Output feedback control for rigid-body attitude with constant disturbances

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Output Feedback Control for Rigid-Body Attitude withConstant DisturbancesJinchang Huab & Honghua Zhangab

a Beijing Institute of Control Engineering, Beijing 100190, Chinab Science and Techonology on Space Intelligent Control Laboratory, Beijing, 100190, ChinaAccepted author version posted online: 07 Nov 2014.

To cite this article: Jinchang Hu & Honghua Zhang (2014): Output Feedback Control for Rigid-Body Attitude with ConstantDisturbances, International Journal of Control, DOI: 10.1080/00207179.2014.971342

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Publisher: Taylor & Francis

Journal: International Journal of Control

DOI: http://dx.doi.org/10.1080/00207179.2014.971342

Output Feedback Control for Rigid-Body Attitude with Constant Disturbances

Jinchang Hu*a,b, Honghua Zhanga,b

a Beijing Institute of Control Engineering, Beijing 100190, China

b Science and Techonology on Space Intelligent Control Laboratory, Beijing, 100190, China

* Corresponding author. Email: [email protected], Telephone: +086 01062744801

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Output Feedback Control for Rigid-Body Attitude with Constant Disturbances

In this article, the control problem of rigid-body attitude under constant disturbances without

angular velocity measurement is solved by the combination of the immersion and invariance

methodology and the dynamic scaling technique. Two observers, which are respectively for

estimating the angular velocity and the disturbance , are constructed by utilizing the immersion

and invariance method. The mismatched term arising from the observers is dominated by the high

gain injection. The control law is a simple proportional-derivative controller plus a disturbance

compensation term, where the estimates of the angular velocity and the disturbance from

observers are used for feedback directly. The overall closed-loop system is shown to be almost

globally asymptotically stable under easy choices of some control parameters. Finally, simulations

are conducted to demonstrate the effectiveness of the proposed control scheme.

Key words: rigid-body attitude control; output feedback control; disturbances elimination; immer-

sion and invariance; dynamic scaling.

I. Introduction

The problem of controlling a rigid-body attitude has long attracted many researchers’ attention due to

its inherent nonlinear dynamics and wide applications ranging from satellites to robot manipulators.

The authors in [1] proved that a proportional-derivative(PD) control law can stabilize the attitude

system. However, some important factors should be taken into account in some engineering

applications. One problem is that disturbances almost exist in all practical operations. Another one is

that the angular velocity might be sometimes unavailable, due to the cost considerations in small

satellites, or that gyros are prone to failure.

Tremendous research efforts have been devoted to dealing with external disturbances. The au-

thors in [1] pointed out that a simple PD law is robust against small amplitude of disturbances. Sliding

mode controller [2] has also been employed to attenuate the disturbances. The authors in [3] and [4]

proposed two independent control schemes that can deal with bounded disturbances and input satura-

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tion. However, care must be taken of the choice of certain parameters as the convergence of the attitude

cannot always be guaranteed. Ref. [5] proposed an adaptive scheme to account for sinusoidal distur-

bances. To address constant disturbances, it is natural to use a PID controller to eliminate it[6-9]. Adap-

tive schemes can also be designed since constant disturbances can be regarded as a kind of parameter[6,

10].

On the other hand, lots of work has also been done on addressing the case without velocity

measurement. To solve this problem, one common way is to design an observer for the angular velocity.

However, it is extraordinarily difficult to accomplish this task due to the existence of Coriolis term. In

this aspect, one remarkable progress was achieved in [11] where the nonlinear feature of the attitude

dynamics has been fully exploited. However, the coupling issue between an observer and a controller is

not considered there. In [12], a controller-observer scheme was proposed for attitude tracking, yet it is

just locally stabilizing. In [13, 14], an intrinsic observer was designed and was later applied in [15] to

the tracking problem of simple mechanical systems. However, the intrinsic observer there is also just

locally convergent and tracking could only be realized for small initial observer errors.

Due to the difficulty in constructing a velocity observer, more attention has been directed to-

wards utilizing passive filters to generate a desired signal to substitute the velocity. Ref. [12] proposed

a lead-filter-based regulation scheme, yet the convergence is just local and the parameters there are dif-

ficult to choose. In [16], passivity was utilized to design a globally convergent controller for attitude

stabilizing, while [17] accounted for the input saturation case. In [18] a nonlinear auxiliary dynamical

system was constructed and almost global asymptotic stability was obtained. The method in [18] was

later applied to attitude synchronization of a group of spacecraft in [19]. In [20] a dynamic partial state

feedback controller was designed to deal with the velocity-free case.

Few work has been done in dealing with parameter uncertainties and/or external disturbances in

the absence of velocity measurement. Ref. [21] used a linear passive filter to construct a controller to

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deal with parameter uncertainty, yet the result was just semiglobally stable. In [22], the linear filter in

[21] was also used, while Chebyshev neural networks were utilized to approximate the nonlinearities

and disturbances; however, only ultimate boundness could be achieved and the result only holds lo-

cally. Ref.[23] proposed a proportional-integral controller by means of a passive linear filter to elimi-

nate constant disturbance when the angular velocity is not available, whereas the proof only held in the

disturbance-free case. In [14], state-dependent external forces were considered for simple mechanical

systems without velocity measurement, yet its method is not applicable to constant disturbance.

