# Adaptive Fixed-Time 6-DOF Coordinated Control of Multiple Spacecraft Formation Flying with Input Quantization

Authors: Shiyu Wang; Ruixia Liu; Lihua Wen
Journal: Complexity
 (2020)

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## Abstract

This paper investigates the fixed-time coordinated control problem of six-degree-of-freedom (6-DOF) dynamic model for multiple spacecraft formation flying (SFF) with input quantization, where the communication topology is assumed directed. Firstly, a new multispacecraft nonsingular fixed-time terminal sliding mode vector is derived by using neighborhood state information. Secondly, a hysteretic quantizer is utilized to quantify control force and torque. Utilizing such a quantizer not only can reduce the required communication rate but also can eliminate the control chattering phenomenon induced by the logarithmic quantizer. Thirdly, a 6-DOF fixed-time coordinated control strategy with adaptive tuning laws is proposed, such that the practical fixed-time stability of the controlled system is ensured in the presence of both upper bounds of unknown external disturbances. It theoretically proves that the relative tracking errors of attitude and position can converge into the regions in a fixed time. Finally, a numerical example is exploited to show the usefulness of the theoretical results.


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## Body

## 1. Introduction

The recent decades have seen an ever increasing research interest in the coordinated control problem of spacecraft formation flying (SFF) due to its successful applications in the space industry such as atmosphere monitoring of the Earth, deep space exploration, and spacecraft on-orbit maintenance [1–5]. As is well known, the coordinated control of attitude and orbit are two equally important technologies. It is essential to achieve the desired attitude and position simultaneously for SFF mission [6, 7]. Owning to the dynamical coupling between orbit motion and attitude motion, these two motions can be considered as a whole six-degree-of-freedom (6-DOF) motion. Recently, the 6-DOF coordinated control of SFF has attracted considerable research attention [8, 9]. However, the control strategies proposed in the aforementioned literature can only guarantee asymptotic stability of the controlled systems [10–13].For coordinated control problems of SFF, fast convergence performance is an important requirement [14–18]. In contrast to asymptotic stabilization controllers, the finite-time stabilization controllers can provide a faster response and better disturbance-rejection ability [19–22]. Therefore, the finite-time controllers have been developed in spacecraft formation control [23–25]. Even though the finite-time control methods can ensure the controlled systems finite-time stabilization, the convergence time relies on the information of initial system states, which gives rise to difficulties in practical applications [26]. To cope with this constraint, the fixed-time stable concept was applied to study finite-time controller design, in that the convergence time is upper bounded regardless of initial system states [27–29]. To date, fixed-time control strategies have been used for various control systems [30, 31] but less attention has been paid to fixed-time 6-DOF coordinated control problem for SFF, especially for external disturbances with unknown upper bounds. Another significant issue in multiple SFF task is that the interspacecraft communication links are not always bidirectional, such as in the unidirectional spacecraft laser communication system. However, in some of the existing results, the coordinated control issue is investigated based on an assumption that the communication topology is undirected.On another research frontier, networked control systems (NCSs) as an active field of research have been applied successfully in various modern complicated engineering processes [13, 32–35] such as unmanned vehicles, nuclear power stations, and aerospace engineering systems. In modern low-cost plug-and-play small spacecraft formation systems, the functional components connected by wireless networked media [36–39]. It is quite common that when the signal is transmitted between the control and actuator module via wireless networks, the SFF systems unavoidably suffer from quantization errors caused by quantization behavior which will degrade the control performance or even lead to instability [40–43]. Thus, it is needed to propose new controllers for SFF where the signal quantization is taken into consideration. Although some research attention has been centered on the quantized control problem of SFF, there is still no result available that considers 6-DOF coordinated control of multiple spacecraft formation in the presence of quantized input control signal. The complexity of the multiple spacecraft formation coordinated control task makes the quantized fixed-time coordinated control a serious challenge.In this paper, we are motivated to deal with the problem of fixed-time 6-DOF adaptive coordinated control for multiple SFF with input quantization under directed communication topology. The main contributions of this paper are highlighted as follows: (1) the communication topology among follower spacecraft is described by a directed graph, which will bring more challenges than the case that the communication topology among follower spacecraft is described by an undirected graph. (2) A novel multispacecraft nonsingular FTTSM based on a 6-DOF dynamic model is designed, on which each spacecraft converges to its desired states while keeping synchronization with other formation spacecraft. (3) A fixed-time adaptive coordinate control strategy is derived to compensate for the effects of hysteretic quantizer and external disturbances on the control performance and guarantee the practical fixed-time stability of the controlled system.The rest of this paper is organized as follows: in section2, the modelling and preliminaries are presented. In section 3, a multispacecraft nonsingular fixed-time terminal sliding mode vector is designed. In section 4, a fixed-time adaptive coordinated control scheme is proposed. An illustrative example and a conclusion are given in Sections 5 and 6, respectively.


