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Attitude cooperative control of spacecraft

formation via output-feed back

Yueyong Lv, Qinglei Hu, Guangfu Ma and Jian Zhang

Department of Control Science and Engineering, Harbin Institute of Technology, Harbin, China

Abstract

Purpose – The purpose of this paper is to propose a decentralized output feedback controller for cooperative attitude regulation of spacecraft

formation in absence of angular velocity feedback.

Design/methodology/approach – The nonlinear relative attitude dynamic and kinematic equations represented by relative quaternion and relative

angular velocity, respectively, are considered in this paper. The lead ﬁlter is employed to synthesize virtual angular velocity signal so that the design of

output feedback controller is achieved. Lyapunov method is adopted to prove the stability of closed-loop system. Considering the external disturbance,

the theory of L2-gain disturbance attenuation is employed to improve the designed controller. Numerical simulations are carried out to verify the

controllers proposed.

Findings – It is found that the closed-loop system can be guaranteed asymptotically stable in absence of external disturbance. When disturbance is

considered, as long as the sufﬁcient condition proposed is satisﬁed, the improved controller can render system uniformly ultimately bounded stable.

Practical implications – The proposed output feedback control scheme can be considered as a fall-back alternative for the case that the angular

velocity sensors fail, or seen as another option for the system without angular velocity sensors at all.

Originality/value – Unlike most classical works in the ﬁeld of output feedback which focus on centralized scheme and neglect the disturbance, the

controller proposed in this paper is able to handle the output feedback control problem of multi-agent formation in a decentralized fashion, so as to

avoid the single failure point of a centralized scheme. Meanwhile, the capability of L2-gain disturbance attenuation is also achieved simultaneously.

Keywords Spacecraft, Controllers, Decentralized output feedback, Cooperative attitude control, Spacecraft formation ﬂying, Attitude regulation,

L2-gain

Paper type Research paper

Introduction

Spacecraft formation ﬂying (SFF) has received extensive

interests in recent years (Bristow et al., 2000). By distributing

the functionality of a monolithic complex spacecraft between a

set of more than one smaller and closely ﬂying spacecraft, the

performance of the whole system can be enhanced greatly.

Comparing with spacecraft constellation that is also based on

the distributing concept (Folta et al., 1998), any of the agent

spacecraft dynamic states are coupled together (Scharf et al.,

2004), while the dynamic states of spacecraft constellation are

mutually independent. In order to satisfy the requirement of

formation deployment so that the formation would work as a

virtual monolithic spacecraft, precise relative attitude control is

identiﬁed as another key issue as control of relative position.

Furthermore, when speciﬁed requirement is made such as

attitude coincident and alignment, it is of particular

signiﬁcance to promote the research on cooperative attitude

control of SFF.

To implement cooperative attitude control, there are three

main control architectures: Leader-Follower (L-F), behavioral

and virtual structure architectures. The L-F architecture is

the most studied in present literatures, in which the

cooperative attitude control of formation is reduced to an

attitude tracking problem for the follower spacecraft itself

(Wang and Hadaegh, 1996; Wang et al., 1999; Kristiansen,

2009; Bondhus, 2005). Although this reduction could bring

the existing attitude control algorithms for a single spacecraft

to SFF and make the stability analysis easy, the follower is a

single point of fault. Additionally, when the leader spacecraft

rotates too fast for the follower to track, the conﬁguration of

the formation would be difﬁcult to keep due to there is no

explicit feedback from the follower to the leader (Wang et al.,

1999). As an alternative to L-F architecture, the basic idea of

virtual structure architecture is to regard the formation as a

whole virtual space structure, and each formation agent is

controlled individually. Comparing with L-F architecture, the

desired state trajectory is generated by the virtual structure

not a speciﬁc spacecraft as in L-F. In Beard et al. (2001) and

Ahn and Kim (2008), the virtual structure scheme was

successfully applied in attitude synchronization. Generally

speaking, the two architectures mentioned above have both

fallen into the category of centralized control. So the problem

of fault sensitive and tracking capability in L-F architecture

still exists in the virtual structure. In order to handle this

problem, explicit feedback was introduced into the virtual

structure (Ren and Beard, 2004a, b), and a cooperative

The current issue and full text archive of this journal is available at

www.emeraldinsight.com/1748-8842.htm

Aircraft Engineering and Aerospace Technology: An International Journal

84/5 (2012) 321–329

q Emerald Group Publishing Limited [ISSN 1748-8842]

[DOI 10.1108/00022661211255502]

This work was supported by the Scientiﬁc Research Foundation for

National Natural Science Foundation of China (Project Number:

61,004072, 61,174200), and Program for New Century Excellent

Talents in University (NCET-11-0801). The author fully appreciates the

ﬁnancial support. The author would also like to thank the reviewers and

the Editor for many suggestions that helped improve the paper.

321

control scheme with each spacecraft’s information was

proposed. Nevertheless, each agent needs to run the

complex computation of virtual structure, and thus heavy

computation ability is required for the onboard computer. As

discussed in Balch and Arkin (1998) and Beard et al. (2001),

the behavioural architecture combines the outputs of multiple

controllers designed for different and possibly competing

behaviours together. Roughly speaking, it is a kind of complex

strategy. In Lawton et al. (2000), the behavioural architecture

is also used to maintain attitude alignment among a set of

spacecraft. An advantage of the behavioural architecture is

the explicit feedback is included, while it is hard to analyze

mathematically (Ren and Beard, 2004a, b). Indeed, all the

architectures stated previously fall into two opposite

categories: centralized and decentralized. Single point of

failure is an insoluble question for the centralized method,

which takes L-F architecture as typical representative.