As far as the authors know, the problem of eliminating constant disturbance without angular

velocity measurement for controlling rigid-body attitude has not been solved thoroughly, and no global

results have be published.

In [24, 25], the immersion and invariance (I&I) methodology (for more information about I&I,

readers are referred to [26] and the references therein) was proposed for output control of general me-

chanical systems. In this method, the nonlinear terms arising from the Coriolis terms were dominated

by the dynamic scaling technique. As the method in [24, 25] needs the solution to certain integrals and

no explicit expressions of the solution can be derived in most cases, [27] overcame this difficulty by a

particular choice of the gain matrix. The methods developed in [24, 25, 27] shed new light on designing

observers for general Euler-Lagrange systems. However, all of them just considered the disturbance-

free case. For systems with parameter uncertainties, [28] proposed a redesign method to estimate the

uncertainties based on I&I. However, the conditions required in [28] are too strict to be applicable for

our problem.

Inspired by the methods developed in [24, 25, 27], the I&I method in conjunction with dynamic

scaling skill is employed in this study to solve the output feedback control for rigid-body attitude with

constant disturbances. The main novelty of this article lies in that the I&I method is used for the first

time to design observers for constant disturbances as well as angular velocity. The overall scheme is in

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the observer-controller form, in which the unit quaternion is adopted for attitude representation. More-

over, almost global asymptotic stability (GAS) is established for the closed-loop system, whereas [24,

25, 27, 28] only considered the observer design problem.

The remainder of this paper is organized as follows. In Section II, the control problem is formu-

lated, and the controller and some preliminaries are provided. Then the main results are given in Sec-

tion III. Section IV provides the elaborate proof of the closed-loop stability. Then simulations are con-

ducted to validate the proposed control strategy in Section V, followed by conclusions in Section VI.

II. Problem formulation and preliminaries

To avoid the singularity in attitude representation, we use the unit quaternion to describe the orientation

of a rigid body. The attitude dynamics of a rigid body with constant disturbances can be written as

0

0 3

1

21

( )2

Tv

v v

q q

q q I q

(1)

J J u d (2)

where 30( , )vq q represents the quaternion with 0q and vq representing the scalar and vector

parts respectively and respect the unit constraint 20 1T

vq q ; 31 2 3: , , represents the angu-

lar velocity with respect to an inertia frame I and is expressed in the body frame B ; 3u denotes the

control torque input; 3d represents the constant external disturbance torque; 3 3J is the inertia

matrix. For 3v , the operator v denotes a skew-symmetric matrix acting on the vector v .

Before we begin, the following assumptions are made:

Assumption 1: The inertia matrix J is constant and perfectly known.

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Assumption 2: The disturbance d is an unknown constant, whereas the upper bound of its 2-norm is

assumed to be known and denoted by Md .

Assumption 3: Only the quaternion 0( , )vq q is measureable.

To simplify the analysis and clarify the main idea, we only consider the attitude stabilization

problem. The problem of output feedback stabilization for rigid-body attitude can thus be stated as fol-

lows:

Consider the attitude system composed of (1) and (2). Under Assumptions 1-3, design the con-

trol input u that only depends on the information of 0( , )vq q , such that for all initial states 0( (0), (0))vq q

and (0) , there is:

0 1q , 0vq , 0 , as t

In our study, the controller is simply designed as

ˆˆp v vu k q k d (3)

where and d denote the estimates of angular velocity and unknown constant disturbances respec-

tively.

It should be pointed out that the forms of and d will be derived in the sequel, and that the

design of and d and the stability analysis are the major difficulties of our study.

To facilitate our analysis, we define:

0:v

qq

q

,

0 3

1

2( ) :1

( )2

Tv

v

qE q

q I q

With the above definitions, (1) can be rewritten as

( )q E q (4)

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Before we proceed, the following lemmas should be introduced.

Lemma 1 The following equalities hold[29]:

1Tq q , 34 ( ) ( )TE q E q I , 0 3 2|| || 1vq I q

Lemma 2 For 4x , the following equality holds

2 2

1|| ( ) || || ||

2E x x

The proof of Lemmas 1 and 2 can be accomplished with direct calculations and hence will be

omitted.

In the following, norms of all matrices and vectors, if the subscripts omitted, will be regarded as

the 2-norm (for matrix, it means the induced 2-norm). Moreover, we define min,J as the smallest ei-

genvalue of matrix J , and for notational convenience, we define 2: || ||J J .

