Tsinghua Science and Technology Tsinghua Science and Technology

Volume 22 Issue 1 Article 8

2017

Attitude Control of Rigid Body with Inertia Uncertainty and Attitude Control of Rigid Body with Inertia Uncertainty and

Saturation Input Saturation Input

Xi Ma Department of Computer Science and Technology, Tsinghua University, State Key Lab of Intelligent Technology and Systems, and Tsinghua National Laboratory for Information Science and Technology (TNList), Beijing 100084, China. Department of Aerospace Engineering, High-Tech Institute of Xi’an, Xi’an 710025, China.

Fuchun Sun Department of Computer Science and Technology, Tsinghua University, State Key Lab of Intelligent Technology and Systems, and Tsinghua National Laboratory for Information Science and Technology (TNList), Beijing 100084, China.

Hongbo Li Department of Computer Science and Technology, Tsinghua University, State Key Lab of Intelligent Technology and Systems, and Tsinghua National Laboratory for Information Science and Technology (TNList), Beijing 100084, China.

Bing He Department of Aerospace Engineering, High-Tech Institute of Xi’an, Xi’an 710025, China.

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Recommended Citation Recommended Citation Xi Ma, Fuchun Sun, Hongbo Li et al. Attitude Control of Rigid Body with Inertia Uncertainty and Saturation Input. Tsinghua Science and Technology 2017, 22(1): 83-91.

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TSINGHUA SCIENCE AND TECHNOLOGYISSNll1007-0214l l08/11l lpp 83–91Volume 22, Number 1, February 2017

Attitude Control of Rigid Body with Inertia Uncertainty andSaturation Input

Xi Ma, Fuchun Sun�, Hongbo Li, and Bing He

Abstract: In this paper, the attitude control problem of rigid body is addressed with considering inertia uncertainty,

bounded time-varying disturbances, angular velocity-free measurement, and unknown non-symmetric saturation

input. Using a mathematical transformation, the effects of bounded time-varying disturbances, uncertain inertia,

and saturation input are combined as total disturbances. A novel finite-time observer is designed to estimate the

unknown angular velocity and the total disturbances. For attitude control, an observer-based sliding-mode control

protocol is proposed to force the system state convergence to the desired sliding-mode surface; the finite-time

stability is guaranteed via Lyapunov theory analysis. Finally, a numerical simulation is presented to illustrate the

effective performance of the proposed sliding-mode control protocol.

Key words: attitude control; inertial uncertainty; angular velocity-free measurement; saturation input; finite-time

observer; sliding-mode control

1 Introduction

The problem of rigid body attitude control has attractedmuch interest over past decades due to its widetheoretical and practical applications. The attitudemodel is represented by a set of two vector equations,namely, the kinematic equation and the dynamicequation[1]. From a practical point of view, the designof an effective and robust attitude control protocolwith high-precision is an important issue for industrialapplication. Due to its inherent highly nonlinearproperties, some nonlinear control methods, such as

�Xi Ma, Funchun Sun, and Hongbo Li are with Departmentof Computer Science and Technology, Tsinghua University,State Key Lab of Intelligent Technology and Systems,and Tsinghua National Laboratory for Information Scienceand Technology (TNList), Beijing 100084, China. E-mail: mayue13@mails.tsinghua.edu.cn; fcsun@mail.tsinghua.edu.cn; hbli@mail.tsinghua.edu.cn.�Xi Ma and Bing He are with Department of Aerospace

Engineering, High-Tech Institute of Xi’an, Xi’an 710025,China. E-mail: sxxaepmx@126.com.�To whom correspondence should be addressed.

Manuscript received: 2016-01-27; revised: 2016-03-07;accepted: 2016-09-23

feedback control[2], sliding-mode control[3], adaptivecontrol[4], backstepping control[5], and robust control[6],have been studied to address the problem withcomplicated conditions, such as external disturbanceand/or dynamic uncertainty and/or saturation input. Formore details on these topics, we refer the readers toRefs. [7–10] and the references therein.