## 2. Modelling and Preliminaries

### 2.1. 6-DOF Dynamic Model

The 6-DOF dynamic model of spacecraft formation is represented as follows [8]:(1)x˙1i=Λx˙1ix2i,Gfix˙2i+Cx2i+Nx1i+τi=ui,i=1,2,…,n,where(2)x1i=ρiqi,x2i=ρ˙iωi,Λx˙1i=I3×303×303×3Tqi2,Gfi=mfiI3×303×303×3Jfi,Cx2i=Cn0ρ˙iωi×Jfiωi,Nx1i=Nρi,n0,R03×1,τi=Fdizi,ui=ufiuti,Tqi=−qviTq0iI3×3+qvi×,Cn0=2n00−10100000,Nρi,n0,R=−n˙0yi−n02xi−2μR3xin˙0xi−n02yi+μR3yiμR3zi,where superscript i stands for the ith follower spacecraft; ρi=xiyiziT represents the relative position vector from the ith follower spacecraft to the leader spacecraft; ωi∈ℝ3 denotes the angular velocity; qi∈ℝ4 is the quaternion defined as qi=q0iqviT, where n0∈ℝ represents angular velocity of the virtual leader spacecraft; q0i is the scalar part and qvi is the vector part; mfi∈ℝ denotes the mass; Jfi∈ℝ3×3 is the inertia matrix; ufi∈ℝ3 represents the control force; uit∈ℝ3 is the control torque; Fdi∈ℝ3 is disturbance force; and zi=ℓi−TiGT, where ℓi∈ℝ3 is disturbance torque and TiGT∈ℝ3 is the gravity gradient torque. The notation ı× for the vector ı=ı1ı2ı3T represents the skew-symmetric matrix as follows:(3)l×=0−l3l2l30−l1−l2l10.It is worth to mention that the attitude and orbit are mutually coupled byTiGT∈ℝ3, which is given as(4)TiGT=3μR^fi×JfiR^fixi2+R+yi2+zi23/2,where R^fi∈ℝ3 is the position unit vector. Since TiGT is much smaller compared with control torque, TiGT is always treated as a disturbance.We define the following error states:(5)e1i=ρi−ρdiqei,e2i=ρ˙eiωei=ρ˙i−ρ˙idiωi−Rqeiωdi,where qei is error quaternion defined as qei=q0eiqveiTT=qei⊗q¯di; Rqei is the rotation matrix from the ith follower spacecraft’s reference frame to its body-fixed frame; and ρdi, ρ.di, ωdi, and qdi are the desired position, desired velocity, desired angular velocity, and desired attitude, respectively.Then the 6-DOF relative error dynamic model of SFF can be expressed by(6)e˙1i=Λe˙1ie2i,Gfie˙2i+Ce2i+Ne1i+τi=ui,where(7)Λe˙1i=I3×303×303×3Tqei2,Ne1i=Nρi,n0,R−ρ¨di03×1,Ce2i=Cn0ρ˙i−ωi×Jfiωi+Jfiωei×ωdi−Rqeiω˙di.


### 2.2. Graph Theory

It is supposed that the information flow amongn follower spacecraft is described by a directed graph G=V,χ,A, where V=V1,V2,…,Vn represents the set of nodes, χ⊆V×V represents the set of edges, and Vi,Vj∈χ represents if and only if node Vi can receive the information of node Vj. In spacecraft 6-DOD coordinated control application, Vi,Vj∈χ represents only the jth spacecraft can obtain the ith spacecraft’s states information. A=aij∈ℝn×n denotes the weighted adjacency matrix of the graph G with entries(8)aij>0,ifVi,Vj∈χ,aij=0,otherwise,where aij is the nonnegative element of A, which denotes communication quality between the ith spacecraft and jth spacecraft. It is noticeable that self-edges are not allowed, meaning that aii=0.The in-degree matrix of the graphG is D with entries(9)D=diagd1,d2,…,dn,where(10)di=∑j=1naij=∑j∈χiaij,i=1,2,…,n.The Laplacian matrixL∈ℝn×n of the graph G is [44](11)L=D−A.


### 2.3. Hysteretic Quantizer

To eliminate the control chattering phenomenon induced by logarithmic quantizer, a hysteretic quantizer is used to quantify control torque and force in this paper, which is similar to [45]. It can be expressed by(12)qut≜uJsgnu,ifuJ1+δ<u≤uJ1−δ,u˙J<0,or,uJ<u≤uJ1−δ,u˙J>0,uJ1+δsgnu,ifuj<u≤uJ1−δ,u˙J<0,or,uJ1−δ≤u≤uJ1+δ1−δ,u˙J>0,0,if0≤u<umin1+δ,u˙J<0,or,umin1+δ≤u≤umin,u˙J>0,qut−,othercase,where uJ=ρ1−iumin,J=1,2,…,n with 0<ρ<1, umin>0, and δ=1−ρ/1+ρ. qut is in the set U=0,±ui,±ui1+δ. The map of the hysteresis quantizer qut for u>0 is illustrated in Figure 1.Figure 1
Map ofqut for u>0.Remark 1.
The parameterρ can be termed as a measure of quantization density. From the definition of ρ, we can see that the smaller parameter ρ is, the coarser the hysteretic quantizer becomes [45]. Therefore, the design of parameter ρ should be based on a criterion to guarantee the control performance with a smaller quantization density. In addition, compared with a traditional logarithmic quantizer, the hysteretic quantizer (10) can avoid oscillations by adding additional quantization levels.


