Attitude coordination of satellite swarms with communication delays

Attitude coordination of satellite swarmswith communication delays

Haizhao Liang, Zhaowei Sun and Jianying Wang

Research Center of Satellite Technology, Harbin Institute of Technology, Harbin, China

AbstractPurpose – This paper aims to investigate the fast attitude coordinated control problem for rigid satellite swarms with communication delays.Design/methodology/approach – Based on behavior-based control approach, the attitude control system is designed to guarantee that the attitudeof the satellite swarm converge to a dynamic reference state in finite time. A fast sliding mode is developed to improve the convergence rate androbustness of the control system. All the effects of communication delays, parameter uncertainties and external disturbances are taken into accountsimultaneously, and the communication topology of the satellite swarm can be arbitrary types. Numerical simulations are provided to demonstrate theanalytic results.Findings – Despite the existence of communication delays, parameter uncertainties and external disturbances, the stability of the closed-loop systemcan be successfully guaranteed and the proposed control strategies are effective to overcome these unexpected phenomena subject to arbitrarycommunication topology.Originality/value – This paper introduces a fast terminal sliding mode control method which can guarantee the fast convergence of the attitude stateof the satellite swarm in the presence of communication delays, switched communication topology, parameter uncertainties and external disturbances.

Keywords Coordination control, Satellite swarm, Sliding mode control, Fast convergence, Communication delays, Communication,Artificial satellites

Paper type Research paper

Nomenclature

Symbolss ¼ modified Rodrigues parametersn ¼ Euler axisu ¼ Euler angle

v ¼ angular velocityI ¼ identity matrixve ¼ angular velocity errorvei ¼ angular velocity error of the ith satellite

se ¼ relative attitude errorsei ¼ relative attitude error of the ith satelliteJi ¼ inertia matrix of the ith satelliteJoi ¼ nominal inertia matrix of the ith satellite

eJ i ¼ perturbation inertia matrix of the ith satellited i ¼ disturbance torques of the ith satellitesi ¼ sliding mode of the ith satellite

G ¼ kinematics matrixMe ¼ rotation matrixR ¼ 2v£Jv2 J M e _vd 2 v£

e M evd

� �Pi ¼ J iG

21e

Qi ¼ _Gevei þ a _sei þ basa21ei _sei

ui ¼ control input of the ith satelliteuti ¼ tracking control action of the ith satellite

uci ¼ coordination control action of the ith satellite

wi ¼ ks jðt 2 TijÞk1=2

kiij ¼ weight parameter of the coordination control action

kjij ¼ weight parameter of the jth satellite’ attitude

ksij ¼ weight parameter

Tij ¼ time delay between the jth and ith satelliteTf ¼ convergence time of FSM

Tl ¼ convergence time of LSMTt ¼ convergence time of TSM

Definitions, Acronyms and AbbreviationsFSM ¼ fast sliding modeTSM ¼ terminal sliding mode

LSM ¼ linear-hyperplane-based sliding modew.r.t. ¼ with respect to

e.g. ¼ for example

Introduction

The recent years have witnessed active investigations forspacecraft formations due to a host of advantages such as cost

reduction and robustness improvement. A satellite swarm isa formation system composed of a set of satellites. Controlling a

swarm of satellites in order to equalize their orientations is calledattitude coordination which has gained an increase of interests

for its important application in space-based interferometry andsynthetic-aperture imaging (Wang et al., 1999). Behavior-based

approach was adopted in Lawton and Beard (2002)and VanDyke and Hall (2006) to solve the attitude

coordination problem of spacecraft formations. A decentralizedcontrol scheme using virtual structure approach was proposed in

Ren and Beard (2004). In Sarlette et al. (2009), the authorsaimed to solve the attitude synchronization problem for a fully

autonomous formation system with restricted informationexchange. Without absolute attitude information, Bai et al.(2008) achieved attitude coordination using relative attitudeinformation only. Dimarogonas et al. (2009) solved the attitude

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Aircraft Engineering and Aerospace Technology: An International Journal

85/3 (2013) 222–235

q Emerald Group Publishing Limited [ISSN 1748-8842]

[DOI 10.1108/00022661311313713]

222

containment control problem such that the attitude states of the

followers could converge to the convex hull of the leaders’orientations. Sliding mode control approach was introduced to

coordination control in Liang et al. (2011) since it possessesbetter robustness than the previous traditional ones. However,

the information flow among the formation must be bidirectionaland the effect of information transmission delay was not

considered in Liang et al. (2011).Although the aforementioned control schemes have shown

adequate reliability in attitude coordination control, all theseaforementioned literature can only guarantee asymptoticalconvergence of the system states, which means that the

control objective is obtained as time tends to infinity.In contrast, finite-time control approaches have been

extensively studied in the field of single system control inGruyich and Kokosy (1999), Feng et al. (2001), Hong et al.(2002), Yu and Man (2002), Yu et al. (2005) and Wu et al.(2011) because it demonstrates some nice features such as

quick response and high control accuracy. For instance, thefinite-time regulation problem of robotic manipulators was

investigated in Gruyich and Kokosy (1999). The authors inYu et al. (2005) realized finite-time tracking of a single roboticmanipulator using terminal sliding mode technique. In

particular, since the requirements of fast attitude maneuverbecome more and more attractive in modern space missions,

finite-time control strategies were utilized to solve the fastattitude control problem of single spacecraft in recent years

(Wu et al., 2011).Thus, far, few works have studied the application of finite-

time control in the field of formation system. The controlproblem for a formation becomes more difficult than that for

a single system due to the information exchange between theagents within a formation. Finite-time attitude control was

introduced in Meng et al. (2010) to solve the containmentcontrol problem of a multiple rigid bodies system. InMeng et al. (2010), a class of control schemes using finite-

time control algorithm were developed to realize finite-timereachability of desired attitude. However, the effects of

external disturbance torques and parameter uncertaintieswere not taken into account, and the communication topology

of the formation was assumed to be an undirected graph withno time delay.

