attitude stabilization of spacecrafts under

8/13/2019 Attitude Stabilization of Spacecrafts Under

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 21, NO. 6, NOVEMBER 2013 2251

Attitude Stabilization of Spacecrafts Under

Actuator Saturation and Partial Loss

of Control Effectiveness

Bing Xiao, Qinglei Hu, and Peng Shi, Senior Member, IEEE

Abstract A practical solution is presented to the problemof fault tolerant attitude stabilization for a rigid spacecraftby using feedback from attitude orientation only. The attitudesystem, represented by modified Rodriguez parameters, is con-sidered in the presence of external disturbances, uncertain inertiaparameters, and actuator saturation. A low-cost control schemeis developed to compensate for the partial loss of actuatoreffectiveness fault. The derived controller not only has thecapability to protect the control effort from actuator saturationbut also guarantees all the signals in the closed-loop system to be

uniformly ultimately bounded. Another feature of the approachis that the implementation of the controller does not require anyrate sensor to measure angular velocity. An example is includedto verify those highly desirable features in comparison with theconventional velocity-free control strategy.

Index Terms Actuator saturation, attitude stabilization, faulttolerant control (FTC), uniformly ultimately bounded, velocity-free.

I. INTRODUCTION

ONCE A spacecraft is launched, it is highly unlikelythat its hardware can be repaired if any fault occurs.Thus any component or system fault cannot be fixed with

replacement parts. This issue can potentially cause a host ofeconomic and safety problems. A classic example was the

accident of GPS BII-07, a spacecraft in the NAVSTAR GPS

constellation developed by the US Department of Defense.

It suffered a reaction wheel failure that led to three-axis

stabilization failure and a total loss of the spacecraft [1]. In

practical aerospace engineering, a common and cost-effective

practice is to plan space missions with the ability to execute

Manuscript received November 19, 2011; revised August 31, 2012; acceptedDecember 10, 2012. Manuscript received in final form December 20, 2012.Date of publication January 23, 2013; date of current version October 15,2013. This work was supported in part by the National Natural Science Foun-dation of China under Grant 61004072, Grant 61273175, Grant 61174058,and Grant 61134001, the Program for New Century Excellent Talents in

University under Grant NCET-11-0801, the Heilongjiang Province ScienceFoundation for Youths under Grant QC2012C024, the National Key BasicResearch Program of China under Grant 2012CB215202, and the 111 ProjectB12018. Recommended by Associate Editor N. E. Wu.

B. Xiao and Q. L. Hu are with the Department of Control Science andEngineering, Harbin Institute of Technology, Harbin 150001, China (e-mail:[email protected]; [email protected]).

P. Shi is with the School of Engineering and Science, Victoria University,Melbourne, 8001 VIC, Australia, and also with the School of Electricaland Electronic Engineering, The University of Adelaide, Adelaide SA 5005,Australia (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCST.2012.2236327

a safe mode transition in which a detected anomaly deacti-

vates actuators and postpones the payload activities until the

situation is resolved by ground intervention. This solution is

sufficient to tolerate a variety of faults so long as the spacecraft

faces no immediate risk and provided the mission can be

continued following a potentially extended safe mode episode.

For instance, the far ultraviolet spectroscopic explorer satellite

went into a safe mode due to failure of two reaction wheels.

The mission engineers managed to develop a new control

law to reestablish fine pointing capability and to recover

the satellite mission by integrating the magnetic torquer

bars [2].

During the critical phases of some high real-time missions,

safe mode is not an option. Consider a military satellite

tasked with providing coverage of a specific high-priority

area. If this satellite goes to safe mode, substantial loss of

ground objectives could occur. As a result, on-line and real-

time fault tolerant control (FTC) design for spacecraft has

received considerable attention [3][5]. In [6], the problem

of automated attitude recovery for rigid and flexible spacecraft

was investigated on the basis of feedback linearization control.

Variable structure reliable control design was presented in [7]

to perform attitude stabilization maneuver. Passive and active

reliable control laws were synthesized with an observer iden-

tifying actuator faults. Another fault-tolerant attitude-tracking

control to compensate for the loss of reaction wheel effec-

tiveness fault was discussed in [8]. Only the disturbance-free

case was considered; the knowledge of the spacecraft inertia

was needed to implement the controller. Cai et al.[9] derived

an adaptive controller to follow the desired attitude trajecto-

ries with thruster failures. Input constraint, uncertain inertia

parameters, and external disturbance were explicitly addressed.

In a related work [10], a sliding-mode-based adaptive fault-

tolerant controller was proposed for a satellite. The designed

scheme can achieve the attitude control in the presence ofunknown, slow-varying satellite mass distribution, and several

fault scenarios of rotating solar flaps. In [11], an adaptive

fault-tolerant attitude-tracking controller was developed for a

flexible spacecraft. Although terminal sliding-mode controller

was designed in [12] to achieve satellite formation flying, the

stability analysis was not carried out in the case of faults. To

recover control capabilities in a faulty situation as quickly as

possible, a terminal sliding mode control scheme was proposed

in [13] to perform the rest-to-rest maneuver of a satellite with

the degradation of actuation effectiveness.

1063-6536 2013 IEEE


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2252 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 21, NO. 6, NOVEMBER 2013

The preceding FTC schemes assume that the measurement

of angular velocity is exactly available by using rate gyros. In

practice, however, the availability of angular velocity measure-

ments is not always satisfied due to either cost limitations or

implementation constraints. The occurrence of faults in sen-

sors also leads to wrong/imprecise measurements of angular

velocity [14]. Hence, the design of efficient and low-cost

attitude control without angular velocity measurements is

becoming a very challenging task. The earliest known result

in the field of velocity-free control was examined in [15] and

[16]. A passivity filter was incorporated to generate a velocity-

related signal from attitude orientation measurements. Subse-

quent extension to this control design scheme was discussed

in [17]. Although the inevitable factors of uncertain system

parameters were explicitly addressed, external disturbance

was not considered. In [18], a new class of proportional

integral controller was derived on the basis of a first-order

passivity filter to achieve attitude stabilization without velocity

feedback. The stability analysis in case of disturbance was

not carried out. In [19], an adaptive controller by employing

a filtering technique was proposed. System uncertainties andexternal disturbance were estimated by using a Chebyshev

neural network.

