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Acta Astronautica

Acta Astronautica 81 (2012) 445–455

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Robust decentralized attitude control of spacecraft formations undertime-varying topologies, model uncertainties and disturbances

Haizhao Liang n, Zhaowei Sun, Jianying Wang

Research Center of Satellite Technology, Harbin Institute of Technology, P. O. Box 3013, Yi Kuang Street, Nan Gang District, 150080 Harbin, China

a r t i c l e i n f o

Article history:

Received 15 November 2011

Received in revised form

12 August 2012

Accepted 15 August 2012Available online 12 October 2012

Keywords:

Decentralized control

Spacecraft formation

Sliding mode control

Robustness

Attitude control

65/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.actaastro.2012.08.017

esponding author. Tel.: þ86 45186 402357 8

6 45186 416457.

ail addresses: waiting_1986@163.com (H. Lia

wei@vip.sina.com (Z. Sun), diandian3866@12

a b s t r a c t

This paper investigates the robust decentralized attitude control problem for spacecraft

formations under uncertain communication topologies. Based on modified Rodrigues

parameters representation and a Lagrange-like model, a class of decentralized attitude

control schemes designed by the use of sliding mode control approach is proposed to

steer the attitude of the spacecraft formation to a time-varying reference states with no

assumption on the information exchange graph. By taking into account the model

uncertainties and external disturbances simultaneously, the proposed robust control

strategies are proven effective to overcome these unexpected phenomena. Finally,

numerical examples are included to demonstrate the effectiveness and robustness of

the developed control schemes.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The concept of spacecraft formation has been a sig-nificant research topic due to a host of advantages such asmission flexibility and cost reduction. The spacecraftformation is a distributed system composed of multiplespacecraft. To perform a flying task cooperatively, theattitude controller making the control decision is impor-tant in achieving a harmonious behavior of the formationsystem. According to the place where the control deci-sions are made, attitude control can be categorized intocentralized and decentralized types. In particular, decen-tralized control for spacecraft formations has been wit-nessed great interest in recent years since it showssuperior fault-tolerance and simpler controllers thancentralized control [1–12]. In decentralized control, if alocal control agent fails for some reason, the failure willbe confined to the region of the failed agent but not affect

ll rights reserved.

505;

ng),

6.com (J. Wang).

the entire system, so the stability of the whole system canstill be maintained. In addition, the global controller ofthe formation system can be decomposed into smallcontrol agents leading to simpler control laws in decen-tralized control. The performance of each local controlagent on its local control task can guarantee the stabilityof the global system. However, mathematical analysis ofdecentralized control is difficult and that is the drawbackof decentralized control.

A number of works have been done in the field ofdecentralized control [1–12]. A class of control laws basedon the behavior-based approach that can guarantee globalstability was proposed in [2]. Decentralized controlschemes using the virtual structure approach were pro-posed in [3]. In [6], by using only relative attitude, relativeangular velocity and absolute angular velocity information,the attitude synchronization problem of a fully autono-mous formation system was solved. A reconfigurationproblem for a satellite formation was considered in [8].The variable structure control method was introduced intothe decentralized control in [9], and the control schemescould overcome the effects of communication delays.However, only the stabilization control problem was

H. Liang et al. / Acta Astronautica 81 (2012) 445–455446

investigated in [9]. It is worth pointing out that a finite-time control algorithm was utilized in [11] to solve thefinite-time attitude containment control problem for amultiple-rigid-bodies system. The developed control lawscould guarantee finite-time reachability of the convex hullformed by the leader’s attitude, but the effects of uncer-tainties and disturbances were not taken into account.

In practical situation, model uncertainties and unex-pected environment disturbances are ubiquitous problemswhich should be taken into account and solved in thedevelopment of controllers for the formation system toachieve an ideal performance. Ignoring environmental dis-turbing torques such as the gravity gradient torque, solarradiation pressure torque and aerodynamic torque canjeopardize the mission. Hence, it is desirable to design robustcontrol schemes that are able to counterbalance the effectsof parameter uncertainties and external disturbances.

Conventional attitude control methods cannot coun-teract the effects of uncertainties and disturbances, andits convergence is at best exponential with infinite set-tling time. Fortunately, the finite-time control approachwhich has been studied in the field of single roboticmanipulator systems [13–16] demonstrates some nicefeatures such as quick response and superior robustness.In particular, since the requirements of fast attitudemaneuver and fast attitude tracking become more andmore attractive in modern space missions, finite-timecontrol strategies were utilized to solve the fast attitudecontrol problem of single spacecraft [17,18] in the lastdecade. Nevertheless, little attention has been dedicatedin the field of multiple-agents systems. Therefore, it issignificant to employ finite-time control strategies tosolve the robust control problem of a formation system.

In addition to the aforementioned issues, the controlproblem for a formation becomes more difficult than thatfor a single system due to the information exchangebetween the agents within a formation. Ref. [11] hasintroduced finite-time control to a multiple-rigid-bodiessystem with an undirected communication topology. Butfuture study on designing control schemes with no con-straint of the communication topology is still necessarysince the undirected communication topology in [11] isan assumption that needs to be checked all the time inpractical situation.

Inspired by these facts, it is desirable to developcontrol schemes for a spacecraft formation, which arerobust against model uncertainties and external distur-bances, and are effective in the presence of arbitrarycommunication topologies. To the best of the authors’knowledge, such type of control problem has not beeninvestigated in the existing literature.