In practical situation, satellite swarms are always subject toenvironment disturbances and parameter uncertainties, which

are ubiquitous problems that should be taken into accountand solved in the development of control laws for the

formation system to achieve an ideal performance. Ignoringparameter uncertainties or environmental disturbance torquessuch as the gravity gradient torque, solar radiation pressure

torque and aerodynamic torque can jeopardize the mission.Hence, it is desirable to design robust control schemes that

are able to counterbalance the effects of parameteruncertainties and external disturbances.

In addition to the aforementioned issues, the informationin a satellite swarm are exchanged and transmitted over

a network link, so time delays always exist in thecommunication links ineluctably. Moreover, the undirected

communication topology used in Meng et al. (2010) is anassumption which needs to be checked all the time, because

the communication topology is not always undirected such asunidirectional satellite laser communication system in Wu andWang (2011). All these effects should be taken into account in

the design of control schemes.

Inspired by these facts, it is an interesting and challenging

problem to design control schemes based on finite-time

control approach for satellite swarms such that the control

system is able to realized finite-time reachability of desire

attitude, robust against model uncertainties and external

disturbances, and effective in the presence of arbitrary

communication topologies. To the best of the Author’s

knowledge, such type of control problem has not been

addressed in the existing literature, which motivates our

current research.The scope of this research focuses on fast attitude

coordination of satellite swarms with communication delays,

parameter uncertainties and external disturbances. This

scenario is important to reorientation, attitude tracking and

observation of moving objectives. The modified Rodrigues

parameters (MRPs) Schaub and Junkins (2003) and Ozgur

Doruk (2009) are used to represent attitude kinematics and

dynamics equations. A manifold called fast sliding mode

(FSM, or FTSM in Feng et al. (2001) and Yu and Man

(2002)) is introduced to improve the robustness of the control

system, and the FSM is proven to possess better convergence

rate than either the linear-hyperplane-based sliding mode

(LSM) or the terminal sliding mode (TSM). Based on the

FSM and behavior-based control method, a class of robust

finite-time control schemes are proposed to guarantee finite-

time convergence of the formation’s attitude states and to

keep certain relative attitude between spacecraft within the

formation. With elaborate treatment of a common Lyapunov

function, it is proven that the developed control strategies are

able to achieve the control objective and overcome the effects

of communication delays, external disturbances and

parameter uncertainties subject to arbitrary communication

topologies. Finally, numerical examples are given to

demonstrate the effectiveness and application of the

proposed control schemes.The paper is organized as follows. In second section,

preliminaries are given. In third section, the main result

is presented, and the corresponding convergence and

robustness analyses of the proposed control laws for the

resulting close-loop system are provided. In fourth section,

numerical simulation results are presented.

Preliminaries

In order to facilitate the theoretical analysis, the following

kinematics equation, dynamics equation, and lemmata are

presented.

Attitude kinematics and dynamics

The MRPs are adopted to describe the attitude of a rigid

satellite. The MRPs are defined as:

s ¼ n · tanðu=4Þ; ð1Þ

where n is the Euler axis, u is the Euler angle. Note that any

three-dimensional attitude description may cause a geometric

singularity. The MRPs’ singularity occurs at u ¼ ^3608.We use s [ R3 to represent the absolute attitude state of

a satellite’s body-fixed reference frame with respect to the

inertial reference frame. The absolute angular velocity v [R3 represents the angular velocity of the body-fixed reference

frame with respect to the inertial reference frame expressed in

the body-fixed reference frame.

Attitude coordination of satellite swarms with communication delays



Volume 85 · Number 3 · 2013 · 222–235

223

The kinematics equation in terms of MRPs is given as

follows:

_s ¼ GðsÞv; ð2Þ

where:

GðsÞ ¼ 1

4{ð1 2 s TsÞI þ 2½s £� þ 2ss T} ð3Þ

in which s£ denotes a skew-symmetric matrix defined as:

s £ ¼

0 2ðsÞ3 ðsÞ2

ðsÞ3 0 2ðsÞ1

2ðsÞ2 2ðsÞ1 0

2664

3775 ð4Þ

and I denotes a 3 £ 3 identity matrix.The relative attitude error se [ R3 is the MRPs from the

desired reference frame to the body-fixed reference frame.

According to formula (2), we have that:

_se ¼ G eve; ð5Þ

where G e ¼ GðseÞ, and ve [ R3 is the relative angular

velocity error. ve can be calculated using:

ve ¼ vM evd ; ð6Þ

where vd [ R3 is the desired angular velocity; M e [ R3£3

denotes the rotation matrix from the desired reference frame

to the body-fixed reference frame, which can be calculated

using:

M e ¼ I 3 24 1 2 s 2

e

� �1 þ s 2

e

� �2s £

e

� �þ 8

1 þ s 2e

� �2s£

e

� �2: ð7Þ

The dynamic model of a rigid satellite is given by the

following differential equation:

J _v ¼ 2v£Jvþ u; ð8Þ

where J [ R3£3 represents the inertia matrix of the rigid

satellite, and u [ R3 is the control torque.According to formula (6), the dynamic model represented

by relative attitude state can be stated as:

J _ve ¼ 2v£Jvþ u2 J M e _vd 2 v£e M evd

� �¼ R þ u;

ð9Þ

where:

R ¼ 2v£Jv2 J M e _vd 2 v£e M evd

� �:

Lemmata

In order to facilitate the development of the control schemes,

the following lemmata are presented.

Lemma 1. (Tang, 1998; Hong et al., 2006). Suppose

Y x; tð Þ 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 [ Rþ and 1 [ 0; 1ð Þ such that_Yþ kY1 is a negative semi-definite function on U , then Y ; 0

is achieved in finite time for a given initial condition xðt0Þ in a

neighborhood of the origin in U.

