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Acta Astronautica

Acta Astronautica 83 (2013) 44–53

0094-57

http://d

$ Fou

(No. 10$$ Tn Corr

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jia_ying

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journal homepage: www.elsevier.com/locate/actaastro

Control analysis of an underactuated spacecraft underdisturbance$,$$

Dongxia Wang, Yinghong Jia n, Lei Jin, Shijie Xu

Department of Guidance, Navigation and Control, School of Astronautics, Beihang University, Beijing 100191, China

a r t i c l e i n f o

Article history:

Received 2 November 2011

Received in revised form

17 September 2012

Accepted 18 October 2012Available online 20 November 2012

Keywords:

Underactuated spacecraft

Periodical oscillation disturbance

Controllability

Weakly positively Poisson stable

Lie algebra

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

x.doi.org/10.1016/j.actaastro.2012.10.029

ndation item: National Natural Science Fou

902003).

his paper was presented during the 62nd IA

esponding author. Tel.: þ 86 01082319751

ail addresses: wdx2008abc@163.com (D. Wa

h@yahoo.com.cn (Y. Jia), jinleibuaa@163.com

@yahoo.com.cn (S. Xu).

a b s t r a c t

Based on the nonlinear controllability theory, this paper analyzes the attitude controllability

for an underactuated spacecraft using two or one thrusters in sequence. In order to provide a

preconditional guide on designing the control law for the underactuated attitude control

system in the actual case, the underactuated spacecraft with respect to the orbit frame is

investigated in the presence of periodical oscillation disturbance. First, attitude dynamic

model was established for an underactuated spacecraft actuated by three thrusters, one or

two of which has failed separately, and the special orthogonal group (SO(3)) is deployed to

describe the attitude motions. Second, Liouville theorem and Poincare’s recurrence theorem

were used to confirm that the drift field is Weakly Positively Poisson Stable (WPPS).

Furthermore, the sufficient and necessary condition of controllability was obtained on the

basis of Lie Algebra Rank Condition (LARC). Finally, according to the case of two available

thrusters, angular velocity stabilization and three-axis attitude stabilization control laws

are designed, and proved by Lyapunov stability theory and LaSalle invariant theorem.

The analysis and simulation results illustrate the feasibility of the proposed control law.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

During the past two decades, underactuated spacecraftattitude control problem has become a new hot researcharea. The stabilization of underactuated system not onlyguarantees safe operations of the spacecraft, but also hasspecial significances for small satellites and deep spaceprobes which have weight, size and cost limitation. Con-trollability is the premise of designing attitude control lawof system. However, for the underactuated spacecraft, thedimension of attitude control system is less than thenumber of degree of freedom, which essentially changedthe characteristics of the originally complete system andcannot ensure the controllability. Therefore, in order to

All rights reserved.

ndation of China

C in cape town.

.

ng),

(L. Jin),

offer a preconditional guide on designing the control law, itis necessary to study the problem of controllability forunderactuated spacecraft attitude control system.

Thruster is a kind of actuator which has multidimen-

sional control effect. In the attitude control system, one

thruster can generate a three-dimension torque, any two

dimensions of whose magnitude exists in a special ratio. For

underactuated rigid spacecraft with thrusters, Controllabil-

ity was first studied in 1983 as in reference [1]. The author

provided sufficient and necessary condition for the controll-

ability of a rigid body in the cases of one, two and three

independent control inputs. It was later demonstrated by

references [2,3], by using geometric control theory, that

small-time local controllability (STLC) of the rigid body

equations can indeed be achieved with only two control

torques. The study of [4] talked about the controllability of

spacecraft system in a central gravitational field, and

suggested that the requirement of positively Poisson stabi-

lity could be replaced by a weakly condition, i.e. the Weakly

Positively Poisson Stable condition (WPPS). In reference [5],

the controllability of the attitude control system was

iX

iY

iZ

iO

0ω

0ω oX

oYoZ

X

YZ

o

Fig. 1. The geometry of the three coordinate frames.

D. Wang et al. / Acta Astronautica 83 (2013) 44–53 45

studied for the underactuated spacecraft with two thrusters,and the necessary and sufficient condition was proved.The references above mainly studied the controllability ofunderactuated spacecraft using thrusters. Moreover, con-trollability of underactuated spacecraft using other actua-tors was also studied. Reference [1] concluded that, for aspacecraft with momentum exchange devices, controllabil-ity is impossible with fewer than three control inputs whenthe total angular momentum is not equal to zero. Reference[6] used nonlinear controllability theory to show that aspacecraft carrying one or more CMGs was controllable onevery angular momentum level set in spite of the presenceof singular CMG configurations, that is, any two stateshaving the same angular momentum could reach each otheron the basis of choosing proper moving trace of the CMGgimbals. This result was used to obtain sufficient conditionson the momentum volume of the CMG array that guaran-teed the existence of gimbal motions which steered thespacecraft to a desired spin state or rest attitude. Inreference [7], the spacecraft with two reaction wheels andnon-zero momentum was not controllable over the entirestate space, but a five-dimensional subspace of feasiblestates was potentially reachable, and the subset was expli-citly defined. Reference [8] considered the controllability ofspacecrafts actuated by three magnetic actuators, whichindicated that the attitude dynamical system was control-lable if the orbital plane did not coincide with the geomag-netic equatorial plane. In reference [9], controllability testsand motion control algorithms for underactuated mechan-ical control systems on Lie groups were provided, and localcontrollability properties were characterized. Reference [10]showed the controllability of underactuated rolling system.