### 2.4. Preliminaries

For deriving the 6-DOF fixed-time coordinated controller, the lemmas are made as follows.Lemma 1.
The hysteretic quantizerqut is decomposed into two parts as(13)qut=Duut+Qt,where Du and Qt satisfy(14)1−δ≤Du≤1+δ,Qt≤umin.
The proof of Lemma1 is similar to Theorem 1 in [45].Lemma 2 (see [44]).
IfG is a directed graph with N nodes, then all the eigenvalues of the weighted Laplace matrix L have nonnegative real part.Lemma 3 (see [46]).
For any matrixM∈ℝm×m, N∈ℝn×n, X∈ℝm×m, and Y∈ℝn×n, then the following equalities hold:(1)
M⊗NX⊗Y=MX⊗NY(2)
If the matricesM and N are invertible, then M⊗N−1=M−1⊗N−1(3)
If the eigenvalues ofM are λ1,…,λm and the eigenvalues of N are μ1,…,μn, then the eigenvalues of M⊗N can be expressed by λiμji=1,…,m;j=1,…,nLemma 4 (see [47]).
For anyx,y∈ℝ, if ν∈ℝ+ and ν>1, then(15)x+yν≤2ν−1xν+yν.Lemma 5 (see [48]).
Ifxi∈ℝ,i=1,2,…,n, and 0<p≤1, then(16)∑i=1nxip≤∑i=1nxip≤n1−p∑i=1nxip.Lemma 6 (see [47]).
Consider the nonlinear system given by(17)x˙=fx,t,f0,t=0,x∈ℝn.
Suppose that there exists a Lyapunov functionVx that satisfies the following condition:(18)V˙x≤−αVxp+βVxgk+υ,x∈Uo,where α, β, p, g∈ℝ+, pk<1, gk>1, and 0<υ<∞. Then, the origin of system (17) is practical fixed-time stable and the residual set of the solution satisfies(19)limt⟶Tx|Vx≤minα−1/pυ1−θk1/kp,β−1/gυ1−θk1/kg,where θ is a scalar satisfying 0<θ≤1. The setting time is bounded by(20)T≤1αkθk1−pk+1βkθkgk−1.


## 2.1. 6-DOF Dynamic Model

The 6-DOF dynamic model of spacecraft formation is represented as follows [8]:(1)x˙1i=Λx˙1ix2i,Gfix˙2i+Cx2i+Nx1i+τi=ui,i=1,2,…,n,where(2)x1i=ρiqi,x2i=ρ˙iωi,Λx˙1i=I3×303×303×3Tqi2,Gfi=mfiI3×303×303×3Jfi,Cx2i=Cn0ρ˙iωi×Jfiωi,Nx1i=Nρi,n0,R03×1,τi=Fdizi,ui=ufiuti,Tqi=−qviTq0iI3×3+qvi×,Cn0=2n00−10100000,Nρi,n0,R=−n˙0yi−n02xi−2μR3xin˙0xi−n02yi+μR3yiμR3zi,where superscript i stands for the ith follower spacecraft; ρi=xiyiziT represents the relative position vector from the ith follower spacecraft to the leader spacecraft; ωi∈ℝ3 denotes the angular velocity; qi∈ℝ4 is the quaternion defined as qi=q0iqviT, where n0∈ℝ represents angular velocity of the virtual leader spacecraft; q0i is the scalar part and qvi is the vector part; mfi∈ℝ denotes the mass; Jfi∈ℝ3×3 is the inertia matrix; ufi∈ℝ3 represents the control force; uit∈ℝ3 is the control torque; Fdi∈ℝ3 is disturbance force; and zi=ℓi−TiGT, where ℓi∈ℝ3 is disturbance torque and TiGT∈ℝ3 is the gravity gradient torque. The notation ı× for the vector ı=ı1ı2ı3T represents the skew-symmetric matrix as follows:(3)l×=0−l3l2l30−l1−l2l10.It is worth to mention that the attitude and orbit are mutually coupled byTiGT∈ℝ3, which is given as(4)TiGT=3μR^fi×JfiR^fixi2+R+yi2+zi23/2,where R^fi∈ℝ3 is the position unit vector. Since TiGT is much smaller compared with control torque, TiGT is always treated as a disturbance.We define the following error states:(5)e1i=ρi−ρdiqei,e2i=ρ˙eiωei=ρ˙i−ρ˙idiωi−Rqeiωdi,where qei is error quaternion defined as qei=q0eiqveiTT=qei⊗q¯di; Rqei is the rotation matrix from the ith follower spacecraft’s reference frame to its body-fixed frame; and ρdi, ρ.di, ωdi, and qdi are the desired position, desired velocity, desired angular velocity, and desired attitude, respectively.Then the 6-DOF relative error dynamic model of SFF can be expressed by(6)e˙1i=Λe˙1ie2i,Gfie˙2i+Ce2i+Ne1i+τi=ui,where(7)Λe˙1i=I3×303×303×3Tqei2,Ne1i=Nρi,n0,R−ρ¨di03×1,Ce2i=Cn0ρ˙i−ωi×Jfiωi+Jfiωei×ωdi−Rqeiω˙di.


## 2.2. Graph Theory

It is supposed that the information flow amongn follower spacecraft is described by a directed graph G=V,χ,A, where V=V1,V2,…,Vn represents the set of nodes, χ⊆V×V represents the set of edges, and Vi,Vj∈χ represents if and only if node Vi can receive the information of node Vj. In spacecraft 6-DOD coordinated control application, Vi,Vj∈χ represents only the jth spacecraft can obtain the ith spacecraft’s states information. A=aij∈ℝn×n denotes the weighted adjacency matrix of the graph G with entries(8)aij>0,ifVi,Vj∈χ,aij=0,otherwise,where aij is the nonnegative element of A, which denotes communication quality between the ith spacecraft and jth spacecraft. It is noticeable that self-edges are not allowed, meaning that aii=0.The in-degree matrix of the graphG is D with entries(9)D=diagd1,d2,…,dn,where(10)di=∑j=1naij=∑j∈χiaij,i=1,2,…,n.The Laplacian matrixL∈ℝn×n of the graph G is [44](11)L=D−A.