Another solution to examine the challenge associated with

the unavailability of angular velocity is to construct model-

based nonlinear observers, as suggested in [20]. In that

approach, a class of velocity observers was introduced to esti-

mate the joint velocities in rigid manipulators. The controller

design was not discussed. In [21], Ren presented a solution to

the distributed cooperative attitude synchronization and track-

ing problems for multiple rigid bodies. The implementation of

the controller was independent of the absolute angular velocity

measurements. However, actuator faults were not considered.

In [22], an observer was designed to estimate the elasticmodes, their rates, and unknown angular velocity. A model-

independent and observer-based attitude control method was

explored in [23]. The controller met the objective of attitude

tracking. However, a full knowledge of inertia matrix was

needed, and it was assumed that the spacecraft is free from

external disturbances. In order to release these two restric-

tions, a new attitude-tracking control scheme without velocity

measurements was developed in [24]. The design was inertia-

independent and robust to external disturbances. In addition

to the above-mentioned two velocity-free design methods,

several other control schemes on the basis of quaternion output

only were also motivated with Lyapunov-based techniques

[25], [26].Although various nonlinear control schemes are available

to accomplish various attitude maneuvers with fault-tolerant

capability guaranteed even in the absence of angular veloc-

ity measurements, few of the previously discussed literature

account for actuator saturation. Due to physical limitations,

spacecraft attitude control system is subject to actuator sat-

uration. The typical control actuators mounted in a space-

craft, such as the reaction wheel and thruster, have an upper

bound on the control torque they can exert on the attitude

system. If the attitude system is not equipped with a novel

control algorithm to solve this kind of nonlinearity, then

actuator saturation may lead to attitude control performance

deterioration or even system instability. Consequently, actuator

saturation is another key issue that needs to be addressed,

and a great deal of attention has been focused on controller

design with saturated actuators [27]. The endeavor to handle

actuator saturation is complex and multidimensional. The

most prominent method among the solutions for actuator

saturation is the antiwindup design, due to its simple structure.

In particular, a simple but effective modified proportional

integralderivative control was presented in [28] to achieve

large-angle attitude control of the spacecraft. In that paper,

the specific advantage for handling actuator saturation with

antiwindup control was verified. An alternative algorithm was

proposed in [29]. A new positive constant gain within the

framework of conventional backstepping-based control design

was derived to reduce the peak control torque. More recently, a

nonlinear adaptive control was investigated in [30]. A feedback

and a feed-forward component were incorporated to solve the

input saturation problem and to accommodate any linearly

parameterized disturbance.

It should be pointed out that almost all research until nowhas dealt with attitude control design either under component

faults, or without angular velocity measurements, or with

actuator saturation. For instance, velocity-free attitude con-

troller was designed in [31] for spacecraft formation flying

subject to actuator constraints, but actuator faults were not

investigated. In [32], although the problem of robot tracking

control with bounded torque input was solved by using output

feedback only, the problem of actuator fault tolerance was

not solved. The integrated design to deal with actuator faults,

unavailable angular velocity, and actuator saturation still needs

further research. Actually, if an unknown fault occurs in an

actuator, then in spite of the fault, the attitude control system

would continue issuing its maneuver that may no longer beachievable by the spacecraft. As a consequence, the required

control effort will quickly saturate the actuator while striving

to maintain the healthy attitude maneuvering performance,

and subsequently will destabilize the attitude. This situation

may quickly become mission critical. On the other hand, direct

high-precision velocity measurement is usually unavailable for

spacecraft in the presence of sensor faults. With a view to

tackle the above-stated challenges and potential problems, we

investigate in this paper the feasibility of performing desired

attitude maneuver by developing a dynamic control scheme

in the presence of partial loss of actuator effectiveness fault,

external disturbances, and uncertain inertia parameters.

The main contributions of this paper are as follows.1) A fault-tolerant control scheme is proposed to compen-

sate for the partial loss of actuator effectiveness fault

without any fault detection and isolation mechanism.

The closed-loop attitude system is uniformly ultimately

bounded stable, when the external disturbances are

bounded.

2) In contrast to the attitude control schemes without

angular velocity measurements available in literature,

the proposed velocity-free control approach can tolerate

actuator fault with attitude successfully stabilized. More-

over, the approach can explicitly account for actuator


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XIAO et al.: ATTITUDE STABILIZATION OF SPACECRAFTS 2253

saturation without any need of magnetic torquers to

dump the excess momentum of reaction wheels, even in

the case of actuator fault. Thus, the objective of efficient,

low-cost, and reliable attitude control design can be met.

The rest of this paper is organized as follows. In Section II,

some necessary notations, definitions, and preliminary results

are first introduced. The mathematical attitude model of a

rigid spacecraft and the control problem formulation aresummarized in Section III. The control solution is presented

in Section IV, and a numerical example of the derived control

design is presented in Section V. Conclusions and future works

are given in Section VI.

