5t,cos0.3t)T.The reaction wheel misalignment angleΔαi  (i=1,2,3,4) can be selected randomly   between -4.5°~+4.5°, and Δβi  (i=1,2,3,4) can be selected randomly  between -5.5°~+5.5°.In this simulation, spacecraft initial parameter is set as follows: initial angular velocityωi(0)=[0.1-0.1-0.05]°/s, i=1,2,3; initial attitude quaternary qi(0)=[0.181-0.2870.792-0.524]T, i=0,1,2,3; the corresponding initial roll angle, pitch angle, and yaw angle are set 0.2°, −0.4° and −0.3° respectively. The control gains are selected by the following list: β=0.7, K=0.05, κ=1.05, ki=1.5  (i=1,3,4,5), as li=1.5  (i=1,3,4,5); moreover, the initial of the adaptive update laws are π^˙1(0)=1.25, k^˙3(0)=0.68, k^˙4(0)=0.42, k^˙5(0)=0.22.To demonstrate the effectiveness of the proposed misalignment compensation and disturbance rejection scheme, a spacecraft is numerically simulated using the proposed control compensation strategy (24).We see in Figures3–8 the controller managed to stabilize the origin equilibrium point in 30 seconds with great pointing accuracy. Indeed, since the knowledge of spacecraft inertia parameters was not required and an implicit integral item was incorporated in the control law design, external disturbance effect on the attitude control performance can be compensated efficiently, and also great robustness to system uncertainties, such as misalignment, can be guaranteed.Figure 3
Time responses of angular velocity.Figure 4
Time response of satellite attitude angle.Figure 5
Time response of control torque.Figure 6
Time response of control torque for reaction wheels.Figure 7
Time responses of quaternion.Figure 8
Time response of sliding mode surface.We can see in Figures3 and 4 the time responses of angle velocity and attitude angle; the proposed control scheme surely realized the high precision stable control in the presence of external disturbance, uncertain moment of inertia, and reaction wheel misalignment, and the pointing accuracy is superior to 0.01°; the attitude stable precision is superior to 0.001°/s. Meanwhile, from Figures 5 and 6, we can see that for compensating the misalignment and other uncertainties, the designed control command of control redundancy configuration for 4 reaction wheels τ is allocated to the three-actual-output torque u, and then the purpose of compensation, attitude high precision control is realized. In addition, the finite-time control validity is shown in Figure 8. And from Figure 8, we can see that the spacecraft attitude control system status has realized the tracking control at tF=30.5. Thereinto, the spacecraft attitude has arrived at slide mode surface s at tF1=25.9; afterwards, the statuses converge to equilibrium point at tF2=5.4 under the normal control τnom(t). The same validity of finite-time attitude compensation control strategy proposed in this paper can be further proved from the time response of the quaternion as shown in Figure 7.From the above illustrated simulation results, it is shown that the proposed scheme can accomplish the attitude stabilization in finite-time in presence of time-varying external disturbances, uncertain inertial parameters, and even reaction wheel installation deviation.


## 5. Conclusions and Future Works

Considering the spacecraft issues about reaction misalignment, external disturbances, and parameters uncertainty, in this paper, a finite-time adaptive attitude compensation control has been proposed. A quantitative installation deviation angle analysis has been done and given out the value range of the reaction wheel misalignment angle. In the end the system stability and engineering practical value have been discussed from the perspective of theory and engineering. Numerical simulation of this novel control strategy was also presented to confirm the advantages and improvements over existing controllers. The case of actuator misalignment mentioned in Section4 had only discussed for four reaction wheel configuration, but this compensation control scheme is suitable for more than that reaction wheel number. Moreover, the actuator faults have not been considered. The latter case should be as one of subjects for future research. Meanwhile, the method optimal control approach combined robust control [37, 38] also can be applied in this field.



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*Source: 268904-2013-12-16.xml*