All of the above achievements mainly focus on thecontrollability of inertia attitudes of spacecraft whileconsidering little about disturbance. However, whenspacecraft operates in the space, what researchers reallycared is the attitude of the body-fixed reference framewith respect to orbit frame. Therefore, for the under-actuated spacecraft actuated by two or one thrustersseparately, controllability of attitude with respect to theorbit frame is studied in the presence of periodicaloscillation disturbance. The goal of this paper is to enrichthe controllability theory of underactuated attitude con-trol system, and to offer a preconditional guide ondesigning attitude control law.

2. Mathematical models

In this section, the attitude dynamic model is estab-lished for a spacecraft actuated by three thrusters in thepresence of periodical oscillation disturbance. And then thedynamic equation is deduced while one or two of thrustersfailed separately. Finally, spacecraft attitude motion isdescribed by the special orthogonal group (SO(3)).

2.1. Dynamic modeling

2.1.1. The general dynamic equation of a spacecraft

actuated by three thrusters

Introducing three coordinate frames: the inertial framedenoted by si OiXiYiZið Þ, has its origin located at the center

of the Earth, with the Zi-axis passing through the celestialNorth pole, the Xi-axis directed towards the vernal equi-nox, and the Yi-axis completing the right-handed rule.Next, we define a local vertical local horizontal (LVLH)orbital reference frame so OXoYoZoð Þ with its origin at themass center of the spacecraft. Here the Xo-axis along theline of intersection of the orbital plane and the localhorizontal plane pointing to the orbital forward direction,the Zo-axis along the local vertical pointing at the centerof the earth, and the Yo-axis is taken the normal to theorbital plane and the direction is confirmed by the right-hand rule with Xo-axis and Zo-axis. The body-fixed framedenoted by framesbðOXYZÞ, of which origin is at the centerof the spacecraft, X,Y ,Z are unit vectors aligned with thethree inertial principal axes of spacecraft. The relationshipof the coordinate frames can be described in Fig. 1.

Then the attitude dynamic equation of a rigid space-craft can be written as

J _x ¼x�JxþTþTd ð1Þ

where x is the angular velocity of the body-fixed framewith respect to the inertial frame. x� is the antisym-metric matrix of the cross product [11]. That is, for anyvector x¼ o1 o2 o3

� �T, there is

x� ¼0 o3 �o2

�o3 0 o1

o2 �o1 0

264

375

and J is the inertia moment matrix of the spacecraft.T¼ TX ,TY ,TZ½ �

T denotes the external torque generated bythrusters along the axes of body-fixed frame of thespacecraft. Td depicts the periodical oscillation distur-bance torque. Without loss of the generality, we set

Td ¼

d1

d2

d3

264

375¼

a1sin O1tð Þ

a2sin O3tð Þ

a3sin O3tð Þ

264

375 ð2Þ

where ai is the amplitude, Oi is the frequency. Obviously,:di:rai, i¼ 1,2,3 .

The magnitude and direction of thrusters torque areshown in Fig. 2, where a1,a2,a3 denote the direction

X

Y

Z

α1α2

α3

β1

β2

β3

γ1γ2

γ3

T 2

T1

T 3

Fig. 2. The magnitude and direction of thrusters torque.

D. Wang et al. / Acta Astronautica 83 (2013) 44–5346

angles between T1 (the torque of the first thruster) andthe axes X, Y, Z of body-fixed frame. Similarly, b1,b2,b3 arethe direction angles between T2 (the torque of the secondthruster) and the axes X, Y, Z of body-fixed frame. g1,g2,g3

are the direction angles between T3 (the torque of thethird thruster) and the axes X, Y, Z of body-fixed frame.Thus, we obtain

TX

TY

TZ

264

375¼

cosa1 cosb1 cosg1

cosa2 cosb2 cosg2

cosa3 cosb3 cosg3

264

375

T1

T2

T3

264

375 ð3Þ

2.1.2. The dynamic equation of a spacecraft actuated by two

thrusters

Suppose the thruster that generates T3 fails, and thenEq. (3) yields to

T¼

TX

TY

TZ

264

375¼

T1cosa1þT2cosb1

T1cosa2þT2cosb2

T1cosa3þT2cosb3

264

375 ð4Þ

Set

u1

u2

" #¼

cosa1 cosb1

cosa2 cosb2

" #T1

T2

" #,

then, Eq. (4) can be simplified as

T¼

TX

TY

TZ

264

375¼

u1

u2

b1u1þb2u2

264

375 ð5Þ

where

b1 ¼cosa3cosb2�cosa2cosb3

cosa1cosb2�cosa2cosb1

,

b2 ¼cosa1cosb3�cosa3cosb1

cosa1cosb2�cosa2cosb1

Therefore, the attitude dynamic equation of the under-actuated spacecraft actuated by two available thrusterscan be written as:

J _x ¼x�Jxþg1u1þg2u2þTd ð6Þ

where

g1 ¼ 1 0 b1� �T

, g2 ¼ 0 1 b2� �T

2.1.3. The dynamic equation of a spacecraft actuated by one

thruster

Suppose the thrusters that generate T2 and T3 fail, andthen Eq. (3) yields to

T¼

TX

TY

TZ

264

375¼

T1cosa1

T1cosa2

T1cosa3

264

375 ð7Þ

Set u1 ¼ T1cosa1, then, Eq. (7) can be simplified as

T¼

TX

TY

TZ

264

375¼

u1

c1u1

c2u1

264

375 ð8Þ

where

c1 ¼ cosa2=cosa1,c2 ¼ cosa3=cosa1:

Therefore, the attitude dynamic equation of the under-actuated spacecraft actuated by one available thruster canbe written as:

J _x ¼x�Jxþh1u1þTd ð9Þ

where h1 ¼ 1 c1 c2� �T

.

2.2. Kinematic modeling

Kinematic equation of spacecraft is a group of first-order differential equation, which determines the varia-tion of attitude parameters over time. In this paper, weuse SO(3), an orthogonal matrix of which determinantequals 1, to describe the motion of the attitude. Thecorresponding kinematic equation can be written as

_R ¼xr�R ð10Þ

where xr is the rotational angular velocity of the body-fixed frame with respect to the orbit frame. The relationbetween x and xr can be expressed as

x¼xrþCboxe ð11Þ

where xe ¼ 0 �o0 0� �T

is the orbital angular velo-city with respect to the inertial frame expressed in thebody-fixed frame. o0 is a constant in circular orbit anddenotes the orbit angular rate. Cbo is the rotation matrixfrom the orbit frame to body-fixed frame [12]. Under theassumption that the attitude changes little, Cbo can beapproximated as the identity matrix.

3. Controllability analysis

3.1. The notation and preliminaries based on nonlinear

control system

The attitude control system for an underactuatedspacecraft is essentially a nonlinear control system.Therefore, the condition for controllability of nonlinearcontrol system need to be introduced first based on thedifferential manifold. For a nonlinear control system, it

D. Wang et al. / Acta Astronautica 83 (2013) 44–53 47

yields

_x ¼ fðxÞþXn

i ¼ 1

giðxÞui; x 2M ð12Þ

where u 2 X, X is the subset of Rn, M is a C1

connected manifold of dimension n. x is the state vector,

f and gi are analytic vector fields on M. Let the multi-

variable system be g¼ ffþPn

i ¼ 1

giui9u 2 Og and LðgÞ be the

smallest Lie algebra that contains g. Thus, it can beconcluded that LðgÞ is equivalent to the Lie algebra

generated by f,g1 � � �gn

� �. When the dimension of

span L f,g1 � � �gn

� �ðpÞ is n, the point is said to be satisfying

the Lie algebra rank condition (LARC). If it is true for everypoint in M, the system satisfies LARC. Then, the followingTheorem 1 is given to derive the sufficient and necessarycondition for controllability of nonlinear system.

Theorem 1. if f(x) is a WPPS vector field, System (12) is

controllable iff the LARC is satisfied, i.e. for any 8p 2M ,

there is

dim½span L f,g1 � � �gm

� �ðpÞ� ¼ dim TpM

� �¼ n ð13Þ

Where TpM is the tangent space at the point p.For the above theorem, the ‘‘if’’ part is proved by Lian

and Wang [4] and the ‘‘only if’’ part is confirmed by Lobry[13]. One can refer to the related reference for the detailsof the proof.

Based on Theorem 1, we can derive the sufficient andnecessary condition for the controllability of the system. Itsprocess is divided into two steps. The first step is to showthat the system is WPPS. Then we can deduce the conditionby which the underactuated attitude system satisfies LARC.According to Theorem 1, the condition is the necessary andsufficient condition of controllability. In the following, weseparately obtain the condition in the cases of two availablethrusters and one available thruster. On the state manifold

M¼ SOð3Þ � R3, let x¼R

x

� , we have

3.1.1. Two available thrusters

Composing Eqs. (6) and (10) to the form of Eq. (12):

fðxÞ ¼xr�R

J�1 x�JxþTdð Þ

" #,g1 ¼

03�1

g1

" #,g2 ¼

03�1

g2

" #ð14Þ

3.1.2. One available thruster

Composing Eqs. (9) and (10) to the form of Eq. (12):

fðxÞ ¼xr�R

J-1 x�JxþTdð Þ

" #, h1 ¼

03�1

h1

" #ð15Þ

Comparing the forms of Eqs. (14) and (15), it is knownthat the drift vector field fðxÞ is the same. Thus, whetherWeakly Positively Poisson Stable (WPPS) or not is thesame conclusion.