I I . NOTATIONS, DEFINITIONS, AN DP RELIMINARY

RESULTS

Let(respectively, +)denote the set of real (respectively,positive real) numbers,mn denote the set of m by n realmatrices, and In nn an n n identity matrix. For amatrix A mn , AT denotes its transpose, A1 for anyleft inverse if A has full column rank, and A2 = ATA.For any vector x(t) = [x1 x2 . . . xn ]T n , we definevectors |x| = [ |x1||x2| . . .|xn| ]T n and|x| = [ |x1| |x2| . . .|xn| ]T n with|xi |= supt0 |xi (t)|,i = 1, 2, . . . , n. We also define ||x||1 =

ni=1 |xi |. The

symbol|| ||denotes the Euclidean norm or its induced norm.The set of all the signals which are bounded on [0, ) isrepresented by L[0, ).

A. Definitions

Our main results are based on the following stability defin-

itions for a given nonlinear system

:x(t)= f(x, t) + g(x, t)d

y=h (x, t) (1)

where f(x, t): n + n are locally Lipschitz andpiecewise continuous int,d s is an exogenous disturbance,whereas y m is the system output. If there exists a closedball B= {x n| ||x|| } such that for all x0 B, thenthere exists an >0 and a number T() satisfying||x|| for all t t0 + T(), it is said that the solution x(x0, t0, t) isuniformly ultimately bounded (UUB) [19].

The following vectors sgn(), ln(), and Tanh() n ,and diagonal matrices Cosh(

) and Sech(

)

n

n are

defined as:

sgn(x)= [sgn(x1) sgn(x2) . . . sgn(xn)]T (2)ln(x)= [ln(x1) ln(x2) . . . ln(xn )]T (3)

Tanh(x)= [tanh(x1) tanh(x2) . . . tanh(xn)]T (4)Cosh(x)= diag(cosh(x1), cosh(x2) , . . . , cosh(xn)) (5)Sech(x)= diag(sech(x1), sech(x2) , . . . , sech(xn )) (6)

where x = [x1 x2 , . . . , xn ]T n, sgn(), ln(), tanh(),cosh(), and sech() are the sign, logarithmic, standard hyper-bolic tangent, cosine, and secant functions, respectively.

B. Preliminary Results

The following results are useful for the proof of the main

results in sequel.

Lemma 1 [33]: For x = [x1 x2 . . . xn]T n , thefollowing expressions hold:

1

2tanh2(xi ) ln(cosh(xi )) (7)

||Sech2(x)|| 1 (8)TanhT(x)Tanh(x) xTTanh(x). (9)

Lemma 2: Forx= [x1 x2 ... xn ]T n , it follows thatxxTsgn(x).

Lemma 3 [34]:Let and be real-valued functions defined

on+, and let b and c be positive constants. If they satisfythe differential inequality

c+ b(t)2, (0)0 (10)and L[0, ), then L[0, ). Moreover, (t) isfurther bounded by

(t) (0)ect + bc||||2. (11)

III. MODEL D ESCRIPTION ANDP ROBLEMF ORMULATION

A. Attitude Model of a Rigid Spacecraft

In this section, the mathematical model of a rigid spacecraft

attitude system, which is given by the attitude kinematics and

spacecraft dynamics, is briefly presented.

1) Spacecraft Dynamics: When all the actuators are fault-

free, with the assumption of rigid body movement, the space-

craft dynamics can be found from Eulers moment equation

as [25]

J= S()J + + d (12)where = [ 1 2 3 ]T 3 denotes the angular velocity ofthe spacecraft with respect to an inertial frame Iand expressed

in the body frame B, J 33 is the total inertia of thespacecraft, = [ u1 u2 u3 ]T 3 is the control torqueinput, d= [ d1 d2 d3 ]T 3 is the external disturbance,and S()= is the skew symmetric vector cross-productoperator.

The actuator failures commonly encountered in spacecraft

attitude system can be termed as [35]: 1) partial loss of

effectiveness (F1); 2) lock-in-place (F2); and 3) float (F3).

In this paper, the spacecraft considered does not have actuator

redundancy. If one of the actuators undergoes F2 or F3, thenit will lead to the loss of three-axis attitude control. These

two catastrophic faults are not the point of this investigation.

Consequently, this paper mainly discusses partial loss of

effectiveness faultF1. Consider actuator fault F1, represented

by a multiplicative matrix E(t). The faulty attitude dynamic

model is given by

J= S()J + E(t)+ d (13)where E(t)= diag(e11(t), e22(t), e33(t)) 33 with 0 0 (26)

2 l1

l2 1> 0 (27)

3KW

4l1l2 1

4 >0 (28)

3l2 KW

4l1 Jmax

3KW

2l1

2

3Cmax

1

41> 0 (29)

1

KP

1

4Kd

l2 KW

l1 2 1

4 22>0 (30)

where 1, 2, and + are prescribed constants specifiedby the designer. Then, the nominal controller (24) guarantees

that the attitude orientation is UUB.

Proof: Refer to the Appendix.Remark 2: Similar to the results in [9] and [24], the control

law (24) is independent on the precise knowledge of inertia

matrix J (particularly time varying and uncertain, due to

onboard payload motion, vibration of flexible appendages,

or fuel consumption). Although the implementation of the

controller needs the upper bounds on J and C, which are

used to determine the control gains in (24), those two bounds

can be approximately estimated or be chosen larger value

before launch. Thus, all the control gains can be determined.

Therefore, from the standpoint of uncertainties and external

disturbances rejection, the derived control law has great sta-

bility robustness.

2) Fault-Tolerant Control Law Design: So far, all the

actuators are assumed to be healthy. This is not a real-

istic assumption in practice. Due to aging of components,

in general, actuator faults occur, especially partial loss of

effectiveness fault. This type of fault may deteriorate attitude

control performance and even result in system instability.