In this paper, we focus on the robust decentralizedattitude control problem of spacecraft formations withuncertain communication topologies, model uncertaintiesand external disturbances. This scenario is important toreorientation, attitude tracking and observation of movingobjectives. Firstly, a Lagrange-like model of the spacecraftattitude motion using modified Rodrigues parameters(MRPs) representation is developed. Then, based on thesliding mode control approach, a class of decentralizedattitude tracking control schemes is proposed to steer the

attitude states of the spacecraft formation to a referencestate in finite time. The proposed control laws are provento be robust against both model uncertainties and exter-nal disturbances with any type of communication topol-ogy. By virtue of the designed sliding surface, theconvergence rate of the sliding mode is theoreticallyproven to be better than the linear-hyperplane-basedsliding mode (LSM) and the terminal sliding mode (TSM).

The paper is organized as follows. In Section 2, pre-liminaries are given. In Section 3, the main result ispresented, and the corresponding convergence androbustness analyses of the proposed control laws for theresulting close-loop system are provided. In Section 4,numerical simulations are presented.

2. Preliminaries

2.1. Notation

Throughout this paper, for a given vector v¼ ½ðvÞ1,. . .,ðvÞm�

T , where ðvkÞ represents the kth component of v, anda real number a, we define

sigðvÞa ¼ ½sgnðvÞ19ðvÞ19a,. . .,sgnðvÞm9ðvÞm9

a�T : ð1Þ

We use JnJ to represent the Euclidean norm of the vectorv, and vi for i¼ 1,2,:::,n to denote a certain parameter orvariable v of the ith spacecraft within the formation.

2.2. Attitude kinematics and dynamics

The MRPs are adopted herein to represent the attitudeof spacecraft. The MRPs are defined as

r¼ nUtanðy=4Þ, ð2Þ

where n is the Euler axis, y is the Euler angle. Note thatthe MRPs can represent the attitude of the spacecraftwithout singularities when y 2 ð�2p,2pÞ since the MRPs’singularity occurs at y¼ 72p [19].

r 2 R3 is used to denote the absolute attitude of thespacecraft, which represents the MRPs of body-fixedreference frame with respect to inertial reference frame.x 2 R3 is used to denote the absolute angular velocity ofthe spacecraft.

The kinematics and the dynamics equations of thespacecraft are given as follows, respectively:

_s ¼ GðrÞx, ð3Þ

J _x ¼�x�Jxþu ð4Þ

with

GðrÞ ¼1

4ð1�rTrÞIþ

1

2r�þ

1

2rrT , ð5Þ

where n� represents a skew-symmetric matrix given by

n� ¼

0 �ðnÞ3 ðnÞ2

ðnÞ3 0 �ðnÞ1

�ðnÞ2 �ðnÞ1 0

264

375 ð6Þ

and I denotes the 3� 3 identity matrix. J 2 R3�3, u 2 R3

represent the inertia tensor and control input of thespacecraft, respectively.

H. Liang et al. / Acta Astronautica 81 (2012) 445–455 447

The representation of the relative attitude fromdesired reference frame to body-fixed reference frame iscalled the absolute attitude error re. The error kinematicsequation can be expressed as follows:

_re ¼Gexe, ð7Þ

where Ge ¼ GðreÞcan be calculated using (5), xe 2 R3 isthe absolute angular velocity error that can be calculatedas

xe ¼x�xd, ð8Þ

where xd 2 R3 is the desired angular velocity.

2.3. Lagrange-like model

As many mechanical systems can be converted to theLagrange form, we introduce the Lagrange equation todescribe the spacecraft attitude motion. According to (4),(7) and (8), the Lagrange-like model representing therelative attitude motion of a spacecraft can be obtainedas follows:

M €reþC _reþG�Te E¼G�T

e u, ð9Þ

where M ¼G�Te JG�1

e , C ¼�M _GeG�1e �G�T

e ðJG�1e _reÞ

�G�1e þ

G�Te xd

�JG�1e , and E¼ J _xdþxd Jxd.

Note that because the control schemes in this researchare designed based on Lagrange expression (9), the sameresult can be extended to other systems if the attitudedynamic model can be stated in the Lagrange form.

2.4. Lemmata

In order to facilitate the development of the controlschemes, the following lemmata are presented.

Lemma 1. (Tang [20]) Suppose Oðx,tÞ is a C1 smooth

positive-definite function (defined on U � Rþ , where U �

U0 � Rn is a neighborhood of the origin). If there are real

numbers k 2 Rþ and e 2 ð0,1Þ such that _OþkOe is a

negative semi-definite function on U , then O� 0 is achieved

in finite time for a given initial condition xðt0Þ in a neighbor-

hood of the origin in U. The settling time satisfies

TrOð0Þ1�e

ðk�keÞ, ð10Þ

where Oð0Þ denotes the initial value of O .

Lemma 2. (Liang et al. [12]) For positive constants Vi , if Vi

satisfies that

ViZ0, i¼ 1,2,:::,n, ð11Þ

then the following inequality holds:

Xn

i ¼ 1

Vi1=2

Z

Xn

i ¼ 1

Vi

!1=2

: ð12Þ

2.5. Problem statement

In this research, we consider an n-spacecraft formationperforming a flying task in which the spacecraft withinthe formation are required to align their attitudes whiletracking a time-varying reference state frdðtÞ,xdðtÞg.