III. Main results

Theorem 1 Consider the attitude system given by (1) and (2). Design the controller as

ˆˆp v vu k q k d (5)

where and d are respectively the observed values of and d , given by:

1ˆ ˆ( , )J K q q (6)

ˆ ˆ( , )dd K q r q (7)

where 3 and 3 are the dynamic parts, 3 4ˆ( , )K q and 3 4ˆ( , )dK q r are the gain ma-

trix, 3 and 4q are the observed values for and q respectively, r is the dynamics scaling

factor. The dynamics of and are designed as

1ˆ

ˆˆ ˆ ˆˆ ˆ ˆ ˆ[ ( , ) ( , ) + ( , ) ( ) ]qJ K q qq K q q K q E q J u d (8)

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ˆ ˆˆ ˆ ˆ ˆ( , ) ( , ) ( , ) ( )q d r d dK q r qq K q r rq K q r E q (9)

where ˆ ˆ( , )q K q and ˆ( , )K q denote the Jaccobi matrices of ˆ( , )K q with respect to q and ,

respectively.

Design the dynamics of and q with high gain injection as

ˆˆ ˆ( ) ( )qq E q q q (10)

ˆˆ ˆ ˆ( )J J u d (11)

Construct the dynamics of r as

min,

ˆ( || || || ||) ( 1)rJ

J J

krr k q r

(12)

The gains and parameters in the above equations are chosen as

ˆ ˆ( , ) 4 ( )TK q k E q (13)

2ˆ ˆ( , ) 4 ( ) /TdK q r E q r (14)

1

4T

Jk k (15)

1

2r zk k (16)

min,

1 1( )4 4

Jr J r

J

k

(17)

2 2 2 2 22

1 2 ˆ(|| || )4q M qr r k d d

r (18)

2 2 2 2ˆ( ) || || Jr k q r (19)

where , r , z , q and are positive scalars that can be adjusted.

Then for all initial conditions, we have the following results:

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(a) The designed observers are converging, i.e., ˆ ( ) ( )t t , ˆ( ) ( )q t q t , ˆ( ) ( )t t , ˆ( )d t d ,

( ) 1r t , as t ;

(b) The attitude system are stabilized, i.e., ( ) (0,0,0)Tvq t and ( ) 0t as t .

Remark 1: The controller in (5) is in fact a PD controller plus a disturbance compensator. It has been

proven in [1] that a PD controller can stabilize a rigid-body attitude system, which inspired us to design

the controller (5) in the case of constant disturbances. The main difficulty is therefore reduced to

designing effective observers for the angular velocity and disturbance. Other kinds of controllers might

be pursued to get better performance, as long as they could guarantee the boundness of the angular

velocity, which will be clear in the sequel.

Remark 2: Compared with [24], the major difference lies in that we construct another observer for

constant disturbance. However, the establishment of this observer is far from easy, which needs a spe-

cial choice of the gain matrix ˆ( , )dK q r .

Remark 3: The dynamics of and are designed to cancel the known terms of the derivative of

ˆ( , )K q q . The structure of the gain matrix ˆ( , )K q is constructed to suppress the nonlinear term

J . Note that ˆ( , )K q is the function of q and , and it should be pointed out that we could not

simply replace in ˆ( , )K q with . If so, it will be very difficult to design the dynamics of to

eliminate , which involves the unknown variable , as can be seen later. To overcome this difficulty,

we need to design another observer (see (11)) for .

Remark 4: It should be pointed out that although the observers are quite complicated, the controller is

very simple. Moreover, we only need to modify the parameters r , z , q and to adjust the con-

vergence speed of the observers, while only the gains pk and vk are needed to choose to adjust the

controller’s performance.

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Remark 5: The structure of the closed-loop system is depicted as Figure 1. It is obvious that the

overall closed loop is in the observer-controller form.

IV. Stability Analysis

A. Derivation of some dynamics

In order to prove Theorem 1, we need to derive some useful dynamics.

Define the estimate errors of angular velocity and quaternion as

ˆ , ˆq q q *

(20)

Differentiating J with respect to time and applying (4) and (8), yields

ˆˆ ˆˆ( , ) ( )

ˆˆˆ( , ) ( )

J K q E q J J d d

K q E q J J d d

(21)

From (6) and (8), we can get

ˆˆ ˆ ˆˆ( , ) ( )J K q E q J u d (22)

Based on (11) and (22), we have

ˆ( )ˆˆ( , ) ( ) ( )

dJ K q E q

dt

(23)

From (4) and (10), and in view of the definition of q in (20), we have:

( ) qq E q q (24)

Define the scaled variable z as

zr

(25)

* Note that q no longer satisfy the unit-norm constraint, which is different from the usual quaternion error from

quaternion multiplication.