## 2.3. Hysteretic Quantizer

To eliminate the control chattering phenomenon induced by logarithmic quantizer, a hysteretic quantizer is used to quantify control torque and force in this paper, which is similar to [45]. It can be expressed by(12)qut≜uJsgnu,ifuJ1+δ<u≤uJ1−δ,u˙J<0,or,uJ<u≤uJ1−δ,u˙J>0,uJ1+δsgnu,ifuj<u≤uJ1−δ,u˙J<0,or,uJ1−δ≤u≤uJ1+δ1−δ,u˙J>0,0,if0≤u<umin1+δ,u˙J<0,or,umin1+δ≤u≤umin,u˙J>0,qut−,othercase,where uJ=ρ1−iumin,J=1,2,…,n with 0<ρ<1, umin>0, and δ=1−ρ/1+ρ. qut is in the set U=0,±ui,±ui1+δ. The map of the hysteresis quantizer qut for u>0 is illustrated in Figure 1.Figure 1
Map ofqut for u>0.Remark 1.
The parameterρ can be termed as a measure of quantization density. From the definition of ρ, we can see that the smaller parameter ρ is, the coarser the hysteretic quantizer becomes [45]. Therefore, the design of parameter ρ should be based on a criterion to guarantee the control performance with a smaller quantization density. In addition, compared with a traditional logarithmic quantizer, the hysteretic quantizer (10) can avoid oscillations by adding additional quantization levels.


## 2.4. Preliminaries

For deriving the 6-DOF fixed-time coordinated controller, the lemmas are made as follows.Lemma 1.
The hysteretic quantizerqut is decomposed into two parts as(13)qut=Duut+Qt,where Du and Qt satisfy(14)1−δ≤Du≤1+δ,Qt≤umin.
The proof of Lemma1 is similar to Theorem 1 in [45].Lemma 2 (see [44]).
IfG is a directed graph with N nodes, then all the eigenvalues of the weighted Laplace matrix L have nonnegative real part.Lemma 3 (see [46]).
For any matrixM∈ℝm×m, N∈ℝn×n, X∈ℝm×m, and Y∈ℝn×n, then the following equalities hold:(1)
M⊗NX⊗Y=MX⊗NY(2)
If the matricesM and N are invertible, then M⊗N−1=M−1⊗N−1(3)
If the eigenvalues ofM are λ1,…,λm and the eigenvalues of N are μ1,…,μn, then the eigenvalues of M⊗N can be expressed by λiμji=1,…,m;j=1,…,nLemma 4 (see [47]).
For anyx,y∈ℝ, if ν∈ℝ+ and ν>1, then(15)x+yν≤2ν−1xν+yν.Lemma 5 (see [48]).
Ifxi∈ℝ,i=1,2,…,n, and 0<p≤1, then(16)∑i=1nxip≤∑i=1nxip≤n1−p∑i=1nxip.Lemma 6 (see [47]).
Consider the nonlinear system given by(17)x˙=fx,t,f0,t=0,x∈ℝn.
Suppose that there exists a Lyapunov functionVx that satisfies the following condition:(18)V˙x≤−αVxp+βVxgk+υ,x∈Uo,where α, β, p, g∈ℝ+, pk<1, gk>1, and 0<υ<∞. Then, the origin of system (17) is practical fixed-time stable and the residual set of the solution satisfies(19)limt⟶Tx|Vx≤minα−1/pυ1−θk1/kp,β−1/gυ1−θk1/kg,where θ is a scalar satisfying 0<θ≤1. The setting time is bounded by(20)T≤1αkθk1−pk+1βkθkgk−1.


## 3. Multispacecraft Nonsingular Fixed-Time Terminal Sliding Mode

In this section, a multispacecraft nonsingular fixed-time terminal sliding mode (FTTSM) vector is proposed to realize the orbit and attitude coordinated control for SFF. The following assumptions are presented regarding the 6-DOF dynamic model.Assumption 1.
The total disturbanceτi is assumed to be bounded due to the fact that magnetic forces, J2 perturbations, gravitation, and solar radiation pressure are bounded.Assumption 2.
The desired angular velocityωdi and its time derivative ω.di are assumed to be bounded. The desired trajectory ρdi and its time derivative ρ.di are assumed to be bounded.In order to achieve coordinated control for multiple spacecraft formation, the sliding mode based on (6) is defined as(21)S=s1,…,siT,where si=si,1,si,2,…,si,kT∈ℝ6×1, i=1,2,…,n and k=1,2,…,6, given by(22)si=biGfie2i+αie1i+∑j=1naijGfie2i−Gfje2j+Gfiαie1i−Gfjαje1j,with αie1i=αi,1e1i1,αi,2e1i2,…,αi,ke1ikT∈ℝ6×1, and(23)αi,ke1ik=sgnσ1isgne1ikp1+σ2isgne1ikp2k1,ifs¯i,k=0ors¯i,k≠0,e1ik>ε,l1ie1ik+l2isgne1ik2,ifs¯i,k≠0,e1ik≤ε,where i=1,…,n, k=1,2,…,6, s¯i=s¯i,1,s¯i,2,…,s¯i,6T, s¯i=e2i+sgnσ1isgne1ip1+σ2isgne1ip2k1, l1i=2−k1σ1iεp1−1/k1+σ2iεp2−1/k1k1, l2i=k1−1σ1iεp1−1/k1+σ2iεp2−1/k1k1, bi>0 is control scheme gain to realize tracking control, aij≥0 is control scheme gain to realize coordinated control between ith and jth formation spacecraft, sgne1ikp1=e1ikp1sgne1ik and sgne1ikp2=e1ikp2sgne1ik, σ1i>0, σ1i>0, p1>0, p2>0k1>0 are designed parameters, sgnσ1isgne1ikp1+σ2isgne1ikp2k1=σ1isgne1ikp1+σ2isgne1ikp2k1sgnσ1isgne1ikp1+σ2isgne1ikp2, ε>0 is small constant, 0<p1k1<1, p2k1>1, and sgn⋅ is sign function.Remark 2.
It is supposed that the communication topology of SFF is undirected; we can obtainaij=aji, which simplifies the design and analysis of the controller. However, for the directed communication topology, aij=aji does not hold. Therefore, compared with bidirectional communication topology, the coordinated control of formation spacecraft under directed communication topology is more challenging.Remark 3.
Note that the multispacecraft FTTSM (21) can be simplified as a modified terminal sliding mode (TSM) designed in [47] if k1=1 and p2=0; moreover, (21) coincides with the modified fast TSM designed in [49] for k1=1. It is worth mentioning that when e1ik converges to the region e1ik≤ε, the multispacecraft FTTSM is converted to the general sliding mode for s¯i,1≠0. Thus, the singularity problem of (21) can be effectively avoided. Moreover, by the choice of l1 and l2, the continuity of αi,k and its first-order time derivative is guaranteed.By Kronecker product, the sliding mode function (21) can be described by(24)S=L+B⊗I6G¯e2+αe1,where L is the weighted Laplace matrix, which is determined by directed topology, B=diagb1,b2,…,bn,e2=e21T,e22T,…,e2nTT, αe1=α1Te11,α2Te12,…,αnTe1nT,G¯=diagGf1,Gf2,…,Gfn.