Taking partial loss of effectiveness fault into consideration,

the following fault-tolerant control law is proposed to perform

attitude stabilization maneuver:

= N+ F (31)where N is the nominal control (24) for the normal system,

and F is the fault-tolerant control part designed and added tocompensate for possible actuator faults effect on the system,

and it is given by

F= sgn

+ 1

Tanh( )T

GTT (1 e0)

e0||N||,

>1. (32)

The stability analysis of the closed-loop system under the

effect of the fault-tolerant control (31) can be stated in the

following Theorem.

Theorem 2: Consider the faulty attitude system governed

by (13) and (15) under partial loss of actuator effectiveness

fault. With the application of the control law (31), suppose

that the design parameters are chosen to hold (26)(30). Then,

the attitude orientation is UUB if the nominal control (24)

guarantees the nominal attitude system UUB.

Proof: When partial loss of effectiveness fault E(t)

occurs, it yields from (16), (A7), and (31) that

J( )

+C(,

)

= Kp Tanh( )

KITanh()

+KWf+d+KdTanh(f)GT(I3E(t))+ GTF. (33)

Then, it has

TJ= TC(, )+ Kp TTanh( ) KI TTanh() + Kd TTanh(f)+KW Tf+ Td+ TGTF T GT(I3E(t)) (34)

1

Tanh( )TJ= 1

Tanh( )T

[C+ Kp Tanh( ) +dKITanh() +KdTanh(f) +KWf]

1

Tanh( )TGT(I3E(t))

+ 1

Tanh( )TGTF. (35)

With the same Lyapunov function candidate defined in

Theorem 1, substituting (34) and (35) into (A6) and following

the same lines as in (A12)(A16) result in

V (1+2)||d||2 m1|| ||2 m2||f||2

m4||Tanh(f)||2 + ( T +1

Tanh( )T)GTF

( T + 1

Tanh( )T)(I3E(t)) m3||Tanh( )||2.(36)

Due to the inequalities (26)(30), it is obtained from (31) and

(32) and Lemma 2 that

V1+2

||d||2 + ( T + 1

Tanh( )T)E(t)F

T + 1

Tanh( )T

I3E(t)

Nm3||Tanh( )||2

(1+2)||d||2( T+ 1

Tanh( )T)GT

1

(1e0)e0

||N||

+( T+1Tanh( )T)GT1(1e0)||N|| m3||Tanh( )||2= (1+2)||d||2 m3||Tanh( )||2

( T + 1Tanh( )T)GT

1

(1 e0)(1 )||N||

(1+2)||d||2 m3||Tanh( )||2. (37)With (37), the result is established using the same argument

as in the proof of Theorem 1. This completes the proof.

Remark 3: The controller is implemented with digital com-

puter in practical aerospace engineering. The value of f in

the time of (k+ 1)T can be approximately estimated by


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using the one-step previous information from attitude sensors

as

f(kT)=f((k+ 1)T) f(kT)

T(38)

whereTis the control update period. By using (22), we can

calculate the value off by

f(kT)=p((k

+1)T)

p(kT)

T

l2 ((k+ 1)T) (kT)

T. (39)

Accordingly, it is obtained from (23) that

= 1l2

fl1

l2f. (40)

Hence, can be numerically derived during the implementa-tion of the control law (31), becausef is obtained with (39),and fis given by the attitude sensors. It can be summarized

that (32) is independent on angular velocity measurements,

although

is involved. On the other hand, Nis also angular

velocity-free. As a result, the proposed fault-tolerant control(31) can be implemented without the need of any rate sensor

to measure angular velocity.

3) Analysis of the Upper Bound of the Control Effort:From

(21), we find that the state of the velocity filter is bounded

by

p l1p + l1l2| |, p(0)0. (41)According to the analysis in the proof of Theorem 2, we have

|| || L[0, ). All conditions in Lemma 3 are satisfied.Hence, the inequality (41) ensures that

p(t)

p(0)el1t

+l2

|

|2

. (42)

The inequality|i | 1, (i = 1, 2, 3) holds from Remark1. Thus, by choosing p(0)= 0, one can evaluate p(t) l2,which leads to||p(t)||

3l2.

With the result in [36], direct calculation shows that the

matrix T( ) in (15) is such that

TT( )T( )=1 + T

4

2I3. (43)

Thus

||(G( )T)1|| = ||T( )|| = 1 + T

4 1

2. (44)

On the basis of the preceding analysis, the upper bound ofthe proposed fault-tolerant control law (31) is summarized in

the following Theorem.

Theorem 3: Consider the developed fault-tolerant control

law (31). Chose the control gains such that (26)(30), and

also satisfying the following inequality:1

2

1 + (1 e0)

e0

3(Kp+KI+ Kd) + (

3 + 1)KWl2

u max. (45)

The control output of each actuator is then rigorously bounded

by the actuator saturation value.

Proof: Together with (44), it leaves the nominal control

law (24) as

|| || 12[

3(Kp+KI+ Kd) +KW||f||]. (46)Next, from (22), f is bounded by

||f|| ||p|| + l2|| || (

3 + 1)l2. (47)

It follows that:|i | ||N| |+ |F i | ||N|| +

(1 e0)e0

||N||

12

1 + (1 e0)

e0

3(Kp+KI+ Kd)

+(

3 + 1)KWl2

(48)

for i = 1, 2, 3, and F i is the i th argument ofF. Thus, with(45) and (48), it can be demonstrated that

|i | u max. (49)Hence, actuator constraints can be satisfied.

From Theorem 2, it is known that the smaller e0 in (32)is selected, more severe actuator faults the controller (31)

can tolerate, and much more fault-tolerant capability results.

In Theorem 3, it is proved that once the value of e0 is

determined (no matter how small the value of e0 > 0 is),

the control with the control gains chosen from (26)(30)

and (45) can always guarantee that the closed-loop system

is UUB. It is further theoretically analyzed that the attitude

orientation is guaranteed to be bounded by|| (t)|| fort T(), and the control effort is such that|i | umax,i= 1, 2, 3. Therefore, when implementing the fault-tolerantcontroller (32), the selection of the value e0 depends on the

tolerable level of actuator faults, which is imposed on the

spacecraft by the designers. For example, if the spacecraftneeds to tolerate 90% loss of control, then one should choose

0< e0 < 0.1. Moreover, the following two results should be

pointed out.