It is worth mentioning that research on the rigid-bodyattitude control problem under practical conditions isongoing. As is well-known, system disturbances areinevitable due to environmental factors[11], which maydeteriorate the system’s stability and force the controlprotocol design to become more intractable. In Ref.[12], an adaptive sliding-mode control scheme forthe attitude control problem with dynamic uncertaintyand external disturbance was proposed. This schemeregulates the relative attitude and angular velocity ofthe rigid body within a desired small region. In Ref.[13], the inertial uncertainty and external disturbancewere lumped together as the total disturbances, andan Extended State Observer (ESO) was introducedto estimate the total disturbances and compensatefor them by designing a sliding-mode controller. Forthe dynamic uncertainty, Chebyshev neural networkswere used as universal approximator to learn unknown

84 Tsinghua Science and Technology, February 2017, 22(1): 83–91

nonlinear functions in Ref. [14]. In addition, a robustcontrol term, using the hyperbolic tangent function, wasapplied to counteract neural-network approximationerrors and external disturbances, which made theproposed controller continuous and hence chattering-free.

In the above literature, the major assumption isthe full states (i.e., attitude and angular-velocity)measurement. However, the above results may notbe used directly to handle the attitude controlproblem, if the angular velocity-free measurement isconsidered. It should also be pointed out that the fullstates measurement assumption is rigorous becausethe angular velocity-free measurement is frequentlyencountered in various engineering systems, such assome low-cost satellites. A feasible way to handle theunmeasured angular velocity is to design an observerto approximate these unknown states. A nonlinearstate observer was designed in Refs. [15, 16] and thesmooth structure property was ensured, meaning thatall the estimated states were in C1 continuity. In Ref.[17], a passivity-based control scheme was proposedto ensure system attitude asymptotic convergencewithout the angular velocity measurement. In Ref. [18],the attitude control problem with angular velocity-free measurement was also considered. A certainty-equivalence passivity-based controller was developedto guarantee the attitude convergence, with an adaptiveobserver to estimate the angular velocity.

All the results proposed in the above literaturefocus on systems without considering the saturationinput. It is well known that saturation control is themost important non-smooth nonlinear property, andshould be explicitly considered in control design.Compared with unrestrained control input, the existenceof saturation input may deteriorate the controlperformance and even result in system instability[19].Lu et al.[20] considered actuator saturation for attitudecontrol problems. In Ref. [21], hyperbolic tangentfunctions were employed to prevent actuator saturation,whereas saturation functions were used in Ref. [22]. InRefs. [23, 24], the actuator saturation was taken intoaccount for the first time and a global saturated finite-time control scheme was proposed.

The main purpose of this paper is to investigatethe rigid-body attitude control problem in the presenceof the external disturbances, the inertial uncertainty,the angular velocity-free measurement, and unknown

non-symmetrical saturation input. Here, the systemuncertain inertia and saturation input are combinedwith external disturbance and a novel finite-timeobserver is used to estimate the angular velocityand the disturbances. An observer-based sliding-mode control protocol is proposed and its finite-timestability is guaranteed by Lyapunov analysis. The maincontributions of this paper are as follows:

(1) In contrast to the existing literature on rigid-body attitude systems, the inertial uncertainty, theangular velocity-free measurement, and the unknownnon-symmetrical saturation input are considered;these are combined with external disturbances andthe effect is estimated by the designed finite-time observer. However, the angular velocity-freemeasurement and input saturation are not considered inRef. [14], and only external disturbance is taken intoaccount in Ref. [3]. The inertial uncertainty, angularvelocity-free measurement, external disturbances, andinput saturation were all explicitly considered whendesigning the control protocol in this paper.

(2) Combined with the designed finite-time observer,an observer-based sliding-mode control protocol isdesigned and the finite-time stability is guaranteedusing Lyapunov theorem analysis. Moreover, theproposed controller is robust against not only structureduncertainty but also unstructured uncertainty.