3.2. WPPS of the underactuated attitude control system

The drift vector field is the vector field of no externalcontrol input of the system. To prove that the drift vector

field of the underactuated attitude control system satis-fies WPPS, first the article uses Liouville theorem toconfirm that the flow generated by the drift vector ofthe underactuated attitude control system is volume-preserving, and then it is concluded that this drift fieldis Weakly Positively Poisson Stable (WPPS) according toPoincare’s recurrence theorem.

In order to show that the drift vector field is a volume-preserving flow, Liouville’s theorem needs to beintroduced.

Theorem 2 [14]. Suppose that f is a vector field and ft is the

flow generated by f. If div f¼0, then ft preserves volume.

According to the definition of divergence, we have

divf ¼ tr@xr

�R

@Rþ@J-1 x�JxþTdð Þ

@x

!

¼ tr@xr

�R

@R

�þtr

@J-1 x�JxþTdð Þ

@x

!¼ 0 ð16Þ

From Theorem 2, the drift vector field f defines avolume-preserving flow. In the case of no external input,the total kinetic energy is constant, i.e.

1

2xTIx¼ C ð17Þ

where C denotes a constant. Considering periodicaloscillation disturbance torque, the state of underactuatedattitude control drift system change in the boundedregion D defined as follows

D¼ xT� � 1

2xTJxþ

X3

i ¼ 1

:di:rCþX3

i ¼ 1

ai&:R:¼ 1

�����)(ð18Þ

Therefore, the drift vector field of underactuated sys-tem yields a volume-preserving flow, and all of the statevariables are in bounded D. According to the Poincare’srecurrence theorem [14], for a volume-preserving, con-tinuous map g on a bounded domain DD IRn, everyneighborhood U of each point in D contains a point qwhich returns to U after repeated application of themapping, i.e., q 2 Up,8n,gnðqÞ 2 Up, the system is WPPS.

The above-mentioned analysis indicates that becauseof no-coupling between Td and x, the volume-preservingcharacteristic of the drift vector field is not affected. Inother words, the WPPS is still satisfied, but the simulta-neously bounded disturbance will reduce the stabilityprecision (because vibrating domain is enlarged, see Eq.(18)) as replacing expense.

3.3. LARC

Obviously, the coefficients of control g for system (14)are different from h for system (15), thus, the tangentspace generate by f,gi

� �for system with two available

thrusters is different from f,hi

n ofor system with one

available thruster. In this section, we deduce the neces-sary and sufficient condition for LARC based on which thesystem (14) and system (15) satisfy Eq. (13) in sequence.

Before calculating the Lie brackets [15], we improvethe pro.3 of reference [16], and derive the followingTheorem 3.

D. Wang et al. / Acta Astronautica 83 (2013) 44–5348

Theorem 3. Let Q ¼ SOð3Þð Þm� Rn , p 2 Q can be para-

meterized as

p¼ A1, � � � , Am, rð Þ

where A1, � � � ,Am 2 SOð3Þ , u1, � � � ,um,a1, � � � ,am2 R3 ,r,v,c 2 Rn . Let vector fields X,Y 2 VðQ Þ at point p be

denoted as

XðpÞ ¼ u1ðpÞð Þ�A1, � � � , umðpÞð Þ

�Am,vðpÞð Þ

YðpÞ ¼ a1ðpÞð Þ�A1, � � � , amðpÞð Þ

�Am, cðpÞð Þ

The flows of X,Ycan be found to be respectively:

fXtðpÞ ¼ et u1ðpÞð Þ

�

A1, � � � , et umðpÞð Þ�

Am, rþtvðpÞ �

fYlðpÞ ¼ el a1ðpÞð Þ

�

A1, � � � , el amðpÞð Þ�

Am,rþl cðpÞ �

Then the Lie bracket of X,Y at p can be computed as

X,Y½ �ðpÞ ¼ ddt

��t ¼ 0

d

dl

����l ¼ 0

fX�tðpÞ3f

Yl ðpÞ3f

Xt ðpÞ

¼ a1�u1ð Þ

�A1�d

dl��0

u13fYl

��A1þ

d

dt��0

a13fXt

��A1, � � � ,

am�umð Þ

�Am�d

dl��0

um3fYl

��Am

þd

dt��0

am3fXt

��Am,

d

dt��0

c fXtðpÞ

��

d

dl��0

v fYl ðpÞ

�!

3.3.1. Two available thrusters

The simplest tangent space is composed of elementslisted as follows

g1, f,g1

� �, g2, f,g2

� �, g3, f,g3

� �ð19Þ

where g3 ¼ f,l1g1þl2g2

� �, l1g1þl2g2

� �� �. According to

the calculating method of Theorem 3, we have

f,gi

� �¼

�gi�R

J-1ðJxÞ��x�Jð Þgi

" #, i¼ 1,2 ð20Þ

g3 ¼ f,l1g1þl2g2

� �, l1g1þl2g2

� �� �¼

03�1

2J�1 l1g1þl2g2

� ��J l1g1þl2g2

� �" #ð21Þ

Let g3 ¼ J�1 l1g1þl2g2

� ��J l1g1þl2g2

� �, then

g3 ¼03�1

2g3

" #ð22Þ

f,g3

� �¼

�g3�R

J�1ðJxÞ��x�Jð Þg3

" #ð23Þ

Thus, the condition ensuring LARC is

span0

g1

" #,

�g1�R

J-1ðJxÞ��x�Jð Þg1

" #,

0

g2

" #,

�g2�R

J-1ðJxÞ��x�Jð Þg2

" #,

(

0

g3

" #,

�g3�R

J-1ðJxÞ��x�Jð Þg3

" #)¼ R6

ð24Þ

Obviously, according to the primary transformationof matrix, the formula (24) is simplified to be

span g1,g2,g3

� �¼ R3, where gi i¼ 1,2,3ð Þ is shown as Eqs.