4) Small Value of e0 Would Not Lead to Weak Control

Power: A conservative selection of e0 may indeed lead to

small control gains. Then, it may follow weak nominal

control effort N. However, as shown in (32), small e0will

result in large fault-tolerant control effort F. This is due to

the term(1 e0)/e0 in F. As a result, the total control power would not be weak, and the control objective can still be

achieved. For instance, assume that N= [ 1 1 1]103 Nmatt

=50 s, whene0

=0.01 and

=1.001 are chosen. One will

haveF= [0.1683 0.1683 0.1683 ] Nm, if the vector sgn()in(32) is equal to[ 111 ]. This leads to the total controleffort= [ 0.1673 0.1673 0.1673 ] Nm. This torque is not aweak control power in practical aerospace engineering.

5) Unacceptably Long Time Would Not be Taken to Stabilize

the Attitude: Because the conservative selection of e0 may

not lead to weak control, as shown in the above analysis, the

closed-loop system would be stabilized within a sufficiently

reasonable amount of time. This is verified by using a numer-

ical example, as presented in Section V. On the other hand, as

shown in the proof of Theorem 2, the time T(), within whichthe attitude is governed to be|| (t)|| , is dependent on


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0 100 200-0.2

0

0.2

0.4

0.6

0.8

t/sec

1

0 100 200-0.8

-0.6

-0.4

-0.2

0

0.2

2

0 100 200-0.2

0

0.2

0.4

0.6

0.8

t/sec

3

(a)

2900 2950 3000-2

-1

0

1

2

3x 10

-6

t/sec

1

2900 2950 3000-3

-2

-1

0

1

2x 10

-6

t/sec

2

2900 2950 3000-2

-1

0

1

2

3x 10

-6

t/sec

3

(b)

t/sec

Fig. 1. Spacecraft attitude orientation with VFFTC (solid line) and UQOFC(dashed line) in the absence of faults and disturbances. (a) Initial response.(b) Steady-state behavior.

0 100 200-0.1

0

0.1

t/sec

1(rad/s)

0 100 200-0.1

0

0.1

0.2

t/sec

2rad/s)

0 100 200

-0.1

0

0.1

t/sec

3(rad/s)

(a)

2900 2950 3000-4

-2

0

2

4x 10

-6

t/sec

1(rad/s)

2900 2950 3000-6

-4

-2

0

2

4x 10

-6

t/sec

2(rad/s)

2900 2950 3000-3

-2

-1

0

1

2

3x 10

-6

t/sec

3(rad/s)

(b)

Fig. 2. Spacecraft angular velocity with VFFTC (solid line) and UQOFC(dashed line) in the absence of faults and disturbances. (a) Initial response.(b) Steady-state behavior.

A. Response With Fault-Free and Disturbance-Free Case

In this case, a relative ideal situation is simulated in which

not only no actuator fault occurs but also there is not any

0 100 200

-0.6

-0.3

0

0.3

0.6

t/sec

1(Nm)

0 100 200

-0.6

-0.3

0

0.3

0.6

t/sec

2(Nm)

0 100 200

-0.6

-0.3

0

0.3

0.6

t/sec

3(Nm)

Saturation Saturation Saturation

(a)

2900 2920 2940 2960 2980 3000-2

0

2x 10

-4

1

(Nm)

2900 2920 2940 2960 2980 3000-2

0

2x 10

-4

2

(Nm)

2900 2920 2940 2960 2980 3000-2

0

2x 10

-4

t/sec

3

(Nm)

(b)

Fig. 3. Time response of (t)with VFFTC (solid line) and UQOFC (dashedline) in the absence of faults and disturbances. (a) Initial response. (b) Steady-state behavior.

external disturbance acting on the spacecraft. We first present

the simulation results when applying VFFTC. It is shown

in Figs. 1 and 2 (solid line) that the velocity-free controller

managed to perform attitude stabilization maneuver with good

control performance. The attitude orientation is stabilized

within 100 s with high pointing accuracy even in the presence

of uncertain inertia parameters (51). Moreover, the results as

illustrated in Fig. 1 (solid line) further verify the conclusion

that the proposed control law (31) can achieve the attitude

control in the absence of actuator faults.

Although the application of UQOFC to the spacecraft can

achieve almost the same attitude control accuracy and stabilityas VFFTC, as shown in Figs. 1 and 2 (dashed line), it requires

180 s to force the spacecraft attitude to a satisfied resolution.

Much more overshoot resulted with UQOFC than that by using

VFFTC. This is due to the fact that the proposed control

scheme can decrease the overshoot by tuning the proportional,

integral, and derivative gains KP , KI, and Kd. It is also

interested to note that both controllers can protect the control

torque from actuator saturation magnitude, as we can see in

Fig. 3 (solid and dashed lines). However, compared with the

control power in Fig. 3, larger control effort of VFFTC is

observed than UQOFC. Indeed, this is due to the fact that fis always activated whether actuator fault occurs or not.


9/13


0 300 600

0

0.2

0.4

0.6

t/sec

1

0 300 600

-0.8

-0.6

-0.4

-0.2

0

t/sec

2

0 300 600-0.1

0

0.3

0.7

1

t/sec

3

(a)

2900 2950 3000-10

-8

-6

-4

-2

0

2x 10

-3

t/sec

2

2900 2950 3000-2

0

2

4

6

8

10x 10

-3

t/sec

3

2900 2950 3000-5

0

5

10

15

20x 10

-3

t/sec

1

(b)

Fig. 4. Spacecraft attitude orientation with VFFTC (solid line) and UQOFC(dashed line) in the presence of faults and disturbances. (a) Initial response.(b) Steady-state behavior.