Lemma 2. (Liang et al., 2011). If V i $ 0; i ¼ 1; 2; . . . ; n,

then the following inequality holds:

Xni¼1

V1=2i $

Xni¼1

Vi

!1=2

: ð10Þ

Lemma 3. Hong et al. (2001). A function V ðxÞ is positive

definite and homogeneous of degree s to the dilation

ðr1; . . . ; rnÞ. If ~VðxÞ is a continuous function such that, for

any fixed x – 0, ( ~Vðrr1x1; . . . ; rrnxnÞ=rsÞ! 0 as r! 0, then

(V þ ~V) is locally positive definite.

Proof. With the conditions, there is a neighbourhood

of the origin, U ¼ {x [ Rn : xi ¼ 1ri zi ; i ¼ 1; . . . ; n; 0 # 1 ,

10; z ¼ ðz1; . . . ; znÞT ; kzk ¼ 1} with 10 . 0 small enough,

such that, for any x [ U:

~VðxÞ�� ¼ ~V 1r1z1; . . . ; 1

rn znð Þ�� # 1

21s

zk k¼1minV ðzÞ:

Therefore:

V ðxÞ þ ~VðxÞ $ 1

21s

zk k¼1minV ðzÞ

holds when x [ U. A

Controllers design

In this section, an n-rigid-satellite swarm performing a flying

task of tracking a dynamic desired reference state coordinately is

considered. The fast attitude coordination problem is to design

the control torque ui for each satellite within the formation, so

that the attitude states of each satellite can converge to the

desired state in the presence of information transmission delays,

model uncertainties, and external disturbances.

Coordination controllers

In the attitude coordination of the satellite swarm, there are

two control objectives that need to be obtained

simultaneously: desired attitude attainment and relative

attitude maintenance. Behavior-based control approach

utilized in Lawton and Beard (2002) and VanDyke and Hall

(2006) is an effective method when there are competing or

even conflict behaviors in control objectives. The behavior-

based control action is determined by a weighted sum of the

control actions for these competing behaviors, and it can

reach a compromise between these control actions. Based on

the basic mind of behavior-based control, the following

attitude coordination control laws are proposed:

u i ¼ uti þ uc

i ; i ¼ 1; 2; . . . ; n; ð11Þ

where:

uti ¼ 2R i 2 P iðdisgnðsiÞ þQiÞ; ð12Þ

uci ¼ 2P i

Xnj¼1

kiijsi 2 kjijs jðt 2 TijÞ þ ksijwjsgnðs iÞ

� �; ð13Þ

where uti and uc

i are the tracking control action

and the coordination control action, respectively; P i ¼ J iG21e ,

Qi ¼ _Gevei þ a _sei þ basa21ei _sei and f j ¼ s jðt 2 TijÞ

�� 1=2; di




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224

satisfies ðdiÞk . 0; k ¼ 1; 2; 3 where we use ð*Þk to denote

the kth component of a vector; s i ¼ _sei þ asei þ bsigðseiÞais the proposed FSM where a; b are positive constant,

and 0:5 , a , 1, sigðseiÞa ¼ sgnðseiÞT jsei ja and jsei ja ¼jðseiÞ1j

a jðseiÞ2ja jðseiÞ3j

ah iT

; kiij . 0, kjij $ 0 and ksij $ 0

are weight parameters where ksij ¼ 0 when kjij ¼ 0; Tij $ 0

represents the communication delay from the jth satellite to

the ith satellite which is assumed to be a time-varying

quantity satisfying _Tij , 1, namely, _Tij [ ð21; 1Þ; s jðt 2 TijÞis the delayed attitude states from the jth satellite to the ithsatellite.

Remark 1. In equation (11), uti is the tracking control

action to drive the attitude of the ith satellite to the desired

state and uci is the coordination control action to

maintain certain relative attitude within the formation.

The overall control action ui is determined by a weighted

sum of the tracking control action and the coordination

control action.Remark 2. Communication delays Tij exist in satellite

swarms due to information exchange. They are mainly

induced by time delays existing in the process of information

collecting, computing, and transmitting. In information

collecting, time delays depend on sensors, e.g. gyros can

provide angular velocity information in every control

period, thus without time delay, but star sensors may need

several control periods to provide attitude information,

which will cause time delays. For computing, due to the

existence of measurement noise and gyro drift, spacecraft

always need attitude estimation, and time delays

mainly depend on the filtering algorithm used in attitude

estimation. For information transmitting, time delays are

usually caused by data packing, data receiving, and data

unpacking.Remark 3. The coordination control action uc

i appended

to the tracking control action is to maintain certain relative

attitude within a spacecraft formation. This term is important

since it makes the satellites within a formation perform as a

whole rather than several individuals, and thus improves the

robustness and reliability of the formation. This term also

enforces an interconnection between satellites in the

formation. Each satellite within the formation without

the coordination control torque will have no feedback

information of other satellites: if a satellite is perturbed by

some disturbances or failed for some reasons, the formation

cannot be maintained. The interdependency of the satellite

within the formation is indicated clearly by the weight

parameters.With control laws equation (11), we have the following

theorem.