(6) and (22).According to the above analysis and Theorem 1, the

necessary and sufficient condition for controllability ofunderactuated spacecraft attitude control system withtwo thrusters is: (l1,l2 2 R,

span g1,g2,g3

� �¼ R3

ð25Þ

Now, we deduce the controllability condition from Eq.(25) in a special case: suppose the axes of the body-fixedframe are along the inertial principal axes, we have thatinertial matrix can decouple into the form:

J¼ diag J1 J2 J3

�, and suppose the thrusters installed

along the body-fixed frame, that is b1 ¼ 0,b2 ¼ 0. We can

obtain that g3 ¼� 0 0 2l1l2 J2�J1

� �=J3

h iTfrom Eq.

(22). Obviously, according to Eq. (25), if and only if

J1aJ2, there is span g1,g2,g3

� �¼ R3. That is to say, under-

actuated spacecraft cannot be symmetric along theunderactuated axis.

3.3.2. One available thruster

The simplest tangent space is composed of elementslisted as follows

h1, f1, h2, f,h2

h i, h3, f,h3

h ið26Þ

where f1 ¼ f,h1

h i, h2 ¼ f1,h1

h i, h3 ¼

hhf,Z1h1þZ2h2

i,

Z1h1þZ2h2

�i. By calculating like in Theorem 3, we

obtain

f1 ¼ f,h1

h i¼

�h1�R

J�1ðJxÞ��x�Jð Þh1

" #ð27Þ

h2 ¼ f1,h1

h i¼

03�1

�J-1 Jh1ð Þ��h1

�J� �

h1

" #ð28Þ

Let h2 ¼�J-1 Jh1ð Þ��h1

�J� �

h1, then

h2 ¼03�1

h2

" #ð29Þ

f,h2

h i¼

�h2�R

J-1ðJxÞ��x�Jð Þh2

" #ð30Þ

h3 ¼ f,Z1h1þZ2h2

h i, Z1h1þZ2h2

�h i

¼03�1

�J�1 J Z1h1þZ2h2

� �� ��� Z1h1þZ2h2

� ��J

� �Z1h1þZ2h2

� �" #

ð31Þ

Let h3 ¼ �J�1 J Z1h1þZ2h2

� �� ���

�Z1h1þZ2h2

� ��JÞ

Z1h1þZ2h2

� �, then

h3 ¼03�1

h3

" #ð32Þ

f,h3

h i¼

�h3�R

J�1ðJxÞ��x�Jð Þh3

" #ð33Þ

D. Wang et al. / Acta Astronautica 83 (2013) 44–53 49

Thus, the condition ensuring LARC is

span0

h1

" #,

�h1�R

J�1ðJxÞ��x�Jð Þh1

" #,

0

h2

" #,

(

�h2�R

J�1ðJxÞ��x�Jð Þh2

" #,

0

h3

" #,

�h3�R

J�1ðJxÞ��x�Jð Þh3

" #)¼ R6

ð34Þ

Obviously, according to the primary transformation ofmatrix, the formula (34) is simplified to bespan h1,h2,h3

� �¼ R3, where hi i¼ 1,2,3ð Þ is shown as Eqs.

(9), (29) and (32).According to the above analysis and Theorem 1, the

necessary and sufficient condition for controllability ofunderactuated spacecraft attitude control system withone thruster is:

span h1,h2,h3

� �¼ R3

ð35Þ

3.4. Controllability comments and guidance on control law

In the process of deducing controllability condition,orbital angular velocity is counteracted, illustrating thatthe controllability condition of attitude with respect toinertial frame is equivalent the condition of attitude withrespect to orbit frame. In addition, the periodical oscilla-tion disturbance enlarge the bounded region of the driftvector field, but do not affect the WPPS, showing that thedisturbance does not affect the controllability of thesystem, but do reduce the stability precision.

With the conclusion that the periodical oscillation dis-turbance does not affect the controllability of the system, thecontrol law can be designed based on nonlinear controltheorem after eliminating periodical oscillation disturbancethrough the internal model principle. In the next section, forunderactuated spacecraft with two thrusters, the angularvelocity stability control law and three-axis stabilizationcontrol law are created by applying the backstepping controlmethod and internal model principle. The global asymptoticstability of the controller is proved by combing the Lyapunovstability theory and LaSalle invariant theorem.