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Substituting (25) into (21) yields

ˆˆˆ( , ) ( )

d d rJz K q E q z z J Jz Jz

r r

(26)

Differentiating d in (7) with respect to time, and in view of (9), we obtain

ˆˆ ˆ ˆ ˆ ˆ( , ) ( , ) ( , ) ( )

ˆ( , ) ( )

q d r d d

d

d K q r qq K q r rq K q r E q

K q r E q

(27)

Substituting (14) into the above equation, and applying the second equation in Lemma 1 and the

definition in (25), we have

2

ˆ4 ( )ˆ ( )

ˆ4[ ( ) ( ) ( )] ( )

4 ( ) ( )

T

T T T

T

E qd E q rz

rz

E q E q E q E qr

z zE q E q

r r

(28)

B. Construction of Lyapunov functions

Based on the above equations, we will construct some Lyapunov functions which will be crucial in the

stability analysis. We first construct the following one:

1

2T

zV z Jz (29)

The function zV is used to prove . Differentiating zV with respect to time, yields

2

ˆˆˆ[ ( , ) ( ) ]

1 ˆˆˆ( , ) ( ) || |||| || ( )

Tz

T T TJ

d d rV z K q E q z z J Jz Jz

r rr

z K q E q z z z d d z Jzr r

(30)

where (26) and the following inequalities have been applied

0Tz z J , 2ˆ ˆ ˆ|| || ||| |||| || || |||| ||T TJz Jz z Jz z

(31)

Define

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1 ˆ ˆ( ) ( )2

TdV d d d d , zd z dV V V (32)

The differentiation of zdV with respect to time is given by

2 ˆ ˆˆˆ( , ) ( ) || |||| || ( ) ( )T T Tzd z d J

z rV V V z K q E q z z d d d z Jz

r r (33)

Note that

2 2 2 22

ˆ ˆ ˆ( ) ( ) 4( ) ( ) ( )

|| ||ˆ(|| || ) || ||

2 1ˆ(|| || ) || || || ||4

T T T

M

M

z zd d d d d E q E q

r rz

d d qr

d d q zr

(34)

where (28), Assumption 2, Lemmas 1 and 2 have been applied.

Substituting (34) into (33), yields

2 2 2 2 22

2 1ˆˆˆ( , ) ( ) || |||| || (|| || ) || || || ||4

T Tzd J M

rV z K q E q z z d d q z z Jz

r r (35)

Substituting (13) into (35) and applying Lemmas 1 and 2, yields

2

2 2 2 22

2 2 2 22

ˆˆ4 ( ) [ ( ) ( ) ( )] ( ) || |||| ||

2 1ˆ (|| || ) || || || ||4

1 2 ˆˆ = [ ( ) || || ] || || (|| || ) || ||4

ˆ 4 ( ) [ ( ) ( )] ( )

T Tzd J

TM

J M

T T T

V k z E q E q E q E q z z

rd d q z z Jz

r r

k z d d qr

rk z E q E q E q z z J

r

2 2 2 22

2

2 ˆˆ [ ( || ||)] || || (|| || ) || ||

( ) || |||| ||

TJ M

T

z

k z d d qr

rk q z z Jz

r

(36)

Notice

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2

2 2

2

ˆ( || ||) || ||

ˆ ˆ( )( )|| || ) || ||

ˆ ˆ|| || || ||

ˆ|| |||| ||

TJ

T TJ

JT T

J

z

z z

z

(37)

min,

2 2

1 1ˆ( || || ( ) || ||)

ˆ( || || ( ) || ||) || || || ||

T

T TrJ

J J

J r

rz Jz

rk r

z Jz k q z Jzr

k q z k z

(38)

where (12) has been employed in (38).

Substituting (37) and (38) into (36), we have

2 2 2 22

2 ˆ( ) || || (|| || ) || || zd r MV k k z d d qr (39)

Moreover, we define the following Lyapunov functions:

1ˆ ˆ( ) ( )

2TV J ,

1

2T

qV q q , min, 2( 1)2

JrV r

(40)

The Lyapunov function V is established to prove ˆ , while qV and rV are for q q and

1r , respectively.

The time derivative of V along (23) is given by

2

2

2 2 2 2 2

ˆ ˆˆ( ) [ ( , ) ( ) ( )]

ˆ ˆˆ|| |||| ( , ) ( ) |||| || || ||

ˆ ˆˆ( ) || |||| |||| || || ||

1ˆˆ|| || [ ( ) || || ] || ||

4

TV K q E q

K q E q

rk q z

z r k q

(41)

where the Young inequality and the following inequality has been used:

ˆ ˆ ˆ|| ( , ) ( ) || 4 ( ) || ( ) ( ) || ( ) || ||K q E q k E q E q k q (42)

The time derivative of qV along (24) is given by

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2

2 2 2 2

[ ( ) ] || |||| ( ) |||| || || ||

1 1 1|| |||| || || || || || ( ) || ||

2 4 4

Tq q q

q q

V q E q q q E q q

r q z q z r q

(43)

while the time derivative of rV along (38) is given by

min,min,

min, 2

min,2 2 2 2 2 2

ˆ( 1){ [ || || ( ) || ||] ( 1)}

ˆ( 1) || || ( ) ( 1) || || ( 1)

1 1ˆ || || ( ) || || ( )( 1)

4 4

rr J J

J J

JJ r

J

JJ r J

J

krV r k q r

r r k r r q k r

r r k q k r

(44)

where the Young inequality has been utilized twice.