## 4. Design of Fixed-Time Adaptive Coordinated Control Scheme

In this section, a fixed-time adaptive 6-DOF coordinated control scheme is presented for multiple SFF with input quantization and external disturbances.To design the control scheme, (6) can be derived as(25)Gfie˙2i+α˙ie1i=hi+Duiui+Qit+τi,with(26)α˙ie1i=σ1ip1k1diagσ1isgne1ip1+σ2isgne1ip2k1−1,⋅diage1ip1−1Λe˙1ie2i+σ2ip2k1diag|σ1isgne1ip1,+σ2isgne1ip2|k1−1diage1ip2−1Λe˙1ie2i,ifs¯ik=0ors¯ik≠0,e1ik>ε,l1iΛx˙1ie2i+2l2idiage1iΛe˙1ie2i,ifs¯ik≠0,e1ik≤ε,hi=−Ce2i−Ne1i−Gfi−e˙2id+Gfiα˙ie1i.Under Assumption1, it can be seen that(27)L+B⊗I61τi+Qit12≤ci,i=1,…,n,where ci are nonnegative constant numbers.The fixed-time adaptive controller is designed as(28)ui=−11−δhi+11−δ∑j=1,j≠inaij+bi−1∑j=1,j≠inaij1−δuj+hj−αisgnsiγ1,−βisgnsiγ2−c^i2ℑi2si,i=1,…,n,(29)c^˙i=−2κiϱic^i+κi2ℑi2si,i=1,…,n,where 0<γ1<1, γ2>1, ℑi>0, κi>0, and ϱi>0 are the controller parameters.Theorem 1.
Consider the 6-DOF control system (6) with the fixed-time coordinated control law (28). If the parameter uncertainty and external disturbance satisfy Assumptions 1-2, then the sliding mode vector si will converge into(30)si≤Δs=minυ1−2/1+γ1υ1−θo2/1+γ1,υ212υ+12−1n+k−1−2/1+γ1υ1−θo2/1+γ2,in fixed time, where βmin=minβi, αmin=minαi, ιmin=minιi, υ1=minαmin2γ1+1/2,ιminγ1+1/22γ1+1/2, υ2=minβmin2γ2+1/2,ιminγ2+1/22γ2+1/2, υ=∑i=1nϱioici2+1+∑i=1n/ℑ22+∑i=1nιi/κiΔi2γ2+1/2−ιi/κiΔi2 , ιi=κiϱi2oi−1/2oi, Δi>0, 0<θ0≤1, oi>1/2.Proof.
We construct the following Lyapunov function candidate:(31)V1=V2+V3,with(32)V2=12STS,V3=12∑i=1nκi−1c˜i2,where c˜i=ci−c^i.
By the Kronecker product, the controller (28) can be rewritten as(33)U=−11−δH+11−δD+B−1⊗I6A⊗I61−δU+H−αsgnSγ1,−βsgnSγ2−c^S.where U=u1,…,unT, H=h1,…,hnT, α=diagα1I6,…,αnI6, α=diagα1I6,…,αnI6,c^=diagc^1/2ℑ12I6,…,c^n/2ℑn2I6.
SinceI6n−D+B−1⊗I6A⊗I6 can be equivalently expressed as(34)I6n−D+B−1⊗I6A⊗I6,=D+B−1⊗I6D+B⊗I6−A⊗I3,=D+B−1⊗I6L+B⊗I6.
It follows from (34) that(35)U=−11−δH−11−δI6n−D+B−1⊗I6A⊗I6−1D+B−1⊗I6,×αsgnSγ1+βsgnSγ2+c^S,=−11−δH−11−δD+B−1⊗I6L+B⊗I6−1D+B−1⊗I6,×αsgnSγ1+βsgnSγ2+c^S,=−11−δH−11−δL+B⊗I6−1αsgnSγ1+βsgnSγ2+c^S.
Considering (24) and (25), it can be shown that(36)V˙2=STS˙,=STL+B⊗I6Gfie˙2i+Gfiα˙ie1i,=STL+B⊗I6H+DUU+Qt+τ,where τ=τ1,…,τnT, DU=diagDu1,…,Dun, Qt=Q1t,…,QntT.
Next, it can be derived that(37)V˙2=STL+B⊗I6L+B⊗I6−1−αsgnSγ1−βsgnSγ2−cS+Qt+τ,=STL+B⊗I6τ+Qt−STαsgnSγ1−STβsgnSγ2−STc^S,≤∑i=1nL+B⊗I6τi+Qitsi−∑i=1nc^i2ℑi2si2−∑i=1n∑k=1pβisikγ2+1,−∑i=1n∑k=1pαisikγ1+1,≤∑i=1nL+B⊗I61τi+Qit12si2ℑ22+∑i=1nℑ22−∑i=1nc^i2ℑi2si2,−∑i=1n∑k=1pαisik2γ1+1/2−∑i=1n∑k=1pβisik2γ1+1/2,≤∑i=1nc˜i2ℑi2si2+∑i=1nℑ22−∑i=1n∑k=1pαisik2γ1+1/2−∑i=1n∑k=1pβisik2γ2+1/2.
In addition, taking the derivative ofV3 yields(38)V˙3=∑i=1nκi−1c˜ic˜˙i,=−∑i=1nc˜i2ℑi2si2+∑i=1n2ϱic˜ic^i.
It is noticed from (37) and (38) that(39)V˙1≤−∑i=1n∑k=1pαisikγ1+1−∑i=1n∑k=1pβisikγ2+1+2∑i=1nϱic˜ic^i+∑i=1nℑ22.