The attitude control problem herein is to design decen-tralized control schemes for a spacecraft formation by theuse of the sliding mode control algorithm and the coordi-nated control approach, such that the attitude of eachspacecraft can converge to the reference state in finitetime and maintain certain relative attitude in the pre-sence of arbitrary communication topologies, modeluncertainties and external disturbances.

3. Attitude control scheme design

In this section, a class of robust decentralized attitudecontrol laws is proposed to solve the attitude controlproblem of spacecraft formations. The proposed controlschemes are effective under arbitrary communicationtopologies and robust against model uncertainties andunexpected disturbances. By employing common Lyapunovfunctions, the convergence for the closed-loop system isproven theoretically.

3.1. Control laws under arbitrary communication topologies

We first propose a fast sliding mode (FSM) to guaran-tee that the system states arriving on it can converge tothe equilibrium in finite time. Using MRPs representation,the sliding surface for the ith agent within the formationis given by

si ¼ _reiþc1reiþc2sigðreiÞa, ð13Þ

where rei is the absolute attitude error of the ith space-craft, c1and c2 are positive constants, a satisfies that0:5oao1. If the attitude states of the ith spacecraftreach the sliding surface, then we have si � 0 whichimplies that

_rei ¼�c1rei�c2sigðreiÞa: ð14Þ

Considering the positive scalar function_V i ¼

12 rT

eirei andcomputing the derivative of

_V i yields

__V i ¼ rT

ei_rei

¼�c1reiTrei�c2

X3

k ¼ 1

9ðreiÞk91þa

r�c2JreiJ1þa

r�_c_V i

ðaþ1Þ=2, ð15Þ

where _c ¼ 2ðaþ1Þ=2c2. According to Lemma 1, it followsthat rei � 0 is achieved in finite time. Then, according to(5), (7) and (14) we know that _rei � 0 which impliesxei � 0 is obtained in finite time.

Next, based on the sliding surface (13), a class ofcontrol schemes is proposed to guarantee that si � 0 canbe obtained in finite time, and as a result, the controlobjective can be achieved in finite time. The control lawsare given as follows:

ui ¼�uFi �uD

i �uCi , i¼ 1,2,. . .,n, ð16Þ

where

uFi ¼�ðG

TeiCi _reþEiÞþGT

eiMiðc1 _reiþc2ara�1ei

_reiÞ, ð17Þ

uDi ¼ qisgn ðsiÞ, ð18Þ

H. Liang et al. / Acta Astronautica 81 (2012) 445–455448

where qi ¼GTeiMidi with ðdiÞk40,k¼ 1,2,3, and

uCi ¼ GT

eiMi

Xn

j ¼ 1

ðkiijsi�kj

ijsjÞ, ð19Þ

where kiij and kj

ij are positive constants.

Remark 1. The proposed controllers (16) have a clearintuitive structure that each of them contains a feed-backterm uF

i , a disturbance-counterbalance term uDi and a

coordination term uCi ; the feed-back term can offset the

nonlinear torque term in the spacecraft equations ofmotion and drives the attitude of the spacecraft to thereference state; with the disturbance-rejection term, thecontrol schemes will perform excellent robustness againstthe model uncertainties and external disturbances; thecoordination term provides feedback information fromthe jth spacecraft to the ith spacecraft, thus enforces theinterconnection between the spacecraft within the for-mation and improves the reliability of the entire system.

Theorem 1. The proposed control schemes (16) can steer

the attitude of the spacecraft formation to the desired state

in finite time with arbitrary communication topologies, if the

following inequalities hold kiij4Zij and 4Zijðk

iij�ZijÞZðk

jijÞ

2

where Zij40 is a constant which will be involved in the

following convergence analysis.

Proof. By substituting control laws (16) into (9), Eq. (9)can be restated as

€rei ¼�ðc1 _reiþc2ara�1ei

_reiÞ�disgnðsiÞ�Xn

j ¼ 1

ðkiijsi�kj

ijsjÞ:

ð20Þ

Consider the following Lyapunov function:

V ¼Xn

i ¼ 1

Vi ð21Þ

with

Vi ¼1

2sT

i si: ð22Þ

It is derived that

_V i ¼ sTi_s i

¼ sTi ð €reiþc1 _reiþc2ara�1

ei_reiÞ: ð23Þ

Substituting (20) into (23), _V i can be calculated as

_V i ¼ sTi �disgnðsiÞ�

Xn

j ¼ 1

ðkiijsi�kj

ijsjÞ

0@

1A

r�X3

k ¼ 1

ðdiÞkð9si9Þk�siTXn

j ¼ 1

ðkiijsi�kj

ijsjÞ

r�JdiJ JsiJ�siTXn

j ¼ 1

ðkiijsi�kj

ijsjÞ: ð24Þ

In light of (24), _V can be calculated as

_V ¼Xn

i ¼ 1

_V i

r�Xn

i ¼ 1

JdiJ JsiJ�Xn

i ¼ 1

Xn

j ¼ 1

siT ðki

ijsi�kjijsjÞ: ð25Þ

Note that the following equation holds:

Xn

i ¼ 1

Xn

j ¼ 1

ðsiT si�sj

T sjÞ ¼ 0: ð26Þ

Then, we have thatPn

i ¼ 1

Pnj ¼ 1 Zijðsi

T si�sjT sjÞ ¼ 0 with

Zij ¼ Zji40. By adding the term Sni ¼ 1S

nj ¼ 1Zijðsi

T si�sjT sjÞ

to the right side of the inequality (25), it can be rewrittenas

_V r�Xn

i ¼ 1

:di: :si:�Xn

i ¼ 1

Xn

j ¼ 1

siT ðki

ijsi�kjijsjÞ

þXn

i ¼ 1

Xn

j ¼ 1

ZijðsiT si�sj

T sjÞ

¼ �Xn

i ¼ 1

:di: :si:

�Xn

i ¼ 1

Xn

j ¼ 1

siT ðki

ijsi�kjijsjÞ�Zijðsi

T si�sjT sjÞ

0@

1A

¼�Xn

i ¼ 1

:di: :si:�Xn

i ¼ 1

Xn

j ¼ 1

ðkijTkijþgijsj

T sjÞ ð27Þ

where kij¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiki

ij�Zij

qsi� kj

ij=2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiki

ij�Zij

q� �sj and gij¼Zij�

ððkjijÞ

2= 4ðkiij�ZijÞÞZ0:

According to Lemma 2, it can be obtained that

_V r�mV1=2ð28Þ

with m¼minðffiffiffi2p

JdiJÞ40,i¼ 1,2,. . .,n.Therefore, according to Lemma 1, si � 0 is achieved in

finite time. For the system states reaching the slidingsurface si � 0, finite-time reachability of the equilibriumcan be proven by choosing the Lyapunov function_V i ¼ ð1=2ÞrT

eirei. According to (15) and Lemma 1, it canbe guaranteed that the control objective frei � 0,xei � 0gis obtained in finite time. Hence, the proof of Theorem 1 iscompleted. &

3.2. Robustness analysis

In practical situation, there always exist uncertaintiessuch that the inertia tensor of the spacecraft cannot beknown exactly. Moreover, the existence of the environ-ment disturbances is an ineluctable problem. Therefore,we will test the robustness of the proposed controlschemes in this section. We restate the Lagrange model(9) by considering the model uncertainties and externaldisturbances

M0 €reþC0 _reþG�Te E0 ¼G�T

e uþG�Te Dd, ð29Þ

where

M0 ¼G�Te J0G�1

e , C0 ¼�M0_GeG�1

e �G�Te ðJ0G�1

e _re�G�1

e

þG�Te xd

�J0G�1e , E0 ¼ J0 _xdþxdJ0xd,

and Ddðre, _re, _xd,xd,DJ,udÞ ¼�G�Te DJG�1

e þG�Te DJG�1

e_Ge

G�1e þG�T

e ðDJG�1e _reÞ

�G�1e �G�T

e xd�DJG�1

e �DJ _xd�xdDJxdþ

ud: The inertia tensor of the spacecraft is calculated asJ ¼ J0þDJ where J0 and DJ denote the nominal andperturbation part of J, respectively. The external distur-bances are represented by ud 2 R3. In practical applica-tions, both the control torques and the desired attitude

H. Liang et al. / Acta Astronautica 81 (2012) 445–455 449

states xd, _xd are bounded, and the term Dd is bounded ifwe assume that the perturbation inertia tensor and theexternal disturbing torque both have upper bounds:DJ irDJi and 9ðudiÞk9r ðtÞk,k¼ 1,2,3 with ðtÞk40.

Theorem 2. The proposed control schemes can solve the

attitude control problem for the spacecraft formation in the

presence of model uncertainties and external disturbances, if

di in uDi satisfies ðdiÞk4ðDdiÞk ¼ ð9M0i

�1G�Tei Ddi9Þk,k¼ 1,2,3

and if kiij4Zij and 4Zijðk

iij�ZijÞZ ðk

jijÞ

2 .

Proof. By substituting control laws (16) into (29), we canobtain that

€rei ¼�ðc1 _reiþc2ara�1ei

_reiÞ�di sgnðsiÞ�Xn

j ¼ 1

ðkiijsi�kj

ijsjÞ

þM0i�1G�T

ei Ddi: ð30Þ

Still consider the Lyapunov function (21), it is derived that

_V i ¼ sTi_s i

¼ sTi ð €reiþc1 _reiþc2ara�1

ei _reiÞ: ð31Þ

Substituting (30) into (31), _V i can be calculated as

_V i ¼ sTi �disgnðsiÞþM0i

�1G�Tei Ddi�

Xn

j ¼ 1

ðkiijsi�kj

ijsjÞ

0@

1A

r�X3

k ¼ 1

ðdi�DdiÞkð si

�� ��Þk�siTXn

j ¼ 1

ðkiijsi�kj

ijsjÞ

r�Jdi�DdiJ JsiJ�siTXn

j ¼ 1

ðkiijsi�kj

ijsjÞ: ð32Þ

Similar with the proof of Theorem 1, it is obvious that

_V r�mV1=2, ð33Þ

where m ¼minðffiffiffi2pÞJdi�DdiJ40, i¼ 1,2,. . .,n. Hence,

according to the analysis similar with Theorem 1, thecontrol objective rei � 0,xei � 0f g can be obtained in finitetime, and this completes the proof. &

Remark 2. The parameter kjij is not only a weight para-

meter but also a parameter that describes the commu-nication topology of the formation system. When kj

ij ¼ 0, itmeans no information transmission from the jth space-craft to the ith spacecraft. By virtue of such a parameter,the communication topology can be undirected or direc-ted, fixed or even time-varying since no assumption on ithas been made in the theoretical analysis. It is worthpointing out that, for any kj

ij, there always exists properconstants Zij and ki

ij4Zij such that the inequality4Zijðk

iij�ZijÞZðk

jijÞ

2 holds. As a result, the finite-time con-vergence of the formation system can be guaranteedunder uncertain communication topologies with the pro-posed control strategies.