Now we construct the total Lyapunov function as follows

t zd q rV V V V V (45)

By applying (39), (41), (43) and (44), the time derivative of tV is given by

2 2 2 22

2 2 2 2 2

2 2 2 2 2

min,2 2 2 2

2 ˆ( ) || || (|| || ) || ||

1ˆˆ || || [ || || ( )] || ||

41 1

ˆ || || ( ) || || || ||4 4

1 1 ( ) || || ( )( 1)

4 4

t r M

q J

Jr J

J

V k k z d d qr

z r q k

z r q r

r k q k r

(46)

which is followed by

min,2 2

2 2 2 2 2 22

2 2 2 2 2

1 1 1( ) || || ( )( 1)

2 4 4

1 2 ˆ [ ( ) (|| || )] || ||4

ˆˆ [ || || ( ) ] || ||

Jt r r J

J

q M

J

V k k z k r

r r k d d qr

r q k r

(47)

Substituting the expressions of rk , k , q and in (16)-(19) into (47), we obtain

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2 2 2 2ˆ|| || || || || || ( 1)t z q rV z q r (48)

We are now in the position to finish our final proof.

C. Proof of (a) in Theorem 1

From (48) and the definition of tV in (45), the following results can be derived easily:

2z L L , 2q L L , 2ˆ( ) L L , d L , 2( 1)r L L (49)

As 2( 1)r L L , we also have r L . Hence from the definition of z in (25), we have

rz L (50)

Moreover, since q L and q q q , there is q L .

Before we proceed, we need to introduce another lemma.

Lemma 3 Consider the attitude system given by (1) and (2). If the control input is designed as (5), then

L (The proof can be found in Appendix A).

As , applying Lemma 3 and (50), we have ˆ L . As ˆ( ) L from (49), we

also have L . In view of (12), it is easy to see that r L . As 2( 1)r L L , and r L , we

have that 1r as t .

Substituting the expressions of ˆ( , )K q in (13) to (26), and based on the previous analysis, it

is not difficult to find that all terms on the r.h.s. of (26) are globally bounded, i.e., z L . As

2z L L , applying the Barbalat’s Lemma, we conclude that ( ) 0z t as t .

Substituting the expression of q in (18) into (24), it is also easy to verify that q L . Since

2q L L , we have that ( ) 0q t by applying Barbalat’s Lemma again.

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Similarly, substituting the expressions of ˆ( , )K q and into (23), we can also verify that

ˆ( ) /d dt L . As 2ˆ( ) L L from (49), the reapplication of Barbalat’s Lemma leads to

ˆ( ) 0 as t .

The remaining task is then to prove that ˆ( ) 0d d as t . To show this, observe that ,

, q , q, and z are all globally bounded as well as their first derivatives. Hence, ˆ|| || and ˆ|| ||q q

are uniformly continuous w. r. t. time. Since r is uniformly continuous w. r. t. ˆ|| || and

ˆ|| ||q q (as can be seen in (12)), one can see that r is also uniformly continuous w. r. t. time. Hence

from (26), one can see that Jz is also uniformly continuous w. r. t. time. As 0z and Jz is uni-

formly continuous, applying Barbalat’s Lemma, we have 0z as t . This means that the r. h. s.

of (26) tends to zero as t . As we have already proven that 0z and all states are globally

bounded, we conclude that ˆ( ) /d d r converges to zero. Since r is globally bounded, we have that

ˆ( ) 0d d as t .

The proof of the first part is thus completed.

D. Proof of (b) in Theorem 1:

Construct the following Lyapunov function

22 0

1( )[( 1) ]

2T T T

p v v v vV k ck q q q J cq J (51)

where 0c . It is easy to check that as long as c is small enough, 2V can be made positive definite.

Define

2ˆ

v vk d d k d (52)

Based on the previous proof, we have 0 and 0d , therefore 2 0 .

From Appendix B, we can also prove that 2 2L .

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The time derivative of 2V along the controller (5) in closed loop with the system (1) and (2) is

given by:

2

0 3

2 22

2 2 21 2 2

1 2

( ) [ ]

1 [ ] [ ( ) ]

23

( ) || || || || ( )23 1 1

( ) || || ( ) || || ( ) || |2 4 4

T Tp v v v p v v v

T Tp v v v v

Tv J p v v

v J p v

V k ck q cq J k q k k d

J k q k k d c q I q J

k c ck q cq

k c ck c q

2|

(53)

where Lemma 1 and the Young inequality have been applied.

Notice that we can always select small enough c>0 and suitable 1 2, such that

1

30

2v Jk c , 22 0pck c

Integrating both sides of (53) on the time interval [0, )t , we have

2 2 22 1 20 0

220

1 2

3( ) ( ) || || d ( ) || || d

21 1

(0) ( ) || || d4 4

t t

v J p v

t

V t k c ck c q

V

(54)

It is not difficult to obtain 2L L and 2vq L L from (54). Applying Barbalat’s Lemma,

we can easily prove that

( ) 0t , ( ) 0vq t , t . □

Remark 6: The first part of Theorem 1 states that for all initial conditions, there is q q . At first

glance, it seems to contradict with the well-known fact: due to the topology obstruction on the three-

sphere 3S , no global asymptotic stability result can be got as long as the controlled flow is continu-

ous[31]. Notice that q q is in fact defined on 3 instead of 3S and q q no longer satisfy the unit-

norm constraint. Hence it is not surprising that we can obtain a global result for q q .