4. Control law design

As we know, SO(3) is the global description of theattitude, it can be expressed in local coordinates para-meters, like Euler angles, Euler–Rodrigues Parameters andCayley–Rodrigues Parameters. Recently, a new parame-terization [17] using a pair of a complex (w) and a realcoordinate (z) was introduced which was shown to havesome significant advantages for attitude control problems[18,19]. In this section, we adopt this standard parameter(w, z) to describe the attitude of spacecraft body-fixedreference frame with respect to orbit reference frame, andthe kinematics gives

_w1 ¼or3w2þ 1þw21�w2

2

� �or1=2þw1w2or2

_w2 ¼�or3w1þ 1�w21þw2

2

� �or2=2þw1w2or1

_z ¼or3þw1or2�w2or1

8><>: ð36Þ

From Eq. (36), we find that the parameter z does notappear in the first two formulas, which means that theproblem can be decomposed into one of controlling onlyw and one of controlling z in some application [20].Therefore, if the actuator failure is about the Z-axis, thenthe dynamics of the underactuated axis can be decoupledfrom the rest of the system, thereby reducing the com-plexity of the problem. Suppose the thrusters are installedalong the axes, that isb1 ¼ b2 ¼ 0, then, Eq. (6) can bewritten as

J1 _o1 ¼ J2�J3

� �o2o3þu1þd1

J2 _o2 ¼ J3�J1

� �o3o1þu2þd2

J3 _o3 ¼ J1�J2

� �o1o2þd3

8><>: ð37Þ

From Eq. (37), we can find that the smaller theabsolute value of J1�J2 is, the greater the difficulty ofthe stabilization control. o3 will be unable to controlcompletely when J1 ¼ J2. This is consistent with theconclusion of controllability that ‘‘underactuated space-craft cannot be symmetric along the underactuated axis’’.Here, we assume that the amplitude of d3 is so small thatcan be ignored. Because the impact is very complex by theunderactuated axis, if there is :d3: cannot be neglected,no three-axis stabilization control can be achieved, unlessadding other actuators, like CMGs, flywheels, magnetictorques. And we will consider this problem in the furtherresearch. Here, angular velocity stabilization control andthree-axis stabilization control are designed.

4.1. Angular velocity stabilization control

The procedure of designing the angular velocity con-trol law : first, invent the virtual control law o1a,o2a tostabilize the angular velocity o3 of the underactuatedaxis. Then, use the proportional control method to obtainthe control law u1a,u2a which will stabilize thesystem (37).

Step 1: consider the candidate Lyapunov function

V1 ¼1

2J3o2

3

Compute the time derivative of V1 and obtain

_V 1 ¼ J3o3 _o3 ¼ J1�J2

� �o1o2o3

In order to make _V 1 negative definite, a control law isderived as

o1a ¼�sign J1�J2

� �sign o3ð Þ

ffiffiffiffiffiffiffiffiffiffio3j j

p

o2a ¼ k1ffiffiffiffiffiffiffiffiffiffio3j j

p

(ð38Þ

where k140, signdð Þ is the sign function.Substituting Eq. (38) into _V 1, we have

_V 1 ¼�k1 J1�J2

�� ��o23r0

Obviously, V1Z0 _V 1r0, and V1 ¼ 0, _V 1 ¼ 0 onlyoccur o3 ¼ 0. And as t-1, there is V1-1. Accordingto the Lyapunov stability theory, we know that when weadopt Eq. (38) as the virtual control law, the third formulaof the system is stabilized globally asymptotically. That is,o3-0.

D. Wang et al. / Acta Astronautica 83 (2013) 44–5350

Step 2: To stabilize the dynamic Eq. (37), we find

u1a ¼�k2J1 o1�o1að Þ� J2�J3

� �o2o3�d1þ J1 _o1a

u2a ¼�k2J2 o2�o2að Þ� J3�J1

� �o3o1�d2þ J2 _o2a

(ð39Þ

where k240.

Proof. substituting Eq. (39) into the first two formulas ofEq. (37), we have

_o1� _o1a ¼�k2 o1�o1að Þ

_o2� _o2a ¼�k2 o2�o2að Þ

(

Obviously, as t-1for anyk240, there iso1�o1a-

0,o2�o2a-0, that is equivalent too1-o1a,o2-o2a.And from the Step 1 we haveo3-0 and theno1a-

0,o2a-0 from Eq. (38). That meanso1- 0,o2-0. Thus,the global asymptotic stability of the control law (38) and(39) follows directly from a straightforward application ofLaSalle invariant theorem [21].