This paper is organized as follows. In Section2, a typical rigid-body attitude control problem isformulated. Preliminaries and the main assumptions arealso given. Section 3 discusses the finite-time observerand the sliding-mode controller for the rigid-bodyattitude control problem under complicated conditions.The elaborate proof of the closed-loop system stabilityis also provided. A numerical simulation is performedto validate the proposed control strategy in Section 4,followed by conclusions in Section 5.

2 Problem Formulation and Preliminaries

The rigid-body attitude control system can be describedby two sets of equations. These are the kinematic anddynamic equations that relate the time derivatives of theangular coordinate to the angular velocity vector, andthe dynamic equation that describes the evolution of thevelocity vector[25, 26] is8̂<̂

:Pq0 D �

12qTv!;

Pqv D12

�q0I3 C q

�v

�!;

J P! D �!�J! C uC d

(1)

Xi Ma et al.: Attitude Control of Rigid Body with Inertia Uncertainty and Saturation Input 85

where q W .q0; qv/ 2 R � R3 represents the unitquaternion vector with q0 and qv being the scalar andvector parts, respectively, and satisfies the constraintq20 C q

Tvqv D 1; ! WD .!1; !2; !3/ 2 R3 is the angular

velocity with respect to the inertia frame I and isexpressed in the body frame B; u 2 R3 is the controltorque input; d 2 R3 is the unknown time-varyingdisturbance; J 2 R3�3 is the symmetric inertia matrix;and � denotes an operator on any vector

a D Œa1; a2; a3�T

such that

a� D

264 0 �a3 a2

a3 0 �a1

�a2 a1 0

375 :To simplify our analysis, we define

qW D

q0

qv

!; E .q/ WD

�12qTv

12

�q0I3 C q

�v

� ! :Therefore, the attitude kinematic equation can be

rewritten asPq D E .q/ !:

Before our progression, the following assumptionsand lemmas are necessary.

Assumption 1 The symmetric positive definiteinertia matrix satisfies J D J0 C�J , where �J

denotes the inertial uncertainty.Assumption 2 The time-varying disturbance d.t/ is

bounded, i.e., d.t/ < dm, where dm is a known positiveconstant.

Remark 1 Many practical attitude control systemscan be described by Eq. (1), such as satellite attitudecontrol systems, spacecraft attitude dynamics, and soon. Assumption 1 is a common assumption regardingthe inertial uncertainty used in most of the literature.Assumption 2 is reasonable while most of the externaldisturbances can be considered as bounded in practicalsituations.

The following lemmas are introduced.Lemma 1 For 8x 2 R4, the following equality

holdskE .x/k2 D

1

2kxk2:

Lemma 2 Assume x1; x2; : : : ; xn > 0 and 0 < p <1, then the following inequality holds:

nXiD1

xi

!p6

nXiD1

xpi :

Now consider the system’s unknown disturbanced and inertia uncertainty �J . The attitude dynamic

equations can be rewritten as.J0 C�J/ P! D �!

� .J0 C�J/! C uC d (2)

Note that .J0 C�J/�1 can be expressed as.J0 C�J/

�1D J�10 C�

QJ :

So, with some translations, system (2) can berewritten as

P! D F C J�10 uCG (3)

whereF D �J�10 !�J0!;

G D J�10 d � J�10 !��J0! �� QJ!�J!C

� QJuC� QJd:

Remark 2 In Eq. (3), F is the known dynamics andG is the unknown dynamics, which can be consideredas the effect of the inertia uncertainty term �J andthe unknown time-varying disturbance d . It should bepointed out that the presence of uncertain inertia andtime-varying disturbances makes the attitude controlproblem more complicated.