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Remark 7: It should be pointed out that as 0vq and 0 , there may be 0 1q .or 0 1q . No-

tice that the equilibrium point 0( 1, 0)q is unstable [1], hence for all initial conditions except a

zero measure set that includes the point 0( 1, 0)q , there will be 0 1q and 0 [5]. This

means that the equilibrium point 0( 1, 0)q of the closed loop is almost globally asymptotically

stable (GAS).

Remark 8: Although we utilize high gain injection for ensuring global stability, our observers are GAS,

which is different from the common high gain observers that can only obtain local or semi-global re-

sults.

Remark 9: At first glance, a controller with integral part can also be utilized to eliminate constant dis-

turbance[8][23], and the control strategy in our study seems to be unnecessarily complicated. It should be

pointed out that due to the nonlinear structure of the attitude system, it is quite difficult to prove the

global stability of the attitude system with a PID controller. To guarantee the closed-loop stability, the

authors in [8] constructed a rather complicated PID-like controller. However, the measurement of

angular velocity is required in [8].

When the angular velocity is not available, the problem becomes more intractable. In this regard,

the authors in [23] employed a nonlinear proportional-integral controller along with a low-pass filter

for controlling a rigid-body attitude system. However, the proof in [23] only holds in the disturbance-

free case.

Different from previous literature, the study realizes constant disturbance elimination without

angular velocity measurement, and presents a rigorous proof of the closed-loop stability, however, at

the price of a more complicated control strategy.

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Remark 10: To deal with the attitude tracking case, we can define the estimate of angular-velocity

tracking error as ˆ ˆ ( ) dR q , where d is the desired angular velocity and is the estimate of

the body angular velocity from the observers in section III. Then we can design the following attitude-

tracking controller:

ˆˆ ( )v v v d d du k k q d R JR JR †

(55)

The proof is similar and hence will be omitted.

V. Simulations

The parameters, gains and initial conditions are chosen as Table 1.

Figure 2 to Figure 7 show the simulation results under the above settings. Figure 2 and Figure 3

show that the rigid body’s attitude converges to (1,0,0,0)Tq and the angular velocity converges to

zero, which verifies the effectiveness of the proposed control scheme. Figure 4 presents the estimation

error of the angular velocity, showing that it is also converging to zero ultimately. Figure 5 shows that

the estimate of the disturbance converges to its true value. Figure 6 is the time history of the control

torque, which shows that the control torque finally balances the disturbance. Figure 7 shows that the

dynamic scaling factor converges to its final value 1.

VI. Conclusions

In this article, we present a systematic method for solving the output feedback control problem of rigid-

body attitude subject to constant disturbances. The core idea is to employ the I&I method to construct

two observers for the angular velocity and the disturbances, respectively. Then the nonlinear Coriolis

torque is dominated by the dynamic scaling skill, while the mismatched part is dominated by the high

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gain injection. The overall closed-loop system is shown to be almost globally asymptotically stable

through a rather lengthy analysis. Compared with existing literature that used ad hoc methods for

output feedback control, the method in the study is systematic and can deal with constant disturbances.

Although the constructed observers are quite complicated, our method can be viewed as a benificial

attempt to solve this intractable problem, and it will be part of our future job to further simplify the

control schemes. Finally, since our control schemes need to know the inertia matrix exactly and

requires the disturbances to be constant, we may also generalize it to the case of unknown system

parameters and state-dependent disturbances.

Appendix A: Proof of Lemma 3

Substituting (5) into (2), yields:

ˆˆ ( )p v vJ J k q k d d (56)

Construct the Lyapunov function as 1TV J with its time derivative along (56) given by

1ˆˆ[ ( )]

ˆˆ[ ( ) ( )]

ˆˆ|| || [ || || ( || || || ||)]

|| || ( || || )

Tp v v

Tp v v v

v p v

v

V J k q k d d

k q k k d d

k k k d d

k

(57)

where we have defined ˆ ˆˆ: || || || || || || || ||p v p vk k d d k k d d .

In view of (49) and (50), we know that ˆ( )d d L and L . Hence, we have L .

Therefore, from the last equality in (57), it is easy to see that L . □

† Note the q is defined as the attitude tracking error by 1

dq q q , which is different from the quaternion observation error defined in (20).

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Appendix B: Proof of 2 2L

To prove 2 2L , notice that

2 2 2 22|| || || || 2 || || 2 || ||v vk d k d (58)

and

2 2 2 2 200 0 0

|| || d ( ) || || d (sup ( )) || || dt t t

tr t z r t z (59)

where max 0sup ( )tr r t , and r L and 2z L have been used.