4.2. Three-axis attitude stabilization control

The three-axis attitude stabilization control isdesigned by two steps: first, the expected angular velocityo1d,o2d is created to stabilize w1,w2,z of the kinematicsEq. (36). Second, according to the dynamic Eq. (37), thecontrol law is created to stabilize the complete system bythe Backstepping method. The detail processing is:

Step 1: To stabilize the kinematics (36), the expressionof the intermediate control law or1d,or2d are obtained as

or1d ¼�k3w1

1þw21þw2

2

þm zþor3=mw2

1þw2

2

w2

or2d ¼�k3w2

1þw21þw2

2

�m zþor3=mw2

1þw2

2

w1

8><>: ð40Þ

Proof. consider the candidate Lyapunov function

V2 ¼w21þw2

2þ1

2z2

Compute the time derivative of V2 and utilize Eqs. (36)and (40), we have

_V 2 ¼ 2w1 _w1þ2w2 _w2þz_z

¼ 1þw21þw2

2

� �w1or1þw2or2ð Þþz or3þw1or2�w2or1ð Þ

¼ �k3w21�k3w2

2�mz2

Obviously, V2Z0, _V 2r0, and V2 ¼ 0 _V 2 ¼ 0 only occurw1 ¼w2 ¼ z¼ 0. And as t-1, there is V2-1. Therefore,system (36) is globally asymptotically stabilized when Eq.(40) is adopted as the expected control law according tothe Lyapunov stability theory.

When w1 ¼w2 ¼ 0, the system (40) is discontinuous.To resolve this problem, and according to Eq. (11), theintermediate control law is rewritten as

o1d ¼�k3w1

1þw21þw2

2

þmsat zþo3=mw2

1þw2

2

w2,a �

o2d ¼�k3w2

1þw21þw2

2

�msat zþo3=mw2

1þw2

2

w1,a �

�o0

8><>: ð41Þ

where sat x,að Þ ¼x, xj joa

signðaÞ, xj j4a

(.

Step 2: To stabilize the systems (36) and (37), wedefine the angular velocity error as g1 ¼o1�o1d,g2 ¼o2�o2d, the expression of control law is derived as

u1 ¼ u10�k4 _a1

u2 ¼ u20�k4 _a2

€a1þO21a1 ¼ g1

€a2þO22a2 ¼ g2

8>>>><>>>>:

ð42Þ

where

u10 ¼�w1 1þw21þw2

2

� �þw2z�k4g1� J2�J3

� �o2o3þ J1 _o1d

u20 ¼�w2 1þw21þw2

2

� ��w1z�k4g2� J3�J1

� �o3o1þ J2 _o2d

ensures the three-axis stabilization in the normal situa-tion; and k4 _a with €aþO2a¼ g is applied to counteract thesinusoidal disturbance, which is proposed according tothe internal model principle. Let k4 _y ¼ k4 _a�x, substitutethe control law (42) into Eq. (37), which in the newcoordinates of error g,y

� �can be rewritten as

J1 _g1 ¼�w1 1þw21þw2

2

� �þw2z�k4g1�k4 _y1

J2 _g2 ¼�w2 1þw21þw2

2

� ��w1z�k4g2�k4 _y2

€y1þO21y1 ¼ g1

€y2þO22y2 ¼ g2

8>>>>><>>>>>:

ð43Þ

Proof. Consider the candidate Lyapunov function

V ¼ V2þ J1g21þ J2g2

2

� �=2þk4 _y2

1þ _y22þy2

1O21þy2

2O22

�=2

Compute the time derivative of V , and apply Eqs. (42)and (43) by utilizing or1 ¼ g1þo1d, or2 ¼ g2þo2dþo0,we obtain

_V ¼ 2w1 _w1þ2w2 _w2þz_zþ J1g1_o1� _o1dð Þþ J2g2

_o2� _o2dð Þ

þk4 _y1 €y1þ _y2 €y2

� �þk4 O2

1y1 _y1þO22y2 _y2

�¼�k3w2

1�k3w22�mz2þg1 J2�J3

� �o2o3þu1þx1�J1 _o1d

�þw1 1þw2

1þw22

� ��w2z

�þg2 u2þx2þ J3�J1

� �o3o1�J2 _o2dþw2 1þw2

1þw22

� ��þw1z�þk4 _y1g1þ _y2g2

� �¼�k3w2

1�k3w22�mz2�k4g2

1�k4g22

Notice that when _V ¼ 0, there are w1 ¼ 0,w2 ¼

0,z¼ 0,g1 ¼ 0,g2 ¼ 0, by which _g1 ¼ 0, _g2 ¼ 0 can bederived through the time derivative of g1,g2. Substituteinto the first two formulas of Eq. (43), then _y1 ¼ 0, _y2 ¼ 0can be derived. €y1 ¼ 0, €y2 ¼ 0 can also be obtained throughthe time derivative of _y1, _y2. Substitute €y1 ¼ 0, €y2 ¼ 0 and_y1 ¼ 0, _y2 ¼ 0 into the last two formulas of Eq. (43),we can have y1 ¼ 0,y2 ¼ 0. Moreover, from Eq. (40),we have or1 ¼ 0,or2 ¼ 0,or3 ¼ 0(or ‘‘from Eq. (40),we have o1 ¼ 0,o2 ¼�o0,o3 ¼ 0’’). Hence the internalmodel principle can counteract the disturbance, and _V ¼ 0only has the unique solution. According to the LaSalleinvariant theorem, system is global asymptotical stability.

Fig. 3. (a) Spacecraft angular velocity. (b) The enlarged spacecraft angular velocity.

Fig. 4. Control torque of angular velocity stabilization.