Remark 3. The coordination control term gives rise to aninterconnection between the spacecraft within the forma-tion. It provides the attitude information of other agents inthe formation to the ith spacecraft so that the informationof the desired attitude and other agents’ attitude are bothtaken into account in the control action of the ith space-craft. Therefore, the attitude states of other spacecraftwithin the formation can affect the attitude of the ith

spacecraft through this term, and it makes the formationperform as a whole instead of several individuals. In casethat the communication topology is disconnected, albeitfinite-time convergence and stability of the formation canstill be guaranteed by the proposed control algorithm, thecontrol performance will be affected significantly since thecoordination control term is deactivated and the space-craft in a formation without the coordination term willhave no feedback information from others.

Remark 4. Theorem 1 is a general conclusion that thecontrollers (16) are always competent and effective underarbitrary communication topologies if ki

ij4Zij and4Zijðk

iij�ZijÞZ ðk

jijÞ

2 hold. On the other hand, if the com-munication network of the formation system is assumed

to be an undirected topology which means kiij ¼ ki

ji and

kjij ¼ kj

ji, then the control objective can easily be obtained if

kiijZkj

ij holds which is a more loose condition to be

satisfied. The proof is similar with the proof of Theorem1 by choosing the parameter Zij ¼ ki

ij=2.

Remark 5. According to Theorem 2 it can be seen that thecontrol parameter d is crucial to the formation in thepresence of parameter uncertainties and disturbing tor-ques. A large d can improve the robustness of theproposed control laws but induce large control torques.c1 and c2 are parameters of the sliding mode. The choicesof d, c1 and c2 are typically limited by the frequency of thelowest unmodeled structural resonant mode, the largesttime-delay and the sampling rate. Hence, the value of thecontrol parameters d, c1 and c2 should be chosen accord-ing to the upper bounds of perturbation inertia tensor anddisturbance torques, system’s mechanical properties andcomputing power. The parameter kj

ij denotes the weightof the attitude information transmitted from the jthspacecraft to the ith spacecraft, and ki

ij represents theweight of the coordination control action. Increasing theweight parameters can give rise to an improvement inrelative attitude control but the tracking performance willbe influenced. According to various space missions, strongweights are preferable for a formation focuses on relativeattitude maintenance, and weak connections are suitablewhen the formation concentrates on desired attitudeattainment [12]. The parameter Zij is introduced in orderto facilitate the stability and convergence analysis, thechoice of which can be any positive constant that satisfieski

ij4Zij and 4Zijðkiij�ZijÞZðk

jijÞ

2 according to kiij and kj

ij.

Remark 6. Because of the term c2ara�1ei

_rei in uFi in the

control law, the control signal could be infinite if rei ¼ 0before the attitude states reach the sliding mode. To solvethis singularity problem, some approaches were devel-oped [21–23]. The two-phase strategy proposed in [21]can be adopted in this paper. The two-phase strategysolves the singularity problem by driving the systemstates to a prescribed region in advance where thesingularity does not occur. In this research, it is to keepthe attitude states away from rei ¼ 0 before si � 0 isreached. Once si � 0is reached, because c2ara�1

ei_rei ¼

�c1c2araei�c2

2ar2a�1ei ¼ 0 when rei ¼ 0, no singularity

will occur.

H. Liang et al. / Acta Astronautica 81 (2012) 445–455450

Remark 7. It should be noted that the proposed controlschemes may induce chattering in control signals due tothe discontinuous sign function used in the control laws.Chattering phenomenon is harmful for the formationsystem and should be alleviated. The saturation function(or any other sigmoid function) which provides a solutionto eliminate the chattering phenomenon through soft-ening the control can be used to replace the sign functionin the control laws. In fact, outside the boundary layer, thecontrol schemes are the same as the original one. Theintroduction of the saturation function is a continuousapproximation to the discontinuous sign function withinthe boundary layer, and the chattering phenomenon iseliminated at the cost of a nonzero steady state error.

3.3. Convergence rate of the sliding surface

In this section, we will show that the convergence rateof the sliding mode (13) is better than that of LSM andTSM. We first restate the three types of sliding modes in ascalar form with states f _x,xg for more intuitive illustration.The scalar form of the FSM is s¼ _xþaxþb sigðxÞa; thescalar form of TSM is s¼ _xþb sigðxÞa; and the scalar formof LSM is s¼ _xþax.

It is well-known that the LSM can only guaranteeasymptotical stability which implies that the systemstates on the sliding surface will converge to the originas time tends to infinity. However, LSM can drive thesystem states to a neighborhood of the equilibrium infinite time because the system states on LSM converge tothe equilibrium exponentially. Therefore, we will com-pare the convergence time for each sliding mode steeringthe system states into a neighborhood of the equilibriumdescribed as Z ¼ fx : 9x9rmg, where m40.

When the system states arrive on the sliding surface,we have that � _x ¼ c1xþc2 sigðxÞa, � _x ¼ c2 sigðxÞa and� _x ¼ c1x for the three sliding mode, respectively. Byintegrating both sides of them, the corresponding settlingtime can be calculated as follows:

Tf sm ¼1

c1ð1�aÞln

c19xð0Þ91�aþc2

c1t1�aþc2

!ð34Þ

for the FSM, and

Ttsm ¼1

c2ð1�aÞð9xð0Þ91�a

�t1�aÞ ð35Þ

for the TSM, and

Tlsm ¼1

c1ln

9xð0Þ9t

� �ð36Þ

for the LSM.