Therefore, we only need to prove 2d L . To this end, establish the Lyapunov function as

2T

t tV V d Jz (60)

where tV is the Lyapunov function in (45), and 0 is small enough to ensure 2 0tV .

The time derivative of 2tV along the closed loop is given by

2

2 2 2 2ˆ|| || || || || || ( 1)

[ 4 ( ) ( ) ]

ˆˆ[ ( , ) ( ) ]

T Tt t

z q r

T T

T

V V d Jz d Jz

z q r

z zE q E q Jz

r r

d rd K q E q z z J Jz Jz

r r

(61)

Since we have proven that all signals are globally bounded, we can always find proper scalars

1 2, 0c c such that

21[ 4 ( ) ( ) ] || ||T Tz z

E q E q Jz c zr r

(62)

2ˆˆ[ ( , ) ( ) ] || |||| ||T r

d K q E q z z J Jz Jz c d zr (63)

Substituting the above results into (61), we have

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2 2 2 2 22 1 2

max

2 2 2

ˆ( ) || || || || || || ( 1) || || || |||| ||

ˆ || || || || ( 1)

t z q r

Tq r

V c z q r d c d zr

q r x Qx

(64)

where : || || || ||T

x z d , and

1 2

2 max

/ 2:

/ 2 /z c c

Qc r

It is easy to verify that Q can always be made positive by small enough , which leads to

2 0tV . Integrating both sides of (64), we can easily derive that 2|| ||d L .

In view of (58) and (59), we have 2 2L .

References

[1] Wen J.T., and Kreutz-Delgado K. (1991), 'The Attitude Control Problem', IEEE Transactions on

Automatic Control, 36, 1148-1162.

[2] Boskovic J.D., Li S.-M., and K.Mehra R. (2001), 'Robust Adaptive Variable Structure Control of

Spacecraft under Control Input Saturation', Journal of Guidance, Control, and Dynamics, 24,

14-22.

[3] Boskovic J.D., Li S.-M., and Mehra R.K. (2004), 'Robust Tracking Control Design for Spacecraft

under Control Input Saturation', Journal of Guidance, Control, and Dynamics, 27, 627-633.

[4] Wallsgrove R.J., and Akella M.R. (2005), 'Globally Stabilizing Saturated Attitude Control in the

Presence of Bounded Unknown Disturbances', Journal of Guidance, Control, and Dynamics,

28, 957-963.

[5] Sanyal A., Fosbury A., Chaturvedi N., and Bernstein D.S. (2009), 'Inertia-Free Spacecraft Attitude

Tracking with Disturbance Rejection and Almost Global Stabilization', Journal of Guidance,

Control, and Dynamics, 32, 1167-1178.

Dow

nloa

ded

by [

Dic

le U

nive

rsity

] at

11:

38 0

8 N

ovem

ber

2014

Accep

ted M

anus

cript


[6] Schaub H., Akella M.R., and L.Junkins J. (2001), 'Adaptive Control of Nonlinear Atttitude Motions

Realizing Linear Closed Loop Dynamics', Journal of Guidance, Control, and Dynamics, 24, 95-

100.

[7] Bang H., Tahk M.-J., and Choi H.-D. (2003), 'Large Angle Attitude Control of Spacecraft with

Actuator Saturation', Control Engineering Practice, 11, 989-997.

[8] Subbarao K. (2004), 'Nonlinear PID-Like Controllers for Rigid-Body Attitude Stabilization', The

Journal of the Astronautical Sciences, 52, 61-74.

[9] Su J., and Cai K.-Y. (2011), 'Globally Stabilizing Proportional-Integral-Derivative Control Laws for

Rigid-Body Attitude Tracking', Journal of Guidance, Control, and Dynamics, 34, 1260-1264.

[10] Ruiter A.H.J.d. (2010), 'Adaptive Spacecraft Attitude Tracking Control with Actuator Saturation',

Journal of Guidance, Control, and Dynamics, 33, 1692-1695.

[11] Salcudean S. (1991), 'A Globally Convergent Angular Velocity Observer for Rigid Body Motion',

IEEE Transactions on Automatic Control, 36, 1493-1497.

[12] Caccavale F., and Villani L. (1999), 'Output Feedback Control for Attitude Tracking', Systems &

Control Letters, 38, 91-98.

[13] Aghannan N., and Rouchon P. (2003), 'An Intrinsic Observer for a Class of Lagrangian Systems',


[14] Maithripala D.H.S., Berg J.M., and Dayawansa W.P. (2004), 'An Intrinsic Observer for a Class of

Simple Mechanical Systems on a Lie Group', Proceedings of the 2004 American Control

Conference, Boston, Massachusetts, Jue 30-July 2, 1546-1551.

[15] Maithripala D.H.S., Berg J.M., and Dayawansa W.P. (2006), 'Almost-Global Tracking of Simple

Mechanical Systems on a General Class of Lie Groups', IEEE Transactions on Automatic

Control, 51, 216-225.

[16] Tsiotras P. (1998), 'Further Passivity Results for the Attitude Control Problem', IEEE Transactions

on Automatic Control, 43, 1597-1600.