From(40)ϱic˜ic^i≤−ϱi2oi−12oic˜i2+ϱioi2ci2and by substituting (40) into (39), we have(41)V˙1≤−∑i=1n∑k=1pαisikγ1+1−∑i=1n∑k=1pβisikγ2+1−∑i=1nιiκic˜i2+∑i=1nϱioici2,−∑i=1nιiκic˜i2−∑i=1nιiκic˜i21+γ1/2+∑i=1nιiκic˜i21+γ1/2+∑i=1nℑ22.where ιi/κi=ϱi2oi−1/2oi.Case 1.
Ifιi/κic˜i2≥1, we have(42)ιiκic˜i21+γ1/2≤ιiκic˜i2.
Substituting (42) into (41) yields(43)V˙1≤−∑i=1n∑k=1pαisik2γ1+1/2−∑i=1n∑k=1pβisik2γ2+1/2∑i=1nιiκic˜i2+∑i=1nϱioici2−∑i=1nιiκic˜i2γ1+1/2+∑i=1nℑ22.
According to Lemma5, it is easy to prove that(44)∑i=1n∑k=1pαisik2γ1+1/2+∑i=1nιiκic˜i21+γ1/2≥υ1V81+γ1/2,where υ1=minαmin2γ1+1/2,ιminγ1+1/22γ1+1/2, αmin=minαi, ιmin=minιi.
By applying (44), (43) can be rewritten as(45)V˙1≤−υ1V11+γ1/2−∑i=1n∑k=1pβisik2γ2+1/2−∑i=1nιiκic˜i2+∑i=1nϱioici2+∑i=1nℑ22.Case 2.
Ifιi/κic˜i2<1, one can obtain(46)ιiκic˜i21+γ1/2−ιiκic˜i2≤1−ιiκic˜i2<1.
Substituting (46) into (41) yields(47)V˙1≤−υ1V11+γ1/2−∑i=1n∑k=1pβisik2γ2+1/2−∑i=1nιiκic˜i2+∑i=1nϱioici2+1+∑i=1nℑ22.
Furthermore, (47) can be rewritten as(48)V˙1≤−υ1V11+γ1/2−∑i=1n∑k=1pβisik2γ2+1/2−∑i=1nιiκic˜i2−∑i=1nιiκic˜i2γ2+1/2,+∑i=1nιiκic˜i2γ2+1/2+∑i=1nϱioici2+1+∑i=1nℑ22.
According to Lemma5, we obtain(49)∑i=1n∑k=1pβisik2γ2+1/2+∑i=1nιiκic˜i21+γ2/2≥υ212γ2+12−1V51+γ2/2,where υ2=minβmin2γ2+1/2,ιminγ2+1/22γ2+1/2, βmin=minβi, ιmin=minιi.
From (49), we have(50)V˙1≤−υ1V11+γ1/2−υ212γ2+12−1V11+γ2/2−∑i=1nιiκic˜i2+∑i=1nιiκic˜i2γ2+1/2+∑i=1nϱioici2+∑i=1nℑ22+1.
Assume that there exists a compact setΠi satisfying(51)Πi=c˜i|c˜i≤Δi,where Δi is an unknown constant.
IfΔi<ζi/ιi, we can obtain(52)ιiκic˜i2<1,ιiκic˜i2γ2+1/2<ιiκic˜i2.
IfΔi≥ζi/ιi, we can obtain(53)ιiκic˜i2γ2+1/2−ιiκic˜i2≤ιiκiΔi2γ2+1/2−ιiκiΔi2.
Denote(54)υ=∑i=1nϱioici2+1+∑i=1nιiκiΔi2γ2+1/2−ιiκiΔi2.
Then, from the above analyses, we can further conclude that(55)V˙1≤−υ1V11+γ1/2−υ212γ2+12−1n+k−1V11+γ2/2+υ.
With Lemma6, system (6) is practical fixed-time stable. Furthermore, si will converge into the region(56)Φ=limt⟶Tssi|V1≤minυ1−2/1+γ1υ1−θo2/1+γ1,υ212υ+12−1n+k−1−2/1+γ1υ1−θo2/1+γ2,in a fixed-time Ts≤1/ϑ1θo1−γ1+12+1/ϑ212γ2+12−1nθoγ2+12−1; that is, si will converge into the region∥si∥≤Δs in a fixed-time Ts.Theorem 2.
Whensi reach the boundary Δs in fixed-time, the tracking error e1i and e2i will converge to(57)e1i<Δe1i=maxε,Δ1ksσ1i1/p1,Δ1ksσ2i1/p2,e2i<Δe2i=maxΔs+l1Δe1ik+l2Δe1ik2,Δs+σ1iΔe1ip1+σ2iΔe1ip2k1,in a fixed time, where i=1,…,n,k=1,2,…,6.Proof.
Ifs¯i=0, we can obtain(58)s¯i=e2i+sgnσ1isgne1ip1+σ2isgne1ip2k1=0,i=1,…,n.
Furthermore, one can obtain(59)ρ˙ei+sgnσ1isgnρeip1+σ2isgnρeip2k1=0,i=1,…,n,ωei+sgnσ1isgnqeip1+σ2isgnqeip2k1=0,i=1,…,n.
Construct the following Lyapunov function candidate:(60)V4=12ρeiTρei+qeiTqei+1−q0ei2,=12ρeiTρei+21−q0ei.
Denote(61)V5=ρeiTρei,V6=21−q0ei.
Taking the derivative ofV5 and V6 yields(62)V˙5=ρeiTρ˙ei,=ρeiT−sgnσ1isgnρeip1+σ2isgnρeip2k1,−σ1iρeiTρei1/k1+p1/2+σ2iρeiTρei1/k1+p21/2k1,≤−σ1i21/k1+p1/2V51/k1+p1/2+σ2i21/k1+p2/2V51/k1+p2/2k1.