Theorem 3. If the system state on the sliding surface is

outside Z such that 9xð0Þ94t, then the convergence rate of

FSM is better than that of either LSM or TSM, in other words,

Tf smoTtsm and Tf smoTlsm.

Proof. We first prove that Tf smoTlsm, then Tf smoTtsm.

(1) Since xð0Þ994t40 and ao1, we know that9xð0Þ91�a

4t1�a. Therefore, we have

1oc19x 0ð Þ91�a

þc2

c1t1�aþc2o

c19xð0Þ91�a

c1t1�a ¼9xð0Þ9t

� �1�a

: ð37Þ

In light of (37), it is derived that

Tf sm ¼1

c1ð1�aÞln

c19xð0Þ91�aþc2

c1t1�aþc2

!

o1

c1ð1�aÞln

9xð0Þ9t

� �1�a

¼1

c1ln

9xð0Þ9t

� �¼ Tlsm: ð38Þ

Therefore, the proof of Tf smoTlsm is completed.(2) Calculating the derivative of Ttsm with respect to

9xð0Þ9 yields

dTtsm

d9xð0Þ9¼

1

c29xð0Þ9�a: ð39Þ

Then, calculating the derivative of Tf sm with respect to9xð0Þ9, we have that

dTf sm

d9xð0Þ9¼

9xð0Þ9�a

c19xð0Þ91�aþc2

: ð40Þ

It is derived that

dTf sm

d9xð0Þ9¼

xð0Þ99�a

c19xð0Þ91�aþc2

o1

c29xð0Þ9�a ¼

dTtsm

d9xð0Þ9: ð41Þ

Since Tf sm ¼ Ttsm ¼ 0 in the case of 9xð0Þ9¼ t, it is obviousthat Tf smoTtsm holds for any 9xð0Þ94t. Hence, the proofof Theorem 3 is completed. &

Remark 8. From the previous proof we can find that theFSM has a globally fast convergence rate that is betterthan either the LSM or the TSM whether the system statesare far away or near to the origin, because there is noassumption on t and Theorem 3 holds for any t40.

Remark 9. From (13) it can be seen that if the systemstate rei is far away from the origin, then the linear part�c1rei prevails over its counterpart �c2 sigðreiÞ

a such thatthe convergence rate depends on the linear part. On theother hand, if the system state is close to the origin, thenthe convergence rate mainly depends on the terminal part�c2 sigðreiÞ

a which makes the control objective beachieved in finite time. The physical interpretation is:when the states are far away from the equilibrium, theapproximate dynamics of FSM becomes _x ¼�c1x whosefast convergence rate when far away from the origin iswell understood; when the states are close to the equili-brium, the approximate dynamics of FSM becomes_x ¼�c2 sigðxÞa whose convergence rate accelerates expo-nentially as the states approach the equilibrium.

4. Numerical examples

The effectiveness of the proposed control schemes isdemonstrated through numerical examples in this sec-tion. A formation system composed of six rigid satellites is

H. Liang et al. / Acta Astronautica 81 (2012) 445–455 451

considered. Two simulations are presented to validate thetheoretical results.

The nominal inertia tensor of each satellite is chosenas [24]

J01 ¼

100 1 2

1 150 1

2 1 200

0B@

1CAkg:m2

J02 ¼

120 1:5 1

1:5 160 1

1 1 190

0B@

1CAkg:m2

J03 ¼

80 0:5 2

0:5 180 0

2 0 180

0B@

1CAkg:m2

J04 ¼

110 1 0

1 130 0:5

0 0:5 240

0B@

1CAkg:m2

J05 ¼

100 1 0:5

1 140 1:5

0:5 1:5 220

0B@

1CAkg:m2

J06 ¼

90 1 1

1 170 0

1 0 200

0B@

1CAkg:m2

In order to highlight the performance of the developedcontrol schemes, the initial attitude states of each satelliteare chosen to be the following large values shown inTable 1.

We consider a formation-flying mission where thespacecraft within the formation are required to align theirattitudes while tracking a desired reference attitude

Table 1The initial attitude states of each agent.

Initial value Agent 1

rð0Þ ð0:3 0:2 �0:3 ÞT

xð0Þðrad=sÞ ð0:35 �0:13 0:27 ÞT

Initial value Agent 4

rð0Þ ð0:4 0:4 �0:2 ÞT

xð0Þðrad=sÞ ð �0:45 0:33 �0:15 ÞT

Table 2

The parameter kjij .

Formation Agent 1 Agent 2 Agent 3

Agent 1 k212ðtÞ k3

13ðtÞ ¼ k212ðtþ1Þ

Agent 2 k121ðtÞ ¼ k2

12ðtþ1:5Þ k323ðtÞ ¼ k2

12ðtþ0:

Agent 3 k131ðtÞ ¼ k2

12ðtþ0:9Þ k232ðtÞ ¼ k2

12ðtþ4:9Þ

Agent 4 k141ðtÞ ¼ k2

12ðtþ3:7Þ k242ðtÞ ¼ k2

12ðtþ6Þ k443ðtÞ ¼ k2

12ðtþ5Þ

Agent 5 k151ðtÞ ¼ k2

12ðtþ2:4Þ k252ðtÞ ¼ k2

12ðtþ1:7Þ k353ðtÞ ¼ k2

12ðtþ0:

Agent 6 k161ðtÞ ¼ k2

12ðtþ1:9Þ k262ðtÞ ¼ k2

12ðtþ0:2Þ k363ðtÞ ¼ k2

12ðtþ4:

defined as [25]

xd ¼

0:1 cosðt=40Þ

�0:1 sinðt=50Þ

�0:1 cosðt=60Þ

0B@

1CArad=s

with the initial value of the desired MRPs as

rd ¼ ð�0:3 �0:4 0:4 ÞT :

The magnitude of the control torque is bounded as9ðuiÞk9r2NUm. The parameters of the control laws arechosen as c1 ¼ 0:6, c2 ¼ 0:8, ki

ij ¼ 8 and kjij is chosen as

Table 2 to describe a time-varying communication topol-ogy. In Table 2

k212ðtÞ ¼

5 for modðt,8Þr6:5

0 for modðt,8Þ46:5,

(

where modðx,yÞ denotes the remainder of dividing x by y.

The simulations are done in the following three cases:

Case 1. The effectiveness and robustness of the proposedcontrol schemes are validated in the presence of a time-varying topology, parameter uncertainties and externaldisturbances. The communication topology is illustratedin Table 2. The perturbation inertia tensor is chosen as

DJ i ¼ 0:3J0i [17] with the consideration of fuel cost onsatellites. The disturbing torque is given by [9]

di ¼

0:2 cosðiþ0:1tÞ

�0:24 sinðiþ0:15tÞ

0:22 cosðiþ0:12tÞ

0B@

1CAN:m

Since the environmental disturbances are mainly inducedby gradient gravity torques, solar radiation torques andaerodynamic torques, the disturbances adopted herein arelarger in magnitude than in the practice according to thesize of the spacecraft in this example.

Agent 2 Agent 3

ð0:1 0:3 0:2 ÞT ð �0:3 0:2 �0:1 ÞT

ð0:52 �0:26 0:33 ÞT ð �0:26 0:22 �0:13 ÞT

Agent 5 Agent 6

ð �0:1 0:3 0:2 ÞT ð �0:2 �0:1 0:1 ÞT

ð0:12 �0:34 �0:28 ÞT �0:36 0:17 0:41� �T

Agent 4 Agent 5 Agent 6

k414ðtÞ ¼ k2

12ðtþ2:8Þ k515ðtÞ ¼ k2

12ðtþ3:3Þ k616ðtÞ ¼ k2

12ðtþ4:8Þ

5Þ k424ðtÞ ¼ k2

12ðtþ2:5Þ k525ðtÞ ¼ k2

12ðtþ3:2Þ k626ðtÞ ¼ k2

12ðtþ2:6Þ

k434ðtÞ ¼ k2

12ðtþ5:6Þ k535ðtÞ ¼ k2

12ðtþ2:1Þ k636ðtÞ ¼ k2

12ðtþ3:6Þ

k545ðtÞ ¼ k2

12ðtþ4:1Þ k646ðtÞ ¼ k2

12ðtþ2Þ

9Þ k454ðtÞ ¼ k2

12ðtþ1:3Þ k656ðtÞ ¼ k2

12ðtþ6:2Þ

8Þ k464ðtÞ ¼ k2

12ðtþ1:1Þ k565ðtÞ ¼ k2

12ðtþ5:3Þ

0 20 40 60 80 1000

20

40

60

80

100

120

140

time (s)

RA

EM

(deg

)

0 50 1000

0.01

0.02

time (s)

RA

EM

(deg

)

Fig. 2. The relative attitude error metric.

0 20 40 60 80 1000

20

40

60

80

100

120

140

160

180

200

time (s)

θ i (d

eg)

θ1θ2θ3θ4

Fig. 3. The error Euler principal angle of each spacecraft.

H. Liang et al. / Acta Astronautica 81 (2012) 445–455452

Case 2. The connection weights are chosen as kiij ¼ 0 and

kjij ¼ 0 to deactivate the coordination control action. The

other control parameters are chosen the same as Case 1.The results are compared with Case 1 to illustrate thecoordination among the agents in the formation and theeffects of the coordination control term on the controlperformance.

Case 3. In this case, three simulations are done by

choosing the connection weights as kiij ¼ 2 and kj

ij ¼ 0:6,

kiij ¼ 4 and kj

ij ¼ 2:8, and kiij ¼ 8 and kj

ij ¼ 5, respectively,

while the other control parameters are held the same asCase 1. The results are presented to investigate the effectsof the coordination control action’s weight on controlperformance.

The performance of the spacecraft formation is mea-sured by an absolute attitude error metric and a relativeattitude error metric. The absolute attitude error metric interms of degree is defined as follows:

AAEMðdegÞ ¼180

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

6

X6

i ¼ 1

y2ei

vuut , ð42Þ

where yei is the error Euler principal angle which can becalculated by (2). The relative attitude error metric indegree is calculated using

RAEMðdegÞ ¼180

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

30

X6

i ¼ 1

X6

j ¼ 1

y2ij

vuut : ð43Þ

Simulation results of the three cases are presented inFigs. 1–9, respectively. Fig. 1 shows the transient behaviorof the absolute attitude error metric, and the relativeattitude error metric is illustrated in Fig. 2. Fig. 3 demon-strates the error Euler principal angle yei of four spacecraftwithin the formation. The control torques applied on eachspacecraft is illustrated by Fig. 4. Then, in Figs. 5 and 6, thetransient responses of the absolute attitude error metricand relative attitude error metric of the spacecraft for-mation in Case 2 are presented. The error Euler principalangle of four spacecraft within the formation is shown in

0 20 40 60 80 1000

50

100

150

200

250

300

time (s)

AA

EM

(deg

)

0 50 1000

0.02

0.04

time (s)

AA

EM

(deg

)

Fig. 1. The absolute attitude error metric.