[17] Akella M.R., Valdivia A., and Kotamraju G.R. (2005), 'Velocity-Free Attitude Controllers Subject

to Actuator Magnitude and Rate Saturations', Journal of Guidance, Control, and Dynamics, 28,

659-666.

Dow

nloa

ded

by [

Dic

le U

nive

rsity

] at

11:

38 0

8 N

ovem

ber

2014

Accep

ted M

anus

cript


[18] Tayebi A. (2008), 'Unit Quaternion-Based Output Feedback for the Attitude Tracking Problem',


[19] Abdessameud A., and Tayebi A. (2009), 'Attitude Synchronization of a Group of Spacecraft

without Velocity Measurements', IEEE Transactions on Automatic Control, 54, 2642-2648.

[20] Bertrand A., Hamel T., Piet-Lahanier H., and Mahony R. (2009), 'Attitude Tracking of Rigid

Bodies on the Special Orthogonal Group with Bounded Partial State Feedback', Joint 48th IEEE

Conference on Decision and Control and 28th Chinese Control Conference, Shanghai, P.R.

China, December 16-18, 2972-2977.

[21] Costic B.T., Dawson D.M., Queiroz M.S.d., and Kapila V. (2001), 'Quaternion-Based Adaptive

Attitude Tracking Controller without Velocity Measurements', Journal of Guidance, Control,

and Dynamics, 24, 1214-1222.

[22] Zou A.-M., Kumar K.D., and Hou Z.-G. (2010), 'Quaternion-Based Adaptive Output Feedback

Attitude Control of Spacecraft Using Chebyshev Neural Networks', IEEE Transactions on

Neural Networks, 21, 1457-1471.

[23] Subbarao K., and Akella M.R. (2004), 'Differentiator-Free Nonlinear Proportional-Integal

Controllers for Rigid-Body Attitude Stabilization', Journal of Guidance, Control, and

Dynamics, 27, 1092-1096.

[24] Astolfi A., Ortega R., and Venkatraman A. (2009), 'A Globally Exponentially Convergent

Immersion and Invariance Speed Observer for N Degrees of Freedom Mechanical Systems',

Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference,

Shanghai, P. R. China, December 16-18, 6508-6512.

[25] Astolfi A., Ortega R., and Venkatraman A. (2010), 'A Globally Exponentially Convergent

Immersion and Invariance Speed Observer for Mechanical Systems with Non-Holonomic

Constraints', Automatica, 46, 182-189.

[26] Astolfi A., Karagiannis D., and Ortega R. (2008), Nonlinear and Adaptive Control with

Applications, London, Springer.

[27] Stamnes Ø.N., Aamo O.M., and Kaasa G.-O. (2011), 'A Constructive Speed Observer Design for

General Euler-Lagrange Systems', Automatica, 47, 2233-2238.

Dow

nloa

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by [

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le U

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rsity

] at

11:

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ovem

ber

2014

Accep

ted M

anus

cript


[28] Stamnes Ø.N., Aamo O.M., and Kaasa G.-O. (2011), 'Adaptive Redesign of Nonlinear Observers',


[29] Lo S.-C., and Chen Y.-P. (1995), 'Smooth Sliding-Mode Control for Spacecraft Attitude Tracking

Maneuvers', Journal of Guidance, Dynamics and Control, 18, 1345-1349.

[30] Hu J., and Zhang H. (2012), 'Output Feedback Control of Rigid-Body Attitude with Input

Saturation', Proceedings of the 31st Chinese Control Conference, Hefei, July 24th-27th.

[31] Bhat S.P., and Bernstein D.S. (2000), 'A Topological Obstruction to Continuous Global

Stabilization of Rotational Motion and the Unwinding Phenomenon', Systems & Control

Letters, 39, 63-70.

Figure 1. Structure of the closed loop

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0 10 20 30 40 50 60 70 80-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

time(s)

quaternion

q0

q1

q2

q3

Figure 2. Quaternion versus time

0 20 40 60 80-0.6

-0.4

-0.2

0

0.2

0.4

time(s)

angular velocity (rad/s)

1

2

3

Figure 3. Angular velocity versus time

Figure 4. Estimation error of the angular velocity versus time

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Figure 5. Estimate of the disturbance versus time

0 20 40 60 80-8

-6

-4

-2

0

2

4

time(s)

control torque(Nm)

u1

u2

u3

Figure 6. Control torque versus time

0 20 40 60 801

1.005

1.01

1.015

1.02

1.025

time(s)

r

r

Figure 7. Dynamic scaling factor versus time

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Table 1 Parameters, gains and initial conditions

2diag(30,25,35)kg mJ , 0.5, 0.5,0.3 N md 8pk , 10vk , 1, 1, 1, 1r q z

0.001 , 1Md (0) 0,0.6,0.8,0T

q , (0) (0.1,0.2,0.3) rad/sT

(0) 1r , (0) 0,0,0T (0) 0,0,0

T , ˆ(0) (0)q q

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Documents

Output feedback control for rigid-body attitude with constant disturbances