0 300 600-0.1

0

0.1

t/sec

1(rad/s)

0 300 600-0.1

0

0.1

t/sec

2rad/s)

0 300 600

-0.1

0

t/sec

3(rad/s)

(a)

2900 2950 3000-4

-2

0

2

4x 10

-6

t/sec

1(rad/s)

2900 2950 3000

-5

0

5

x 10-5

t/sec

2(rad/s)

2900 2950 3000

-5

0

5

x 10-5

t/sec

3(rad/s)

(b)

Fig. 5. Spacecraft angular velocity with VFFTC (solid line) and UQOFC(dashed line) in the presence of faults and disturbances. (a) Initial response.(b) Steady-state behavior.

B. Response With Actuator Fault and Disturbances Case

On-orbit spacecraft is inevitably under the effect of external

disturbances. Hence, a large constant external disturbance

0 300 600

-0.6

-0.3

0

0.3

0.6

t/sec

1(Nm)

0 300 600

-0.6

-0.3

0

0.3

0.6

t/sec

2(Nm)

0 300 600

-0.6

-0.3

0

0.3

0.6

t/sec

3(Nm)

Saturation SaturationSaturation

(a)

2900 2920 2940 2960 2980 3000

-0.02

-0.0198

-0.0196

1

(Nm)

2900 2920 2940 2960 2980 3000-0.02

0

0.02

0.04

2

(Nm)

2900 2920 2940 2960 2980 3000-0.02

-0.01

00.01

t/sec

3

(Nm)

(b)

Fig. 6. Time response of (t)with VFFTC (solid line) and UQOFC (dashedline) in the presence of faults and disturbances. (a) Initial response. (b) Steady-state behavior.

torque specified by d(t)= [ 0.020.01 0.01 ]T is imposed inthis section. Moreover, the following partial loss of actuator

effectiveness fault is considered.

1) Actuator Fault Scenario:The fault scenarios occur under

these situations: 1) the reaction wheel mounted in line with

the roll axes decreases 50% of its normal value after 5 s;

2) the actuator mounted in line with the pitch axes loses its

power of 40% in 10 s; and 3) the reaction wheel mounted in

line with the yaw axes undergoes 50% loss of effectiveness

in 15 s. These are fairly severe faults that, if not compensated

for, will cause overall attitude system instability, which will

be discussed later.

When the VFFTC is implemented to the attitude system,

the time responses of attitude orientation and angular velocity

are presented in Fig. 4 (solid line) and Fig. 5 (solid line),respectively. The driving torque is shown in Fig. 6 (solid line).

As expected, we clearly see that the proposed fault-tolerant

control scheme managed to compensate for the partial loss of

effectiveness fault. High attitude pointing accuracy and attitude

stability are still guaranteed without angular velocity measure-

ments. As shown in Fig. 6 (solid line), the control torque of

each reaction wheel is still within its maximum allowable limit

even in the presence of external disturbances and uncertain

inertia parameters. This is achieved by introducing the fault-

tolerant part f in (32). Due to the limited control power of

each reaction wheel, it demands longer time to stabilize the

attitude in the presence of actuator fault. As we can see in


10/13


TABLE III

PERFORMANCESUMMARYUNDER D IFFERENTCONTROLS CHEMES

Control Actuator

Status Control Schemes

Performance Healthy VFFTC (31) UQOFC [25]

AttitudeRoll

Normal 2.0 106 3.0 106

Control

Fault 2.0 106 0.2

AccuracyPitch

Normal

3.0

106

1.0

106

Fault 4.0 105 0.008

YawNormal 2.0 106 2.0 106

Fault 6.0 105 0.008

Slew Roll

Normal 4.0 106 1.5 106

Rate

Fault 3.0 106 4.0 106

Accuracy Pitch

Normal 4.0 106 2.0 106

(rad/s)

Fault 2.5 106 7.0 105

YawNormal 2.5 106 2.0 106

Fault 1.0 106 8.0 105Attitude

Stabilization Normal 100 180

time (s) Fault 500 Infinity

Fig. 4(a) (solid line), the whole attitude stabilization maneuver

is performed nearly 500 s.

The application of UQOFC leads to the attitude orientation

and angular velocity shown in Figs. 4 and 5 (dashed line). As

pointed out, that this control law can stabilize the attitude with

angular velocity eliminated only in the absence of external

disturbance and actuator fault. Therefore, when disturbance

d(t) and actuator fault are introduced in the system, UQOFC

failed to achieve attitude stabilization control as we can see

in Fig. 4 (dashed line), although a relative higher slew rates

accuracy and limited control effort were still observed in

Figs. 5 and 6 (dashed line), respectively.

C. Summary of Simulation Results

On the basis of the above illustrated simulation results, the

steady attitude stability and control accuracy under VFFTC

and UQOFC are summarized in Table III.

1) In the absence of actuator fault and external disturbance,

both the proposed methodology and the scheme devel-

oped in [25] can accomplish the attitude stabilization

maneuver without the measurements of angular velocity.

Almost the same attitude pointing accuracy and attitude

stability are achieved. However, the developed controller

demands less time to govern the attitude, the comparisonwith [25] shows that our solution provides a faster

response.

2) When actuator fault and external disturbances are con-

sidered, our presented control law can successfully

perform the attitude stabilization maneuver. While the

controller [25] failed to stabilize the attitude, although a

high slew rates accuracy is achieved.

3) When the developed strategy is implemented to the

spacecraft, comparison with the results of the actuator

normal case and the fault case shows that the slew rates

accuracy is met with the same order of magnitude, and

only the attitude control precision decreases one order

of magnitude in the presence of actuator fault.