Remark 3 Compared with the previous work inRef. [13], we consider a more complicated condition.However, control saturation and model uncertainty areinevitable in some practical applications, which makesthe control problem more intractable. Additionally,from a practical viewpoint, focus on synchronizationtime is more important than achieving synchronization,which implies better robustness and disturbancerejection properties. Here, we consider the finite-timecase instead of the asymptotic converging case.

3 Main Result

Our aim is to design a control protocol u for system(1) under some practical conditions to make the attitudesystem stable, i.e.,

q0 !˙1; qv ! 0; ! ! 0; as t !1:

3.1 Novel extended state observer

A feasible way of handling an unknown state is todesign an observer to approximate its real value. TheESO mentioned in Ref. [27] shows high efficiency whenaccomplishing the nonlinear dynamic estimation. Inaddition, this ESO regards the system disturbance asthe extended state for estimation. Inspired by Ref. [28],system (1) can be rewritten as8̂<̂

:Pqv D f .q/ !;

P! D F C J�10 uCG;PG D g .t/

(4)

86 Tsinghua Science and Technology, February 2017, 22(1): 83–91

where f .q/ D1

2

�q0I3 C q

�v

�and the unknown

function g .t/ is the derivative of the total disturbanceG. The finite-time observer is then proposed as follows:8̂̂̂̂<̂̂ˆ̂̂̂:

POqvD f .q/ O! C �1

�je1j

˛1Cje1jˇ1

�sign .e1/ ;

PO!D OF C J�10 uC OGC�2

�je1j

˛2Cje1jˇ2

�sign .e1/ ;

POGD �3

�je1j

˛3Cje1jˇ3

�sign .e1/

(5)where e1 D qv � Oqv is the estimation error, 0:75 <˛1 < 1; ˛2 D 2˛1 � 1; ˛3 D 3˛1 � 2; ˇ1 D ˛

�11 ; ˇ2 D

˛�11 C ˛1 � 1; and ˇ3 D ˇ�11 C 2˛1 � 2.From the attitude system (4) and the designed

extended state observer (5), the corresponding observererror dynamics can be obtained as8̂̂̂<̂ˆ̂:Pe1 D f .q/ e2 � �1

�je1j

˛1 C je1jˇ1

�sign .e1/ ;

Pe2 D F � OF C e3 � �2

�je1j

˛2 C je1jˇ2

�sign .e1/ ;

Pe3 D g .t/ � �3

�je1j

˛3 C je1jˇ3

�sign .e1/

(6)where e2 D ! � O! and e3 D G � OG.

Based on the fact that the function F is continuousand differentiable for ! 2 R3, according to thedifferential mean value theorem, the following equationcan be obtained:

F � OF D F 0 .�/ e2; � 2 .!; O!/ :

Let a denote F 0 .�/, then substituting the aboveequation into Eqs. (6) yields8̂̂̂<̂ˆ̂:Pe1 D f .q/ e2 � �1

�je1j

˛1 C je1jˇ1

�sign .e1/ ;

Pe2 D ae2 C e3 � �2

�je1j

˛2 C je1jˇ2

�sign .e1/ ;

Pe3 D g .t/ � �3

�je1j

˛3 C je1jˇ3

�sign .e1/

(7)The finite-time convergence of the estimation errors

in Eq. (7) can be presented in the following theorem.Theorem 1 Consider the attitude control system

(4) and the observer defined in Eq. (5), then the stateestimation errors e1; e2, and e3 will converge to aresidual set U governed byU D f.e1; e2; e3/ jke1k 6 �; ke2k 6 �; ke3k 6 �g

where � is defined by the observer parameter.Proof See Theorem 1 in Ref. [28]. �Remark 4 It should be noted that the estimation

term OG is the total disturbance’s estimation and isused to compensate for the uncertainty in the controlprotocol. Compared with the observer designed inRefs. [29, 30], one of the most advantages is that it can

approximate the real-time action of system uncertainty,which is more practical in engineering applications.