Fig. 7. Finally, Figs. 8 and 9 show the absolute attitudeerror metrics and relative attitude error metrics withvarious connection weights.

The simulation results validate the mathematical ana-lysis of the developed control schemes in the precedingsection. The trajectories in Figs. 1–3 show fine attitudecontrol performance and rapid transient response of boththe absolute and relative attitude errors in the presence ofa time-varying communication topology, parameteruncertainties and large disturbing torques. From Figs. 1and 2 it can be seen that the absolute and relative attitudeerror metric both fall to the tolerance in 27 s. The attitudetracking accuracy is higher than 0.031, and the relativeattitude control accuracy is able to reach 0.011. It can beseen from Fig. 3 that the spacecraft within the formationcompletes the flying mission cooperatively, which makesthe formation perform as a whole. As shown in Fig. 4, thecontrol torques are bounded as 9ðuiÞk9r2NUm, and theupper bound is reached for only a few times at thebeginning of the tracking procedure.

The results presented in Figs. 5–7 verify the analysis inRemark 3, and the indispensability of the coordination

0 50 100−2

0

2

u 1 (N

m)

0 50 100−2

0

2

u 2 (N

m)

0 50 100−2

0

2

u 3 (N

m)

0 50 100−2

0

2

u 4 (N

m)

0 50 100−2

0

2

time (s)

u 5 (N

m)

0 50 100−2

0

2

time (s)

u 6 (N

m)

(*)1(*)2(*)3

Fig. 4. The control torques.

0 20 40 60 80 1000

50

100

150

200

250

300

time (s)

AA

EM

(deg

)

0 50 1000

0.05

0.1

time (s)

AA

EM

(deg

)

Fig. 5. The absolute attitude error metric without coordination terms.

0 20 40 60 80 1000

20

40

60

80

100

120

140

time (s)

RA

EM

(deg

)

0 50 1000

0.05

0.1

time (s)

RA

EM

(deg

)

Fig. 6. The relative attitude error metric without coordination terms.

H. Liang et al. / Acta Astronautica 81 (2012) 445–455 453

control term is demonstrated. In the absence of thecoordination control action, it can be seen from Figs. 5and 6 that although finite-time reachability of the equili-brium can still be guaranteed by the proposed controlschemes, the settling time increases significantly. Thecontrol accuracy is also affected; the tracking error islarger than 0.051 and the relative control error is largerthan 0.031. As shown in Fig. 7, the spacecraft in theformation without the coordination term performs asseveral individuals. By comparing Fig. 7 with Fig. 3, thedistribution and coordination of the spacecraft formationare illustrated.

The effect of connection weights on control performanceis demonstrated by Figs. 8 and 9. As shown in Figs. 8 and 9,

all these weight parameters are able to guarantee finite-time convergence of the formation system. Nevertheless, asthe weight parameters decrease, both the absolute attitudecontrol accuracy and relative attitude control accuracydescend accordingly. These figures illustrate the relationshipbetween control accuracy and the weight of relative attitudecontrol action; an improvement in control performance ofthe formation system can be achieved by increasing theconnection weights of the control schemes. However, it isworth pointing out that overlarge weight parameters couldinduce instability of the formation system since the avail-able control torques is always bounded in practice. There-fore, the upper bound of control torques should be takeninto account when choosing connection weight parameters.

0 20 40 60 80 1000

20

40

60

80

100

120

140

160

time (s)

θ i (d

eg)

θ1θ2θ3θ4

Fig. 7. The error Euler principal angle of each spacecraft without

coordination terms.

0 20 40 60 80 1000

50

100

150

200

250

300

time (s)

AA

EM

(deg

)

0 50 1000

0.02

0.04

time (s)

AA

EM

(deg

)

k =8,k =5

k =4,k =2.8

k =2,k =0.6

Fig. 8. The absolute attitude error metric with different weights.

0 20 40 60 80 1000

20

40

60

80

100

120

140

time (s)

RA

EM

(deg

)

0 50 1000

0.02

0.04

time (s)

RA

EM

(deg

)

k =8,k =5

k =4,k =2.8

k =2,k =0.6

Fig. 9. The relative attitude error metric with different weights.

H. Liang et al. / Acta Astronautica 81 (2012) 445–455454

5. Conclusions

In this paper, we solved the attitude control problemfor the spacecraft formation with uncertain communica-tion topologies, model uncertainties and external distur-bances. Using sliding mode control, a class of robustdecentralized attitude control schemes based on MRPsrepresentation and a Lagrange-like model was proposed toguarantee finite-time convergence of the attitude states ofthe spacecraft formation. The developed control schemeswere effective under arbitrary communication topologiesand robust against both the model uncertainties andexternal disturbances. Finally, numerical examples of asix-rigid-satellite formation were performed in the pre-sence of a switching communication topology, large dis-turbances and model uncertainties. The effectiveness androbustness of the proposed control strategies were vali-dated, and the indispensability and effects of the coordi-nation control action were also demonstrated by thesimulations.

Acknowledgments

The authors express their gratitude to the reviewersand editor for very useful and constructive comments thathelped substantially improve the quality of the paper.

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