Theorem 1. If kiij . bij and 4ð1 2 _TijÞðkiij 2 bijÞbij $ kjij

� �2

holds, where bij $ 0 is a constant that will be involved in

the following stability analysis and bij ¼ 0 when kjij ¼ 0, then

the controller equation (11) can solve the attitude coordination

problem of the satellite swarm in the presence of information

transmission delays.Proof. We consider the following candidate Lyapunov

function:

V ¼Xni¼1

1

2sTi si þ

Xni¼1

Xnj¼1

Z t

t2Tji

bijsTi sidx: ð14Þ

Computing the derivative of V , we have that:

_V¼Xni¼1

sTi _siþXni¼1

Xnj¼1

bij sTi s i2 ð12 _TijÞsTj ðt2TijÞs jðt2TijÞh i

¼Xni¼1

sTi €seiþa _seiþbasa21ei _sei

� �

þXni¼1

Xnj¼1

bij sTi si2 ð12 _TijÞsTj ðt2TijÞs jðt2TijÞh i

¼Xni¼1

sTi G ei J21i J i _veiþQ i

� �

þXni¼1

Xnj¼1


:

ð15Þ

Then we substitute the relative dynamics equation (9) into

equation (15) and have that:

_V ¼Xni¼1

sTi G eiJ21i ðR i þ u iÞ þQi

� �

þXni¼1

Xnj¼1

bij sTi s i 2 ð1 2 _TijÞsTj ðt 2 TijÞs jðt 2 TijÞh i

¼Xni¼1

sTi 2disgnðsiÞ þG eiJ21i uc

i

� �

þXni¼1

Xnj¼1

bij sTi s i 2 ð1 2 _TijÞsTj ðt 2 TijÞs jðt 2 TijÞh i

# 2Xni¼1

dik k sik k2Xni¼1

Xnj¼1

ksijwj sik k

2Xni¼1

Xnj¼1

6Tij 6ij þ lijsTj ðt 2 TijÞs jðt 2 TijÞ

� �

# 2Xni¼1

dik k sik k2Xni¼1

Xnj¼1

ksijwj sik k;

ð16Þ

where the k*k represents the Euclidean norm of a vector, and:

6ij ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikiij 2 bij

qsi 2

kjij

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikiij 2 bij

q s jðt 2 TijÞ; ð17Þ

lij ¼ bijð1 2 _TijÞ2kjij

� �2

4 kiij 2 bij

� �0B@

1CA . 0: ð18Þ

With the following definitions:

V f ¼Xni¼1

1

2sTi si ; ð19Þ

Vd ¼Xni¼1

Xnj¼1

Z t

t2Tij

bijsTi sidx; ð20Þ

Vs ¼Xni¼1

Xnj¼1

ksijwj sik k; ð21Þ




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225

there is: _Vf # 2Pn

i¼1 dik k sik k2 Vs 2 _Vd ¼ 2Pn

i¼1 dik k�sik k2 ðVs þ _VdÞ holds. From the definition of Vs and Vd

it is easy to find that Vsðsi ; s jðt 2 TijÞÞ is homogeneous

of degree s ¼ 3=2 w.r.t dilation ðr1 ¼ 1; r2 ¼ 1Þsince Vsðrsi ; rs jðt 2 TijÞÞ ¼ r3=2Vsðs i ; s jðt 2 TijÞÞ, and_Vdðs i ; s jðt 2 TijÞÞ has a higher order than Vsðsi ; s jðt 2 TijÞÞsuch that it is homogeneous of degree s ¼ 2 w.r.t the dilation

ðr1 ¼ 1; r2 ¼ 1Þ. According to Lemma 3 and an analysis

similar to Hong et al. (2002), we have:

limr!0

_Vdðrs i ; rs jðt 2 TijÞÞr3=2

¼ limr!0

r2 _Vdðsi ; s jðt 2 TijÞÞr3=2

¼ limr!0r1=2 _Vdðsi ; s jðt 2 TijÞÞ

¼ 0:

Hence, it is safely to derive that Vs þ _Vd is locally positive

definite according to Lemma 3. Then, according to Lemma 2,

the following inequality holds:

_Vf # 2Xni¼1

dik k s ik k

# 2dV1=2f ;

ð22Þ

where d ¼ minð dik kÞ; i ¼ 1; 2; . . . ; n. Therefore, Vf ; 0 can

be obtained in finite time according to Lemma 1 which means

si ; 0 is a terminal attractor that can be achieved in finite

time. Once the attitude states of the ith satellite arrive on the

sliding mode si ; 0, then the equation _sei ¼ 2asei 2

bsigðseiÞa holds. Considering the candidate Lyapunov

function V_

i ¼ ð1=2ÞsTeisei and computing the derivative of

V_

i , we have that:

V__i ¼ sT

ei _sei

¼ 2asTeisei 2 b

X3

k¼1

seið Þk�� 1þa

# 2b seik k1þa

# 2cV_ðaþ1Þ=2

i ;

ð23Þ

where c ¼ 2ðaþ1Þ=2b. According to Lemma 1, sei ; 0 can be

achieved in finite time. Then, _sei ; 0 which means G eivei ;0 can also be obtained in finite time which implies that:

GTeiG eivei ¼

1 þ sTeisei

4

�vei ¼

1

4vei ; 0;

therefore, vei ; 0 can also be achieved in finite time. Hence,

the control objective {sei ; 0; vei ; 0} is achieved in finite

time. ARemark 4. When the weight parameter k

jij ¼ 0 and

ksij ¼ 0, it means no information transmission from the jth

satellite to the ith satellite. In this sense, the parameter kjij and

ksij will perform two tasks simultaneously:1 a weight parameter of the coordination control action; and2 a parameter to describe the communication topology.

In this setting, the communication topology of the formation

system can be any type and the convergence of the satellite

swarm can still be guaranteed. Furthermore, if kjij and ksij are

switching between zero and a positive constant during

the coordination, the communication topology will be a time-

varying type.Remark 5. It has been proven that Theorems 1 is effective

subject to arbitrary communication topologies: if kiij . bij and

4bijð1 2 _TijÞðkiij 2 bijÞ $ ðk jijÞ

2, then the control laws equation

(11) are competent. On the other hand, if the information

transmission delays are constant instead of time-varying and the

communication network is fixed to be an undirected topology

which means kiij ¼ kiji and kjij ¼ k

jji, the control objective can be

obtained easily if kiij $ kjij which is a milder condition. The proof

is similar with the proof of Theorems 1 by choosing the

parameter bij ¼ kiij=2.