3.2 Control protocol design and stability analysis

To facilitate the analysis, a translation form isconsidered that

x D ! C �qv;

where � is a positive constant.Then, the attitude control system (4) can be

transformed asPx D F C J�10 uCG C �f .qv/ ! (8)

It has been proven in Ref. [31] that for system(8), lim

t!1qv .t/ D 0 can be achieved if there exists a

control law u.t/ for Eq. (8) ensuring limt!1

x .t/ D 0

with any initial state kq .0/k D 1. This means whilelimt!1

x .t/ D 0, limt!1

qv .t/ D 0 is also achieved as

well as limt!1

! .t/ D 0 due to Eq. (1).

Similarly, if the system states limt!1

qv .t/ D 0 and

limt!1

! .t/ D 0 are satisfied by the control law u.t/, the

state limt!1

x .t/ D 0 can also be guaranteed.Remark 5 Note that, it is only proven that qv !

0; ! ! 0. The state q0 is not considered. This isdue to the fact that q0 ! ˙1 once qv ! 0 from theconstraint condition q20 C q

Tvqv D 1. In the physical

space, these two equilibrium points q D Œ1; 0; 0; 0�T

and q D Œ�1; 0; 0; 0�T correspond to the same physicalpoint[32]. Hence, it is only needed to guarantee thatqv ! 0.

3.2.1 Sliding-mode control protocol designIn the following, we introduce the sliding-modevariables,

S D Cx;

where C 2 R3�3.Without losing generality, we assume matrix C is of

full rank and matrix CJ�10 is nonsingular.The sliding-mode control protocol is now given by

uD�J0C�1�C OFCC OGC�Cf .q/ O!Cc1SCc2S

vC �

(9)where OF and OG are the estimation of system termsF and G, respectively; c1 > 0; c2 > 0, and 0 < v < 1are control parameters and D Œ 1; 2; 3�

T arehyperbolic tangent functions defined by Ref. [33].

i D �i tanh�ku�iSi

"

�; k D 1; : : : ; 3;

where ku D 0:2785, �i is a positive constant satisfying�i > �max .C /

��2 C

�1C

�

2

���

and �max .C / is the

Xi Ma et al.: Attitude Control of Rigid Body with Inertia Uncertainty and Saturation Input 87

maximum eigenvalue of matrix C . It is easy show thatthe robust term �i has the following property:

�max .C / kSk��2 C

�1C

�

2

����

nXiD1

Si i 6 3"

(10)Remark 6 The structure of the closed-loop system

is depicted in Fig. 1. Note that the proposed controlprotocol (9), the compensating terms OF ; OG, and f .q/ O!are used to eliminate the effect of system nonlinearterm F and unknown terms G and �f .q/ !; therobust term is used to counteract the effects of theestimation errors. And, the sliding-mode feedbackterm c1S C c2S

v is used to drive the system statesto zero. Furthermore, compared with Ref. [14], theproposed controller is continuous with respect to timeand chattering-free, by avoiding the use of the signumfunction.

3.2.2 Stability analysis of closed-loop systemIn this section, the stability of the closed-loop system(8) can be established by the following theorem.

Theorem 2 Considering Assumptions 1 and 2, andthe proposed sliding-mode protocol (9). Then for anyinitial conditions, the finite-time stability of closed-loopsystem (8) can be achieved.

Proof Consider the following Lyapunov functioncandidate,

V D1

2STS (11)

By integrating the designed control protocol (9),the time derivative of the Lyapunov function can beobtained:PV D ST PS D

STC PxDST�CF CCJ�10 uCCGC�Cf .q/ !