D. Wang et al. / Acta Astronautica 83 (2013) 44–53 51

5. Simulation results

Assume that a rigid spacecraft installed with twothrusters is flying at a 400-km near-circular earthorbit. We can calculate that the orbital angular velo-city is 0.0011 rad/s. The constant inertial matrix of

spacecraft is J¼ diag 449:5 264:6 312:5� �

kg m2. The

sinusoidal disturbance torque is assumed to be Td ¼

0:01sin 0:7 tð Þ 0:02sin 0:8 tð Þ 0:001sin 0:1 tð Þh iT

NUm.

The numerical simulation of angular velocity stabilizationand attitude stabilization is performed on the under-actuated spacecraft separately. The parameters used insimulations are as follows.

(1)

Angular velocity stabilization controlInitial angular velocity o1,o2,o3½ �ð0Þ ¼ 0:1,�0:11,½

0:1�rad=s; control parameters k1 ¼ 5in Eq.(38) andk2 ¼ 10in Eq. (39).

(2)

Three-axis attitude stabilization control

Initial attitude parameters w1,w2,z½ �ð0Þ ¼ �1,1,1:27½ �

and initial angular velocity o1,o2,o3½ �ð0Þ ¼ 0:1,½

�0:11,0�rad=s; control parameters k3 ¼ 0:05,m¼ 1,a¼ 0:1 in Eq. (41) and k4 ¼ 0:01 in Eq. (42).

Figs. 3 and 4 show the simulation results of angularvelocity stabilization control. Fig. 3a illustrates that thecontrol law can stabilize angular velocity of the under-actuated spacecraft with good performance, and Fig. 3bindicates that the angular velocity of the spacecraft withrespect to the inertial frame close to zero at 500 s. This isconsistent with the purpose of angular velocity stabiliza-tion in the Section 4.1 that ‘‘o1-0,o2-0,o3-0’’. FromFig. 4, we can see that the control law has a slight burr,which is caused by the interference torque.

Figs. 5–7 show the simulation results of three-axisattitude stabilization control, which indicate that thecontrol law can stabilize the angular velocity and attituderapidly. The effects caused by sinusoidal disturbance canbe eliminated by the internal model principle. Fig. 5aprovides the angular velocity time history subsequentlysteady, and from Fig. 5b we can see that the final value is[0; �0.0011; 0] rad/s. In this situation, the body-fixedframe is coinciding with the orbital frame, and theangular velocity of the body-fixed frame with respect tothe inertial frame is the orbital angular velocity. Thissimulation result is consistent with the purpose of three-axis attitude stabilization control in the Section 4.2 that‘‘w1 ¼ 0,w2 ¼ 0,z¼ 0,or1 ¼ 0,or2 ¼ 0,or3 ¼ 0’’, which isindicating that ‘‘o1-0,o2-�o0,o3-0’’, whereo0 ¼ 0:0011 rad=s at a 400-km near-circular earth orbit.

Note: the above three-axis stabilization control simu-lation is carried out under the assumption of o3 ¼ 0.Actually, o3 will change in the process of attitude adjust-ing. Substitute xd of Eq. (41) into Eq. (37), we cancalculate _o3 (omit the expression here) which is smallvalue, and will eventually converge to zero. However, theimpact of _o3 still cannot be avoided. Therefore, switchingcontrol law is applied to control the system: when o3 islarge, we can use Eq. (38) to reduce its impact on system;

Fig. 5. (a) Spacecraft angular velocity. (b) The enlarged spacecraft angular velocity.

Fig. 6. Spacecraft attitude. Fig. 7. Control torque of three-axis stabilization.

D. Wang et al. / Acta Astronautica 83 (2013) 44–5352

and then stabilize the attitude of the system by thecontrol laws (41) and (42). Repeat this process until thestable range.

6. Conclusions

In this paper, the controllability of attitude controlsystem is studied for an underactuated spacecraft withtwo or one thrusters. Considering the effect of orbit angularvelocity and periodical oscillation disturbance, the necessaryand sufficient condition for underactuated spacecraft atti-tude control system is obtained. In the process of deducingcontrollability condition, two conclusions are obtained: first,controllability condition of attitude with respect to inertialframe is equivalent the condition of attitude with respect toorbit frame. Second, the periodical oscillation disturbancedoes not affect the controllability of the system, but doreduce the stability precision. With these conclusions of theattitude controllability for underactuated system, the con-trol law can be designed based on nonlinear control

theorem after eliminating periodical oscillation disturbancethrough the internal model principle.

Based on this guidance on control law, the angularvelocity stabilization control law and three-axis stabiliza-tion control law are created separately by the backsteppingcontrol method and internal model principle. The globalasymptotic stability of the controller is proved by combingthe Lyapunov stability theory and LaSalle invariant theo-rem. Moreover, the numerical simulation is provided,which proved the control law feasibility once again.

This article only focuses on the controllability of rigidspacecraft. For the flexible spacecraft, the attitude con-trollability still needs further study.

Acknowledgments

The authors would like to thank for the support fromthe National Natural Science Foundation of China (No.10902003).

D. Wang et al. / Acta Astronautica 83 (2013) 44–53 53

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