Remark 6. Note that there exists a possible singularity

before theattitude states arrive at theTSM.To solve this problem,

numbers of approaches were developed in Li and Huang (2010),

Wu et al. (1998), Yang and Yang (2011) and Stonier and

Stonier (2004). We can adopt the two-phase strategy in Wu et al.

(1998) to overcome the singularity problem. The idea is to find a

region V in the state space such that any trajectory starting from

this region will not incur the singularity problem and the set of

switching manifolds sn21 ¼ 0; sn22 ¼ 0; . . . ; s0 ¼ 0 in Wu et al.

(1998) are reached sequentially. In this work, it is to keep the

trajectory away from any of sei ¼ 0 before si ¼ 0 is reached

and the region V can be defined as V ¼ {sei : sei . 0} > {sei :

si . 0} or V ¼ {sei : sei , 0} > {sei : si , 0} according to an

analysis similar with Wu et al. (1998). Then we can design the

two-phase control strategy.Thefirstphase is a control law to bring

the state into the region V in order to avoid the singularity

problem, and the second phase is to switch to the TSM control to

realize finite time convergence if the region V is reached. Once

si ¼ 0 is reached, the control laws will ensure no occurring of

singularity.

Remark 7. It is well-known that the LSM can only

guarantee asymptotical stability which means

sei ; 0;vei ; 0f g can be achieved when t !1. However,

LSM can drive the system states to a neighborhood of the

equilibrium in finite time because the system states on LSM

converge to the equilibrium exponentially. Therefore, we will

compare the convergence time for each sliding mode

steering the states into a neighborhood of the equilibrium

described as N ¼ x : xj j # mf g, where m . 0. The scalar

form of FSM is s ¼ _xþ axþ bsig xð Þa; s ¼ _xþ ax for LSM;

and s ¼ _xþ bsig xð Þa for TSM. The convergence time of the

FSM, LSM, and TSM can be calculated as:

Tf ¼1

að1 2 aÞ lna xð0Þj j12aþb

am12a þ b

!;

Tl ¼1

aln

xð0Þj jm

�; and

Tt ¼1

bð1 2 aÞ xð0Þj j12a2m12a

� �;

where xð0Þ is the initial value of the state x, respectively.Since xð0Þj j . m . 0 and a , 1, we know that

xð0Þj j12a. m12a. The following inequality holds:

1 ,a xð0Þj j12aþb

am12a þ b,

a xð0Þj j12a

am12a¼ xð0Þj j

m

�12a

: ð24Þ

In light of equation (24), it is derived that:




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226

Tf ¼1

að1 2 aÞ lna xð0Þj j12aþb

am12a þ b

!

,1

að1 2 aÞ lnxð0Þj jm

�12a

¼ 1

aln

xð0Þj jm

�¼ Tl :

ð25Þ

Calculating the derivative of Tt with respect to xð0Þj j yields:

dTt

d xð0Þj j ¼1

bxð0Þj j2a: ð26Þ

Then, calculating the derivative of Tf with respect to xð0Þj j,we have that:

dTf

d xð0Þj j ¼xð0Þj j2a

a xð0Þj j12aþb: ð27Þ

It is easy to find that:

dT f

d xð0Þj j ¼xð0Þj j2a

a xð0Þj j12aþb

,1

bxð0Þj j2a

¼ dTt

d xð0Þj j :

ð28Þ

Since Tf ¼ Tt ¼ 0 when xð0Þj j ¼ m, it is obvious that Tf , Tt

holds for any xð0Þj j . m. Hence, it is proven that theconvergence rate of FSM is better that that of both the LSMand the TSM, namely, Tf , Tl and Tf , Tt.

Remark 8. From the previous analysis we can find thatthe FSM has a globally faster convergence rate than either

the LSM or the TSM whether the initial states are faraway or near to the origin, because Theorem 2 holds for anym . 0.

Next, numerical examples are shown here to validate theanalytic result. According to different m, the time differencesðTl 2 Tf Þ and ðTt 2 Tf Þ with respect to the initial states are

shown in Figures 1 and 2.Figure 1 shows the time difference ðTl 2 Tf Þ w.r.t. the

initial states according to different m. Figure 2 shows the time

difference ðTt 2 Tf Þ w.r.t. to the initial states according todifferent m. We can find that, ðTl 2 Tf Þ . 0 and ðTt 2 Tf Þ .0 all the time which validates the analytic result. In Figure 1,

for a fixed initial value xð0Þ, the time difference becomeslarger with the decrease of m; for a fixed m, the time differenceincreases slowly with the increase of the initial value of thestate. In Figure 2, for a fixed initial value xð0Þ, the time

difference shows slight variety with the decrease of m; but for afixed m, the time difference becomes larger obviously with theincrease of xð0Þ.

Robustness discussion

In practice, there always exist uncertainties such that theinertia tensor of the satellite cannot be known exactly, and theenvironment disturbances are ineluctable problems. To testthe robustness of the proposed control strategies, we consider

the parameter uncertainties and external disturbancessimultaneously in the following analysis. The inertia tensor

of the spacecraft is assumed to be calculated as J ¼ J 0 þ DJ

where J 0 and DJ denote the nominal and perturbation part

of J , respectively, (Doruk and Kocaoglan, 2008). The

external disturbances are represented by d [ R3. Then the

dynamics model can be expressed as:

J 0 _ve ¼ R0 þ uþ d þ DFf ; ð29Þ

where:

DFf i ¼2 ðve þ vdÞ£DJ ive 2 DJ i _ve 2 ðve þ vdÞ£DJ ivd

2 DJ i _vd :

We assume that the perturbation inertia tensor is supposed to

be bounded as DJ i # D�J i and the external disturbance torques

is supposed to be bounded as jðdÞkj # y; k ¼ 1; 2; 3, where yis a positive constant. In practice, the desired attitude states

and the control torque all have upper bounds so the term

DFf i is bounded.Similar with the control laws (equation (11)), the

controllers can be modified as:

Figure 1 Time difference of Tl 2 Tf

0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

12

14

16

18

20

x

Tl-T

f (s)