�D

ST�C QF C C QG C �Cf .q/ e2 � c1S � c2S

v � �

(12)where QF D F � OF ; and QG D G � OG

Due to the property of the observer, we can obtain

Sliding Controller

Observer

qw

Controller law design

Attitude dynamics (1) (2)

Dynamic of w

Dynamic of G

Dynamics of q

Attitude dynamics

( )q E q

J J u d

w

w w w´

ì =ïí

= - + +ïî

( )( )q Eq E ( )( )( )( )wì =q Eq Eq E

J JJ JJ JJ JJ J´J JJ JJ JJ JJ JJ JJ JJ JJ JJ JJ J

( )

( )

1

0

q E q

J F J u G

G g t

w

w -

=ìï

= + +íï =î

( )( )q E qq E q( )( )( )wq E qq E q

0J F JJ F J

0J F JJ F JJ F JJ F JJ F JJ F JJ F JJ F J

00

( )( )G g t( )( )G g tG g t

( )( )1

0 1 2

ˆ ˆv

vu J C CF CG cf q c S c Ss w y-= - + + + + +

( )vS C qw s= +

Fig. 1 Closed-loop system structure.

C QF 6 �max .C /�2; C QG 6 �max .C /�;

kCf .q/ e2k 6 0:5�max .C /�:

Then, we can obtain thatPV 6 �max .C /

��2 C�C 0:5��

�kSk�

ST .c1S C c2SvC / 6

�c1STS � c2S

TSv C 3" (13)

From Lemma 2, the following inequality is obtained.

�c2STSv D �c2

3XiD1

SvC1i 6

�2vC1

2 c2

0:5

3XiD1

S2i

! vC12

D � 2VvC1

2 (14)

where 2 D 2vC1

2 c2 > 0. Then, Formula (13) can berewritten as

PV 6 � 1V � 2VvC1

2 C 3 (15)

where 1 D 2c1 and 3 D 3" are positive constants.Then, Formula (15) can be expressed as two forms:

PV 6 �� 1 �

3

V

�V � 2V

vC12 (16)

PV 6 � 1V �

� 2 �

3

VvC1

2

�V

vC12 (17)

From Formula (16), if the condition 1 � 3

V> 0 is

satisfied, then the finite-time stability is still guaranteed,which implies that V 6

3

1in finite time. Then, the

system state x converges to

kxk 61

�min .C /

s2 3

1D

1

�min .C /

s3"

c1:

From Formula (17), if the condition 2 � 3

VvC1

2

> 0

is satisfied, then the finite-time stability is still

guaranteed, which implies that V <

� 3

2

� 2vC1

in finite

time. Then, the system state x converges to

kxk 61

�min .C /

vuut2

� 3

2

� 2vC1

D

p2

�min .C /

� c23"

�vC1:

From Formulas (16) and (17), it is concluded that thedesigned system state x could converge to the boundedregion in finite time, which implies the system state x isfinite-time stable. Furthermore, the control parametersc1 and c2 and hyperbolic tangent function parameter

88 Tsinghua Science and Technology, February 2017, 22(1): 83–91

" depend on the size of the bounded region, that is,the smaller bounded region, the smaller " and biggercontrol parameters c1 and c2 are required. �

Remark 7 In system (4), the proposed sliding-mode protocol (9) can be guaranteed bounded motion ina bounded region. What is more, the size of the boundedregion can be adjusted by tuning the observer andcontrol parameters. Thus, observer parameter selectionis also vital, because it determines not only theperformance of the observer but also the motion of theattitude tracking system.3.2.3 Attitude control with input saturationIn practical engineering, input saturation is the mostimportant non-smooth nonlinear property, and shouldbe explicitly considered in control design. Here,we propose a sliding-mode control protocol forthe attitude system (4) in the presence of externaldisturbance, inertial uncertainty, angular velocity-free measurement, and unknown non-symmetric inputsaturation. Considering the unknown input saturationconstraints, the control input uk can be expressed as

uk D

8̂<̂:umax; if vrk > umaxI

vrk; if umin < vrk < umaxI

�umin; if vrk > umin

(18)

where k D 1; 2; 3 and vr D Œvr1; vr2; vr3�T2 R3 is

the designed control input command. umin and umax arethe unknown parameters of the control input saturation.Here, umin ¤ umax denotes the non-symmetric inputsaturation.