Similarly, taking the derivative ofV6 yields(63)V˙6=−2q˙0ei,=qievTωei,=−qievTsgnσ1isgnqeip1+σ2isgnqeip2k1,=−σ1iqievTqiev1/k1+p1/2+σ2iqievTqiev1/k1+p2/2k1,≤−σ1i0.51/k1+p1/2V61/k1+p1/2+σ2i0.51/k1+p2/2V61/k1+p2/2k1.Then, from (62) and (63), we can obtain that system error states ρei,qei converge into regions 0,0, at the same time q˙0ei∈−1,1 converges to 1 in fixed-time by using Lemma 6.Case 4.
Ifs¯i≠0 and e1ik<ε, which implies that ∣e1ik∣ has converged to the region e1ik<Δe1ik=ε in a fixed time. Then, from (23), we have(64)e2ik+l1ie1ik+l2isgne1ik2=ϖik,ϖik≤Δs.
Moreover, it is easy to see that(65)e2ik<Δs+l1iΔe1ik+l2iΔe1ik2,which means that e2ik converges to the region e1ik<Δe1ik=ε in a fixed time.Case 5.
Ifs¯i≠0 and e1ik>ε, we can obtain(66)s¯i=e2i+sgnσ1isgne1ip1+σ2isgne1ip2k1≠0,i=1,…,n,which means(67)s¯i=e2i+sgnσ1isgne1ip1+σ2isgne1ip2k1=ϖik,ϖik≤Δs.
After a simple transformation, (67) can be rewritten as(68)e2i+1−ϖiksgnσ1isgne1iγ1+σ2isgne1iγ2k1sgnσ1isgne1iγ1+σ2isgne1iγ2k1=0.
When1−ϖik/sgnσ1isgne1ip1+σ2isgne1ip2k1>0, (66) is still kept in the form of the FTTSM as in Case 1, which implies that(69)e1i<Δ1ksσ1i1/p1,e1i<Δ1ksσ2i1/p2.
Thus, system statee1i will converge into the region(70)e1i<Δe1i=maxΔ1ksσ1i1/p1,Δ1ksσ2i1/p2,in a fixed time.
On the other hand, from (67), we can obtain that system state e2i will converge to the region(71)e2i<Δe2i=Δs+σ1iΔe1ip1+σ2iΔe1ip2k1,in a fixed time.
Furthermore, we can conclude that(72)e1i<Δe1i=maxε,Δ1ksσ1i1/p1,Δ1ksσ2i1/p2,e2i<Δe2i=maxΔs+l1Δe1ik+l2Δe1ik2,Δs+σ1iΔe1ip1+σ2iΔe1ip2k1.Thus, e1i and e2i,i=1,2,3, will converge to the regions Δe1i and Δe2i in a fixed time, respectively.Remark 4.
Based on the multispacecraft FTTSM results, the property of graph theory and adaptive technique, a fixed-time 6-DOF coordinated control strategy is designed under directed communication topology. Subsequently, the system tracking error states can be guaranteed to converge their desired trajectories in a fixed time even with external disturbances and quantized control input. Note that this small region is determined by the controller parametersαi>0, βi>0, and κi>0. Thus, this small region is adjustable and can be reduced as needed.Remark 5.
By employing the adaptive method, the precise information of the external disturbance and parameters uncertain is not needed for the controller (28) design. Moreover, it is no required to make an additional assumption about interspacecraft communication topology in the designed fixed-time coordinated controller. Hence, the designed controller is suitable for any communication topology. Even if there is no communication link between the formation spacecraft, this proposed controller can still guarantee the practical fixed-time stability of each formation spacecraft.