µ = 10–1

µ = 10–3

µ = 10–5

µ = 10–7

Figure 2 Time difference of Tt 2 Tf

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

7

8

9

10

x

Tt-T

f (s)

µ = 10–1

µ = 10–3

µ = 10–5

µ = 10–7




Volume 85 · Number 3 · 2013 · 222–235

227

ui ¼ ut0i þ uc

0i

¼ 2R0i 2 P 0iðQ0i þ d0isgnðsiÞÞ

2 P 0i

Xnj¼1

kiijs i 2 kjijs jðt 2 TijÞ þ ksijwjsgnðsiÞ

� �;

ð30Þ

where P 0i ¼ J 0iG21e , and d0i satisfies:

d0i $ DFf i ¼ G eiJ210i ðjDFf i j þ yÞ: ð31Þ

Corollary 1. If kiij . bij and 4bij kiij 2 bij

� �$ k

jij

� �2

, then

the proposed control schemes can steer the attitude states of

the satellite swarm to the desired state fast in the presence of

communication delays, model uncertainties and disturbances.Proof. We still consider the Lyapunov function (14). It is

derived that:

_V¼Xni¼1

sTi _siþXni¼1

Xnj¼1

bij sTi s i2 ð12 _TijÞsTj ðt2TijÞs jðt2TijÞh i

¼Xni¼1

sTi G eiJ21i J i _veiþQ0i

� �

þXni¼1

Xnj¼1


:

ð32Þ

Substituting equation (29) into equation (32) gives:

_V ¼Xni¼1

sTi G eiJ210i R0i þ u0i þ d þ DFfð Þ þQ0i

� �

þXni¼1

Xnj¼1

bij sTi si 2 ð1 2 _TijÞsTj ðt 2 TijÞs jðt 2 TijÞh i

¼Xni¼1

sTi 2d0isgnðsiÞ þ DFf i þG eiJ210i u

ci

� �

þXni¼1

Xnj¼1

bij sTi si 2 ð1 2 _TijÞsTj ðt 2 TijÞs jðt 2 TijÞh i

# 2X3

k¼1

Xni¼1

ðd0i 2 DFf iÞk ðsiÞk�� 2Xn

j¼1

ksijwj ðs iÞk�� " #

2Xni¼1

Xnj¼1

6Tij 6ij þ lijsTj ðt 2 TijÞs jðt 2 TijÞ

� �

# 2Xni¼1

1i sik k2 Vs;

ð33Þ

where 1i ¼ kd0i 2 DFf ik. Similar with the proof of Theorem 1,

we have that:

_Vf # 2Xni¼1

1i sik k

# 21V1=2f ;

ð34Þ

where 1 ¼ minð1iÞ; i; j ¼ 1; 2; . . . ; n. Therefore, it is easily to

find that the control objective {sei ; 0; vei ; 0} can be

achieved in finite time and this completes the proof. A

Numerical simulations

In this section, numerical simulations of a formation

composed of four satellites in a low-earth orbit (LEO) with

a 400 km orbit altitude are presented to investigate the

effectiveness and robustness of the proposed control

approaches. Two scenarios will be considered in this section:1 verify the stability characteristics and robustness of the

proposed control laws in the presence of uncertainties,

disturbances, communication delays, and a time-varying

topology; and2 vary control parameters to analyze their effects on control

performance.

The nominal inertia tensor of each satellite is chosen as

(Ali et al., 2010) to represent four rigid satellites with various

inertia tensors:

J 01 ¼

10 0 4

0 15 0

4 0 20

0BBB@

1CCCAkg·m2; J 02 ¼

12 1 0:5

1 14 3

0:5 3 22

0BBB@

1CCCAkg·m2;

J 03 ¼

15 0:8 2

0:8 19 1

2 1 21

0BBB@

1CCCAkg·m2; J 04 ¼

13 0:4 0

0:4 14 0:8

0 0:8 18

0BBB@

1CCCAkg·m2:

A formation flying mission where the four satellites are

required to align their attitudes while tracking the attitude of

another satellite is taken into account. The desired attitude

states are defined by (Boskovic et al., 2004):

vd ¼

0:1 cos t=40ð Þ20:1 sin t=50ð Þ20:1 cos t=60ð Þ

0BB@

1CCArad=s

with the initial attitude:

sdð0Þ ¼ 20:5 20:4 0:3� �T

:

Before attitude coordination, the four satellites are assumed to

be performing different flying tasks, thus with various initial

attitude states which are given as follows:

v1ð0Þ¼ 0:45 20:43 0:77� �T

rad=s;s1ð0Þ¼ 0:2 0:220:2� �T

;

v2ð0Þ¼ 0:52 20:26 0:33� �T

rad=s;s2ð0Þ¼ 0:3 0:2 0:3� �T

;

v3ð0Þ¼ 20:26 0:22 20:13� �T

rad=s;s3ð0Þ¼ 20:2 0:1 20:1� �T

;

v4ð0Þ¼ 20:37 20:19 20:23� �T

rad=s;s4ð0Þ¼ 0:4 20:2 0:1� �T

:

Due to fuel cost, measuring deviation, and other factors, the

inertia tensor cannot be obtained accurately. In addition,

external disturbances such as gravity gradient torque, solar

radiation pressure torque, and aerodynamic torques exist

ineluctably in practice. A spacecraft formation in LEO is

mainly affected by the gravity gradient torque, while the

disturbances such as the solar radiation pressure torque will

be dominant for a formation in high-earth orbits such as the

geostationary orbit. Taking into account all these factors,

the perturbation inertia tensor and the external disturbances

are chosen as:




Volume 85 · Number 3 · 2013 · 222–235

228

DJ 1 ¼ 0:3J 01; DJ 2 ¼ 0:4J 02; DJ 3 ¼ 0:25J 03; DJ 4 ¼ 0:35J 04;

d1¼ 0:12 20:16 0:12� �T

N·m; d2¼ 0:14 20:12 0:17� �T

N·m;

d3¼ 0:11 20:08 0:12� �T

N·m; d4¼ 0:09 20:15 0:1� �T

N·m:

Communication delays Tij are mainly induced by time delays

existing in the process of information collecting, computing,

and transmitting. When using star sensor to provide attitude

information, communication delays could be several control

period, and gyros could provide angular velocity information

with less delays. Moreover, communication delays also

depend on filtering algorithms and time for data packing

and unpacking. To test the validity of the proposed control

schemes, we choose the communication delays as follows:

T 12 ¼ T21 þ 0:2 ¼ T 13 þ 0:1 ¼ T31 2 0:4 ¼ T14 2 0:1

¼ T41 2 0:2 ¼ T 23 þ 0:3 ¼ T32 þ 0:6 ¼ T24 þ 0:5

¼ T42 þ 0:4 ¼ T 34 2 0:3 ¼ T43 þ 0:7

with T 12 ¼ 0:8 þ 0:5 sinðt=10Þcosðt=5Þ in seconds.The magnitude of the control torque is bounded as

ðuiÞk�� # 1N · m. The tuning of the control parameters a; b

and a is typically done experimentally, and their values are

limited by the frequency of the lowest unmodeled structural

resonant mode, the largest time-delay in actuators and the

sampling rate in practical situation. Furthermore, there is a

trade-off between control accuracy and settling time: high

control accuracy can be obtained by large control parameters

but at a cost of long settling time. Therefore, the value of the

control parameters a; b and a should be chosen according to the

system’s mechanical properties, the limitations on the actuators

and the available computing power. In this numerical example,

the control parameters are chosen as a ¼ 0:6; b ¼ 0:8 and

a ¼ 5=7. The weight parameter kiij of the ith satellite is chosen as

kiij ¼ 6. The parameter kjij and ksij is given by k

jij ¼ 100ksij, and:

k212ðtÞ ¼

4 for modðt; 7Þ # 6

0 for modðt; 7Þ . 6

(;

k212ðtÞ ¼ k3

13ðtþ 1Þ ¼ k414ðtþ 1:3Þ ¼ k1

21ðtþ 2:2Þ ¼ k323ðtþ 0:3Þ

¼ k424ðtþ 0:6Þ ¼ k1

31ðtþ 2Þ ¼ k232ðtþ 0:4Þ ¼ k4

34ðtþ 2:6Þ¼ k1

41ðtþ 3Þ ¼ k242ðtþ 1:7Þ ¼ k3

43ðtþ 0:8Þ

to describe a time-varying communication topology, where

modðx; yÞ calculates the remainder of dividing x by y.Simulation results of Scenario 1 are shown in Figures 3-6. The

performance of the satellite swarm is illustrated using the absolute

value of the Euler angle uj j which can be calculated using

equation (1). Figure 3 shows ueij j of each satellite within the

formation. Figure 4 shows uij�� between the ith and the jth agents

in the formation. By increasing the disturbance torque to twice,

thetransientresponseof ueij j and uij�� areshowninFigures5and6.

The simulation results validate the stability analysis of the

proposed control schemes. The plots in Figures 3 and 4 show

fine relative attitude control performance and attitude

maintenance performance in the presence of large initial

attitude errors, external disturbances, parameter uncertainties,

a time-varying communication topology, and information

transmission delays. Both the relative attitude errors and the

attitude maintenance errors fall to the tolerance in 38 s.

Although the disturbance torques is increased to twice, no

obvious variety in the transient process between Figures 3-4 and5-6 validates favorable robustness of the developed control

strategies.Results of Scenario 2 are shown in Figures 7-10 to illustrate

the effects of various control parameters on control

performance. Investigating the performance of different

coordination connections requires that some simulationparameters be held constant so that meaningful comparisons

can be made. At first, the effects of control parameters a and bon

control performance are investigated. By increasing the controlparameters a ¼ 0:6, b ¼ 0:8 to a ¼ 1:2, b ¼ 1:6 while keeping

the other control parameters constant, the transient response of

relative attitude errors and attitude maintenance errors areshown in Figures 7 and 8. Then, the effects of the weight

parameter kiij and kjij are investigated by simulations. By keeping

the other control parameters constant, three simulations are

done with the weight parameters kiij ¼ 0 and kjij ¼ 0, kiij ¼ 3 and

kjij ¼ 2, and kiij ¼ 6 and k

jij ¼ 3, respectively. Results

demonstrating the transient behavior of relative attitude errors

and attitude maintenance errors are shown in Figures 9 and 10.From Figures 7 and 8 it can be seen that the control

accuracy of both attitude tracking and relative attitude

maintenance is improved significantly, but the settling time

is increased from 38 to 50 s. The result illustrates that byincreasing the control parameters a and b, higher control

accuracy is able to be obtained at a cost of longer settling

time. As shown in Figures 9 and 10, when the weightparameters kiij and k

jij is decreased from 6 to 3 and 4 to 2,

respectively, the performance of attitude tracking is not

influenced but the attitude maintenance errors are increased.When kiij ¼ 0 and k

jij ¼ 0 which means that the attitude

maintenance control term is deactivated, each satellite in the

formation tracks the desired attitude without interconnectionwith others. It can be seen from Figures 9 and 10 that the

relative attitude errors are not affected while the attitude

maintenance errors are larger. In addition, since there is noinformation exchange, the satellite swarm performs as several

individuals, and the settling time of both attitude tracking and

attitude maintenance increased to 55 s.