For further analysis, with considering the effect of theunknown non-symmetric input saturation, the attitudedynamics (4) can be transformed as8̂<̂

:Pqv D f .qv/ !;

P! D F C J�10 vr C J�10 •uCG;

PG D g .t/

(19)

where •u D u � vr .Since the upper and the lower limits of the

non-symmetric input saturation are unknown, •u isunknown. To handle the unknown term J�10 •u, wedefine the compound unknown disturbance as

NG D G C J�10 •u (20)Considering Eqs. (19) and (20), we obtain8̂<̂

:Pqv D f .qv/ !;

P! D F C J�10 vr C NG;PNG D g .t/

(21)

Due to the unknown compound disturbance NG, theobserver (5) is modified as

8̂̂̂<̂ˆ̂:POqv D f .qv/ O! C �1

�je1j

˛1 C je1jˇ1

�sign .e1/ ;

PO!D OF C J�10 uC ONG C �2

�je1j

˛2 C je1jˇ2

�sign .e1/ ;

PONGD�3

�je1j

˛3Cje1jˇ3

�sign .e1/

(22)where ONG is the observer estimation of NG.

Based on the output of the finite-time observer, thesliding-mode attitude control protocol is designed as

vrD�J0C�1�C OFCC ONGC�Cf .qv/ O!Cc1SCc2S

�C

�(23)

The above design procedure and analysis can besummarized in the following theorem, which containsthe results for the attitude control dynamics (4) withunknown non-symmetric input saturation.

Theorem 3 Considering Assumptions 1 and 2 andthe attitude dynamics (19) with saturation input (18).With the finite-time observer in Eq. (22) and theproposed sliding-mode control protocol (23), the stateof the dynamic system (20) can converge to a boundedregion in finite time.

Proof The proof is similar to Theorem 2 and isomitted here. �

Remark 8 For Eq. (20), the effect of unknownnon-symmetric input saturation is treated as a part ofthe total disturbances, which is approximated usingthe designed finite-time observer (18). Although theattitude control dynamics have possibility of unknownnon-symmetric input saturation, the finite-time stabilityis still guaranteed via Lyapunov theory analysis.

4 Illustrative Example

In this section, the performance of the proposedcontrol scheme is investigated with a given numericalsimulation. Consider the rigid-body system (1) with thenominal inertia matrix,

J0 D

264 20 1:2 0:9

1:2 1:7 1:4

0:9 1:4 15

375 kg �m2

and uncertain inertia�J D diag Œsin .0:5t/ ; 2 sin .0:6t/ ; 3 sin .0:4t/� kg�m2:

The unknown disturbances are described asd .t/D Œ0:5 sin .0:1t/; 0:7 sin .0:2t/; 0:3 sin .0:3t/�N�m:

The initial attitude orientation of the unit quaternionis q .0/ D Œ0:3;�0:2;�0:3; 0:8832�T, the initial valueof angular velocity is ! .0/ D Œ0:2; 0:3; 0:5�Trad=s, andthe initial values of the state observers are Oq .0/ DŒ0; 0; 0; 1�T and O! .0/ D .0; 0; 0/Trad=s.

Xi Ma et al.: Attitude Control of Rigid Body with Inertia Uncertainty and Saturation Input 89

The simulation parameters are c1 D 2; c2 D 1:5; v D0:6; � D 0:5; umin D 5; umax D 10, and C D I3, whichcan regulate the convergence rate of the state trajectory.

The angular velocity, unit quaternion, and totaldisturbance estimation errors are shown in Figs. 2–4.Figure 5 shows the unit quaternion finally convergingto q D Œ1; 0; 0; 0�T, which shows that the proposedcontrol scheme is effective. Figures 6 and 7 showthe history of the sliding-mode surface and the controltorque. The sliding-mode surface versus time convergesin accordance with the simulation result and alsovalidates the stability analysis in Theorem 3. It shouldbe noted that the design parameters c1; c2, and "

determine the band of the bounded region, and we canchoose approximate c1; c2, and " to be small enough toguarantee motion along the sliding surface.