VI . CONCLUSION

Although many nonlinear attitude control approaches were

available for spacecraft attitude control design in literature,

none of them have addressed fault tolerance, actuator satu-

ration, velocity-free control, and robustness, simultaneously.In this paper, we developed a novel angular velocity-free

control scheme to achieve attitude stabilization maneuver in

the presence of partial loss of effectiveness fault, uncertainties

in the inertia parameters, external disturbances, and actuator

saturation. The control approach guaranteed the closed-loop

attitude system to be uniformly ultimately bounded stable.

Angular velocity sensors were not needed to implement the

control law, also magnetic torquers were not required to be

mounted to dump the excess momentum of reaction wheels.

The objective of developing a low-cost attitude control system

for spacecraft was realized. It thus let the developed con-

trol scheme be cost-effective for microsatellite applications.

However, the drawback of the scheme remains its dependenceon the lower bound e0 of the actuator fault. As some of

future works, extension of the approach with elimination of

the requirement of such lower bound should be carried out.

The lock-in-place and float faults also need to be addressed.

Further, extension to the fault-tolerant attitude control with

finite-time convergence should be investigated to perform time

critical aerospace mission, in which the attitude is stabilized

in finite time.

APPENDIX

PROOF OFT HEOREM 1

The proof uses the elements of Lyapunov stability theoryand is organized as follows: we first design a candidate

Lyapunov function, which is globally positive and radically

unbounded in the states; then we prove that the time derivative

of this Lyapunov candidate is negative definite along trajecto-

ries generated by (12) and (15). Finally, Barbalats lemma is

invoked to show that the closed-loop attitude system is stable.

A. Lyapunov Function Candidate

Consider the Lyapunov candidate function of the form

V(t)= 12

TJ 1

Tanh( )TJ + Kdl2

3

i=1ln[cosh(fi )]

+Kp3

i=1ln[cosh(i )] +

1

2

Tanh( )0

sT KI

Cosh2()ds+ KW2l2

Tf f (A1)

where

Tanh()0

sT KICosh2()ds =

3i=1

tanh(i )0

KIcosh2(i )si dsi . (A2)


11/13


Because KI and cosh2(i ) are positive, it follows that:

Tanh( )0

sT KICosh2()ds > 0 =0 3. (A3)

Before proceeding with the time derivative computation of

the Lyapunov function, its positive definiteness is shown first.

Using (7) of Lemma 1 and P1, one can get from (26) that

1

4 TJ 1

Tanh( )TJ+ Kp

2

3i=1

ln

cosh(i )

= 14

2

Tanh( )

TJ

2

Tanh( )

1 2

Tanh( )TJTanh( ) + Kp2

3i=1

ln

cosh(i )

Kp2

3i=1

ln

cosh(i ) 1

2Tanh( )TJTanh( )

3

i=1

Kp4

Jmax 2

tanh2(i ) >0. (A4)In view of (A3) and (A4), rearranging (A1) yields

V 14

TJ+ Kdl2

3i=1

ln

cosh(fi )

+ KW2l2

Tf f

+ Kp2

3i=1

ln

cosh(i )+ 1

2

Tanh()0

sT KICosh2()ds

14

TJ + Kd2l2

3

i=1tanh2(fi )+

Kp

4

3

i=1tanh2(i )

+ KW2l2

Tf f+1

2

Tanh()0

sT KICosh2()ds > 0 (A5)

for[ T T T Tf]T = 0. Hence, it can be concluded thatthe selected Lyapunov function candidate V is continuously

differentiable, radially unbounded, and positive definite in the

states ,, , and f.

B. Stability Analysis

The time derivative of Vcan be calculated as

V

=

1

2 T J

+ TJ

1

Tanh( )T J

1

Tanh( )TJ 1

(Sech2( ) )TJ

+ Kdl2

TfTanh(f) +Kp TTanh( ) +KW

l2 Tff

+ KI 2

Tanh()TCosh2()d[Tanh()]

dt. (A6)

Substituting (24) into (16), one can use E(t) I3 for actuatorfault-free case to establish

J( )+ C(, )= Kp Tanh( ) KITanh()+KdTanh(f) +KWf+d. (A7)

Through laborious yet relatively straightforward algebra

followed by the application of (23), and (25), (A7) yields

TJ= TC(, )+ Kp TTanh( ) KI TTanh()+ Kd TTanh(f) +KW Tf+ Td (A8)

Tanh()TCosh2()d[Tanh()]

dt

=2

Tanh()

T

Tanh()T

Tanh( ) (A9) TfTanh(f)= (l1 Tf+ l2 )Tanh(f). (A10)With the inequality

Tanh( )TTanh(f)1

4||Tanh( )||2 + ||Tanh(f)||2

(A11)

imposing the bound||Tanh( )||

3, and using P1P3, the

time derivative of V in (A6) is simplified as

V= 1

Tanh( )TC(, ) +

Sech2( )

TJ

1

Tanh( )Td

Kd

Tanh( )TTanh(f)

Kp

Tanh( )TTanh( )+ Td+

KW T+KW

l2 Tf

f

KW

Tanh( )TfKdl1

l2 TfTanh(f)

1

3Cmax+Jmax

|| ||2

Kp

Kd

4

||Tanh( )||2

1

Tanh( )Td 1

2 l1 Kd

l2Kd

||Tanh(f)||2

KW

Tanh( )Tf+

KW T +KW

l2 Tf

f+ Td.