Conclusions

In this paper, we solved the fast attitude coordination problemfor the satellite swarms in the presence of information

transmission delays, parameter uncertainties and external

disturbances. Based on a FSM and behavior-based controlmethod, a class of robust finite-time control laws was proposed

to guarantee fast coordination of the satellite swarms in the

presence of these unexpected phenomena. By virtue of two well-designed weight parameters, the control schemes are effective

subject to different topologies: full-connected or connected,

directed or undirected, fixed or time-varying. The convergencerate of the FSM was theoretically proven to be better than either

the LSM or the terminal sliding mode. Finally, numerical

simulations were performed in the presence of a switchingcommunication topology, communication delays, large

disturbances and parameter uncertainties. The effectiveness

and robustness of the proposed control strategies was validated,and the indispensability and effects of the coordination control

action were also demonstrated by the simulations.




Volume 85 · Number 3 · 2013 · 222–235

229

Figure 3 Relative attitude errors

0 50 10010–5

100

105

time (s)

|θe1

| (de

g)|θ

e3| (

deg)

|θe2

| (de

g)|θ

e4| (

deg)

0 50 10010–5

100

105

time (s)

0 50 10010–5

100

105

time (s)

0 50 10010–5

100

105

time (s)

Figure 4 Attitude maintenance errors

0 50 100

10–5

100

105

time (s)

0 50 100

time (s)

0 50 100 0 50 100

0 50 100 0 50 100

|θ12

| (de

g)

10–5

100

105

|θ14

| (de

g)

10–5

100

105

|θ24

| (de

g)

10–5

100

105

|θ34

| (de

g)

10–5

100

105

|θ23

| (de

g)

10–5

100

105

|θ13

| (de

g)




Volume 85 · Number 3 · 2013 · 222–235

230

Figure 5 Relative attitude errors with increased disturbances

0 50 10010–5

100

105

time (s)

|θe1

| (de

g)|θ

e3| (

deg)

|θe2

| (de

g)|θ

e4| (

deg)

0 50 10010–5

100

105

time (s)

0 50 10010–5

100

105

time (s)0 50 100

10–5

100

105

time (s)

Figure 6 Attitude maintenance errors with increased disturbances

0 50 100

10–5

100

105

time (s)0 50 100

time (s)

0 50 100 0 50 100

0 50 100 0 50 100

|θ12

| (de

g)

10–5

100

105

|θ14

| (de

g)

10–5

100

105

|θ24

| (de

g)

10–5

100

105

|θ34

| (de

g)

10–5

100

105

|θ23

| (de

g)

10–5

100

105

|θ13

| (de

g)




Volume 85 · Number 3 · 2013 · 222–235

231

Figure 7 Relative attitude errors with various control parameters

0 50 100

a = 0.6, b= 0.8

a = 1.2, b= 1.6

10–5

100

105

time (s)

|θe1

| (de

g)|θ

e3| (

deg)

|θe2

| (de

g)|θ

e4| (

deg)

0 50 10010–5

100

105

time (s)

0 50 10010–5

100

105

time (s)0 50 100

10–5

100

105

time (s)

Figure 8 Attitude maintenance errors with various control parameters

10–5

100

105

time (s) time (s)

0 20 40 60 80 100 0 20 40 60 80 100

0 20 40 60 80 100

0 20 40 60 80 100 0 20 40 60 80 100

0 20 40 60 80 100

|θ12

| (de

g)

10–5

100

105

|θ14

| (de

g)

10–5

100

105

|θ24

| (de

g)

10–5

100

105

|θ34

| (de

g)

10–5

100

105

|θ23

| (de

g)

10–5

100

105

|θ13

| (de

g)




Volume 85 · Number 3 · 2013 · 222–235

232

Figure 9 Relative attitude errors with various weight parameters

10–5

100

105

Kij

i

= 0

Kij

i

= 3

Kij

i

= 6

time (s)

0 20 60 8040 100

0 20 60 8040 100

time (s)

|θe1

| (de

g)

10–5

100

105

0 20 60 8040 100time (s)

|θe2

| (de

g)

10–5

100

105

|θe3

| (de

g)

0 20 60 8040 100

time (s)

10–5

100

105

|θe4

| (de

g)

Figure 10 Attitude maintenance errors with various weight parameters

0 50 100

10–5

100

105

time (s)

0 50 100

time (s)

0 50 1000 50 100

0 50 100

|θ12

| (de

g)

10–5

100

105

|θ14

| (de

g)

10–5

100

105

|θ24

| (de

g)

10–5

100

105

10–5

100

105

0 50 10010–5

100

105

|θ34

| (de

g)|θ

23| (

deg)

|θ13

| (de

g)

Kij

i = 0 K

ij

i = 3 K

ij

i = 6




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Further Reading

Jin, E., Jiang, X. and Sun, Z. (2008), “Robust decentralized

attitude coordination control of spacecraft formation”,

Systems & Control Letters, Vol. 57, pp. 567-577.

About the authors

Haizhao Liang received his Bachelor and Master degree

from Harbin Institute of Technology. Since 2010, he has been a

PhD candidate of Research Center of Satellite Technology,

Harbin Institute of Technology. Now he is an exchange

PhD student supported by the China Scholarship Council with

the Department of Electronic and Information at Politecnico di

Milano, Italy. His research interests include spacecraft

attitude control, cooperative control of formation system and




Volume 85 · Number 3 · 2013 · 222–235

234

robust control. Haizhao Liang is the corresponding author andcan be contacted at: [email protected]

Zhaowei Sun received the Master’s degree in generalmechanics from Beijing Institute of Technology, China, in1988, and the PhD in spacecraft design from Harbin Instituteof Technology, Harbin, China, in 2002. Since 1997 he hasworked as a Professor at Harbin Institute of Technology,Harbin, China. His research interests include spacecraft

system design and simulation technique, spacecraft dynamicsand control technique.

Jianying Wang received her Bachelor degree from HarbinInstitute of Technology. Since 2008, she has been a PhDcandidate of Research Center of Satellite Technology, HarbinInstitute of Technology. Her research interests includespacecraft formation control, 6-DOF control of spacecraftand finite-time control.




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Attitude coordination of satellite swarms with communication delays