Time (s)

-2.0

-1.0

0

1.0

Ang

ular

vel

ocit

y es

tim

atio

n er

ror

(rad

/s)

1

2

3

0 2 4 6 8 10 12 14 16

www

˜˜˜

Fig. 2 Angular velocity estimation error versus time.

Time (s)

0

0.2

0.4

0.6

0.8

Uni

t qu

ater

nion

err

or

q̃0

q̃v1

q̃v2

q̃v3

0 5 10 15 20 25 30

Fig. 3 Unit quaternion estimation error versus time.

0���������� ��1����������� 2�������� ����3��������� ��4��� �� ������5����������� �6��� ��������7� ��� � �� � ��� � ��

Time (s)

-120

-80

-40

0

40

Tot

al d

istu

rban

ce e

rror

(N

·m)

G̃1

G̃2

G̃3

Fig. 4 Total disturbance estimation error versus time.

Time (s)0 5 10 15 20 25 30

-0.10

0.10.20.30.40.50.60.70.80.9

Uni

t qu

ater

nion

err

or

q̃0

q̃v1

q̃v2

q̃v3

Fig. 5 Unit quaternion versus time.

Time (s)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Sli

ding

mod

e su

rfac

e

s1

s2

s3

0 5 10 15 20 25 30

Fig. 6 Sliding-mode surface versus time.

Time (s)0 5 10 15 20 25 30

-60

-40

-20

0

20

40

60

80

100

Con

trol

tor

que

(N·m

)

u1u2u3

Fig. 7 Control torque versus time.

5 Conclusion

In this paper, the rigid-body attitude control problemwith time-varying disturbance, inertial uncertainty, andsaturation input is considered, as the angular velocitycannot be directly used in the control law. A finite-time observer is designed to obtain the real systemstate and total disturbance estimation information. Anobserver-based sliding-mode control law is investigatedto force the system converge to a bounded region oforigin. A numerical simulation is presented to verifythe effectiveness of the proposed method.

Acknowledgment

The work was supported by the National Natural ScienceFoundation of China (No. 61403399).

90 Tsinghua Science and Technology, February 2017, 22(1): 83–91

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Xi Ma received the BS and MS degreesfrom Xi’an High-tech Institution, China,in 2010 and 2013, respectively. Heis currently working towards the PhDdegree at Department of Computer Scienceand Technology, Tsinghua University,China. His research interests includecooperative control of multi-agent system,

robust control, and nonlinear control.

Fuchun Sun received the BS andMS degrees from Naval AeronauticalEngineering Academy, Yantai, China, in1986 and 1989, respectively, and PhDdegree from Tsinghua University, China,in 1998. He was with Department ofAutomatic Control, Naval AeronauticalEngineering Academy. From 1998 to 2000,

he was postdoctoral fellow with Department of Automation,Tsinghua University. He is currently a professor with Departmentof Computer Science and Technology, Tsinghua University. Hisresearch interests include intelligent control, neural networks,fuzzy systems, variable structure control, nonlinear systems,

and robotics. Dr. Sun is the recipient of the excellent DoctoralDissertation Prize of China in 2000 and the Choon-GangAcademic Award by Korea in 2003, and was recognized asDistinguished Young Scholar in 2006 by the National NaturalScience Foundation of China.

Hongbo Li received the PhD degreefrom Tsinghua University, China, in 2009.He is currently an assistant professorwith Department of Computer Scienceand Technology at Tsinghua University.His research interests include networkedcontrol systems and intelligent control.

Bing He received the BS, MS, andPhD degrees from Xi’an High-techInstitution, China, in 2005, 2008, and2012, respectively. He is currently anassistant professor with Xi’an High-techInstitution. His research interests includenonlinear control and flight vehicledesign.