(A12)

In particular, using the Youngs inequality, it follows:

Tf1

4||f||2 + || ||2 (A13)

Td 141

T+1dTd1

Tanh( )Td (A14)

14 22

Tanh( )TTanh( ) +2dTd. (A15)

Hence, for the last three items on the right-hand side of (A12),

it can be found that

KW

Tanh( )T

f+ (KWT

+KW

l2 Tf) f

l1 KW4l2

||f||2 +l2 KW

l1 2||Tanh( )||2 + KW Tf

KWl2

(l1f+ l2 )Tf

= 3l1 KW4l2

Tf f+l2 KW

l1 2||Tanh( )||2

= 3KW4l1l2

||f||2 3l2 KW

4l1|| ||2 3KW

2l1 Tf

+ l2 KWl1 2

||Tanh( )||2


12/13


3l2 KW

4l13KW

2l1

2|| ||2

3KW4l1l2

14

||f||2

+ l2 KWl1 2

||Tanh( )||2. (A16)

By using (26)(30) and (A14)(A16), the total derivative of

V along the closed-loop trajectories of (A12) can be further

established as

V 1

(

3Cmax+Jmax)|| ||2 Kp

Kd

4

||Tanh( )||2

+ l2 KWl1 2

||Tanh( )||2 12

2 l1 Kdl2

Kd||Tanh(f)||2

3l2 KW

4l13KW

2l1

2|| ||2+ 1

4 22Tanh( )TTanh( )

3KW

4l1l2 1

4

||f|| +

1

41 T+1dTd+2dTd

=1+2

||d||2

Kp

Kd

4 l2 KW

l1 2 1

4 22

m3||Tanh( )||2

1

2 l1 Kdl2

Kd

m4

||Tanh(f)||23KW

4l1l2 1

4

m2

||f||2

3l2 KW

4l1 Jmax

3KW2l1

2

3Cmax

1

41

m1

|| ||2

(1+2)||d||2 m3||Tanh( )||2. (A17)

Due to d L[0, ), then there exists a D0 > 0 such that||d|| D0. From inequality (A17), V is bounded as

V

(1

+2)D

2

0m3

||Tanh( )

||2. (A18)

It is seen from (A18) that V < 0 when are outside of theset

D

||Tanh( )|| (A19)

where = D0

(1+2)/m3. Equation (A19) implies thatV(t) decreases monotonically outside the set D. Hence, all

the signal in the closed-loop system are bounded. Moreover,

one can choose small enough 0 to guarantee that

limt || || limt ||Tanh( )|| D . (A20)

It can be concluded from (A20) that, there exists

a T() > 0 such that|| (t)|| for t T(). Thisshows that the attitude is UUB from Definition 2. Thereby,

the proof is completed.

ACKNOWLEDGMENT

The authors would like to thank the associate editor and

reviewers for their very constructive comments and sugges-

tions which have greatly helped to improve the quality and

presentation of the manuscript of this paper.

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Bing Xiao received the B.S. degree in mathematics

from Tianjin Polytechnic University, Tianjin, China,in 2007, and the M.S. degree in engineering from theHarbin Institute of Technology at Harbin, Harbin,China, in 2010, where he is currently pursuing thePh.D. degree in control science and engineering.

His current research interests include spacecraftattitude control, fault diagnosis, and fault tolerantcontrol for spacecrafts.

Qinglei Hu received the B.Eng. degree from theDepartment of Electrical and Electronic Engineer-ing, Zhengzhou University, Zhengzhou, China, in2001, and the M.Eng. and Ph.D. degrees withspecialization in controls from the Department ofControl Science and Engineering, Harbin Instituteof Technology, Harbin, China, in 2003 and 2006,respectively.

He was a Post-Doctoral Research Fellow withthe School of Electrical and Electronic Engineering,

Nanyang Technological University, Singapore, from2006 to 2007. From 2008 to 2009, he was with the University of Bristolas a Senior Research Fellow. He is currently an Associate Professor withthe Harbin Institute of Technology. His current research interests includevariable structure control and applications, spacecraft fault tolerant controland applications, and spacecraft formation flying. He has authored or co-authored more than 60 papers in journals and conferences.

Dr. Hu was a recipient of the Royal Society Fellowship for years 2008 to2009. He was an Associate Editor of the Journal of the Franklin Institute.

Peng Shi (M95SM98) received the B.Sc. degreein mathematics from the Harbin Institute of Technol-ogy, Harbin, China, the M.E. degree in systems engi-neering from Harbin Engineering University, Harbin,the Ph.D. degree in electrical engineering from theUniversity of Newcastle, Newcastle, Australia, thePh.D. degree in mathematics from the Universityof South Australia, Adelaide, Australia, and theD.Sc. degree from the University of Glamorgan,Pontypridd, U.K., in 1982, 1985, 1994, 1998, and2006, respectively.

He was a Lecturer with Heilongjiang University, Harbin. He was a Post-Doctoral Researcher and a Lecturer with the University of South Australia. Hewas a Senior Scientist with the Defence Science and Technology Organisation,Edinburgh, Australia. He was a Professor with the University of Glamorgan.He is currently a Professor with Victoria University, Melbourne, Australia,and the University of Adelaide, Adelaide, Australia. He has authored or co-

authored several papers in journals and conferences. His current researchinterests include system and control theory, computational and intelligentsystems, and operational research.

Dr. Shi is a fellow of the Institution of Engineering and Technology, U.K.,and the Institute of Mathematics and its Applications, U.K. He is on theeditorial board of a number of international journals, includingAutomatica, theIEEE TRANSACTIONS ON AUTOMATIC CONTROL, the IEEE TRANSAC-TIONS ONF UZZYS YSTEMS, the IEEE TRANSACTIONS ONS YSTEMS , MANAN D CYBERNETICSPART B : CYBERNETICS, and the IEEE T RANSAC-TIONS ONC IRCUITS ANDS YSTEMSPARTI: REGULARPAPERS.

attitude stabilization of spacecrafts under

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