## 5. Illustrative Example

To validate the proposed coordinated controller, we give an illustrative example in this section. The communicate topology of three follower spacecraft is described in Figure2, in which “Sat ii=1,2,3” denotes the ith formation spacecraft. The leader spacecraft is specified to a circular orbit with a radius of 6878 km and orbit angel velocity is n0=1.11×10−3 rad/s.Figure 2
The directed communication topology of three follower spacecraft.The weighted Laplace matrixL is designed as(73)L=10−1−1100−11.The inertia matrix and mass are(74)J1=1010.4180.20.40.27kg.m2,J2=110.50.40.590.20.40.27kg.m2,J3=90.50.70.53.50.30.70.28kg.m2,mfi=50kg,∀i=1,2,3.The desired attitude quaternion and attitude angular velocity are(75)qdi=1000T,ωdi=000Trad/s,∀i=1,2,3.The initial attitude quaternion and attitude angular velocity are(76)q10=0.33170.30.4−0.8T,ω10=0.07−0.05−0.04Trad/s,q20=0.26460.20.5−0.8T,ω20=0.07−0.05−0.04Trad/s,q30=0.55680.20.4−0.7T,ω30=0.06−0.05−0.05Trad/s.The relative position and velocity are initialized:(77)ρi0=−1016050Tm,vi0=000Tm/s,∀i=1,2,3.In order to form a triangle of three follower spacecraft, the desired relative position and velocity are specified as(78)ρ1d=01000Tm,ρ2d=02000Tm,ρ3d=0150503Tm,vdi0=000Tm/s,∀i=1,2,3.The external disturbance of torque and force are specified as(79)zi=0.06cos0.3t0.01sin0.1t0.06cos0.2tTNm,Fit=0.001cos0.02t0.002sin0.02t−0.001sin0.02tTN,∀i=1,2,3.The maximum of input control force and torques are limited to 5 N and 1 Nm, respectively.The following performance indexes are defined to describe the synchronization accuracy and tracking accuracy of the proposed control law (28):(80)ϖ1=∑i=13ρ˜i2,ϖ2=∑i=13∑j∈Niρ˜i,j2,ϖ3=∑i=13qveiTqvei,ϖ4=∑i=13∑j∈Niqi,jveTqi,jve,where ρ˜i=ρi−ρdi, ρ˜j=ρj−ρjd, ρ˜i,j=ρ˜i−ρ˜j, and qi,jve is the vector part of qi,je=qi,j0eqi,jveTT=qei⊗qej. From the definition of ϖ1, ϖ2, ϖ3, and ϖ4, it can be shown that smaller ϖ1 and ϖ3 can guarantee the better formation and attitude tracking performance, respectively; smaller ϖ2 and ϖ4 can ensure the better formation and attitude synchronization performance, respectively.The control scheme (28), adaptive updating law (29), and hysteretic quantizer (12) parameters are chosen as γ1=0.9, γ2=1.1, a12=0.5, a23=0.5, a31=0.5, αi=0.1, βi=1, p1=0.4, p2=1.5, k1=2, ci=0.1, bi=0.04, κi=0.000002, ℑi=0.5, ϱi=0.000002,∀i=1,2,3, δ=0.25, and umin=0.00001.The simulation results of the controller (28) are shown in Figures 3 to 9, where Figure 3 depicts the relative attitude error and Figure 4 plots the relative angular velocity error. It can be observed that the relative attitude errors converge to near zero within 50 s, which has a fast convergence rate. The relative position error and relative velocity error are shown in Figures 5 and 6, respectively. It can be seen that the relative position errors converge to near zero about 85 s, which has a fast convergence rate. The quantized control force and torque are shown in Figures 7 and 8, respectively. The curves of the performance indexes ϖ1 to ϖ4 are depicted in Figure 9. As seen from the simulation results, the proposed fixed-time control strategy provides a good performance of tracking and synchronization.Figure 3
The curves of the relative attitude error. (a) Sat1. (b) Sat2. (c) Sat3.
(a)(b)(c)Figure 4
The curves of the relative angular velocity error. (a) Sat1. (b) Sat2. (c) Sat3.
(a)(b)(c)Figure 5
The curves of the relative position error. (a) Sat1. (b) Sat2. (c) Sat3.
(a)(b)(c)Figure 6
The curves of the relative velocity error. (a) Sat1. (b) Sat2. (c) Sat3.
(a)(b)(c)Figure 7
The curves of the quantized control force. (a) Sat1. (b) Sat2. (c) Sat3.
(a)(b)(c)Figure 8
The curves of the quantized control torque. (a) Sat1. (b) Sat2. (c) Sat3.
(a)(b)(c)Figure 9
The curves of the performance indexesϖ1 to ϖ4.
(a)(b)


## 6. Conclusion

In this paper, the fixed-time 6-DOF coordinated control problem has been studied for multiple spacecraft formation with input quantization under directed communication graph. A fixed-time adaptive coordinated control strategy is designed by using multispacecraft FTTSM vector such that, in the presence of the upper bounds of unknown external disturbances, the controlled system is practical fixed-time stable and, at the same time, the tracking errors converge to their desired trajectories in a fixed time. Compared with the existing finite-time stabilization controllers, the designed adaptive fixed-time coordinated controller in this paper is more suitable for practical engineering application due to its convergent time regardless of initial system states. An illustrative example is given to illustrate the performance of the presented fixed-time coordinated controller. It was shown that the presented controller not only can ensure each spacecraft’s convergence to its desired states but also can provide the desired synchronization and tacking performance. Future study will focus on the extension of the presented controller under time-varying communication topology and communication time delay.



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