Consequently, the centralized method is only suit for the

case that there are only few agents in the formation. Even

though some proposed method could work for a n-agent

formation and n could be arbitrarily large (Wong et al., 2002),

it can be converted to a set of two-body problem essentially.

While, it should be noted that all the literatures mentioned

above have utilized feedback of all states, i.e. full state

feedback control, which has naturally made an assumption

that each agent knows not only its own angular velocity but

also its neighbouring agents’. However, such assumption

cannot be always satisﬁed due to either cost limitation or

implementation considerations. As a remedy to such

situation, output feedback control methods, or methods

without angular velocity, have been proposed by several

researchers (Kristiansen, 2009; Bondhus, 2005; Ren and

Beard, 2004a, b; Wong et al., 2002; Caccavale and Villani,

1999; Wei, 2007; Turner, 2009; Lawton and Beard, 2002; Wu

et al., 2010; Abdessameud and Tayebi, 2009). Caccavale and

Villani (1999) presented two alternative schemes for attitude

tracking of single spacecraft in the absence of angular velocity

feedback. One scheme is based on model-based observer,

while the other is based on a lead ﬁlter. Especially what

deserves to be mentioned is that the lead ﬁlter approach can

be seen as a classical method for solving the output feedback

problem, and we try to employ this ﬁlter in this work. In Wei

(2007), the proposed distributed attitude synchronization

approach has taken advantage of Modiﬁed Rodriguez

Parameters (MRP) for attitude representation, and it is not

suitable for the design of global control law due to singularity of

MRP. Obviously, a simple method to handle this problem is to

utilize another attitude representation in which there is no

singularity. Fortunately, quaternion (Turner, 2009) is such one

kind of attitude representation we need. Lawton and Beard

(2002) presented an attitude alignment law which requires a

bidirectional ring communication topology and relies on a

restrictive domain of attraction for convergence analysis. By

contrast, in Wu et al. (2010), the only assumption of the inter

communication topology is that the communication links are

bidirectional. Meanwhile, it is should be noted that all the above

literature about output feedback did not consider the affect of

external disturbance torque. However, in practical application,

the disturbance could not be neglected. For the disturbance

attenuation, the theory of L2 stability (Khalil, 2002) provides

a feasible approach.

Inspired by the analysis above, the main contribution of this

work lies in proposing a decentralized output control law for

cooperative attitude regulation of SFF. Since the attitude is

represented by means of quaternion, the design of global

control law has been brought into practice. Due to the

employment of lead ﬁlter, a pseudo velocity vector is produced

so that only the quaternion feedback signal is taken by the

controller, i.e. output feedback. Furthermore, when the

external disturbance is considered, the controller is improved

to convert the closed-loop system the capability of L2-gain

disturbance attenuation. The rest of this paper is organized as

follows. The next section states attitude and relative attitude

dynamics and kinematics of spacecraft together with basic

L2-gain disturbance attenuation theory. Followingly, control

laws and closed-loop system stability analysis are derived.

Then, numerical simulation and comparison to verify the

performance of the proposed controller are described. Finally,

the paper is completed with some concluding comments.

Mathematical model

Attitude dynamics and kinematics

For a n-agent (n $ 2) spacecraft formation, the dynamics of

the ith agent spacecraft in the formation is described by Euler

differential equation for the attitude rotation and is given by

Shuster (1993):

¼ 2Sð

ÞJ

þ u

þ d

; i ¼ 1; 2· · · ð1Þ

where J

[ R

3£3

denotes the symmetric positive-deﬁnite inertia

matrix of the ith spacecraft,

[ R

is the angular velocity,

and u

and d

are control and disturbance torque vector,

respectively. Accordingly, the kinematic equation using

quaternion q

¼½



is expressed as (Shuster, 1993),

where 1

is a three dimensional vector:

T ðq

Eðq

; i ¼ 1; 2· · · ð2Þ

where the quaternion q

is deﬁned with the constraint q

¼ 1,

and the quaternion operator Eð· Þ :R

! R

3£3

is deﬁned as:

Eðq

Þ¼

þ Sð1

Þð3Þ

Here, the matrix operator S( · ) represents a skew-symmetric

matrix which is equivalent to the vector cross product operation

as follows:

SðaÞb ¼ a £ b; ;a ; b [ R

ð4Þ

with the following property holds:

SðaÞ

¼ 0; ;

– 0 ð5Þ

According to the deﬁnition, the following property of E(·)

is easy to prove.

Property 1. For the quaternion operator deﬁned in

equation (3), it holds the property as:

Eðq

; ;

– 0 ð6Þ

Attitude cooperative control of spacecraft formation

Yueyong Lv, Qinglei Hu, Guangfu Ma and Jian Zhang

Aircraft Engineering and Aerospace Technology: An International Journal

Volume 84 · Number 5 · 2012 · 321 – 329

322

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