ARTICLE IN PRESS

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Control Engineering Practice 14 (2006) 811–818

www.elsevier.com/locate/conengprac

Robust control of the missile attitude based on quaternion feedback

Chanho Song�, Sang-Jae Kim, Seung-Hwan Kim, H.S. Nam

Agency for Defense Development, 3-1-3, P.O. Box 35-3, Yusung, Daejon, 305-600, Republic of Korea

Received 1 January 2003; accepted 4 April 2005

Available online 15 June 2005

Abstract

In this paper, a robust control scheme based on the quaternion feedback for attitude control of missiles employing thrust vector control

is proposed. The control law consists of two parts: the nominal feedback part and an additional term for ensuring robustness to the plant

uncertainties. For the proposed control scheme, a stability analysis is given and the performance is shown via computer simulation.

r 2005 Elsevier Ltd. All rights reserved.

Keywords: Quaternion feedback; Attitude control; Uncertainty; Stability analysis

1. Introduction

In general, most attitude control schemes of tacticalmissiles are based on the Euler angle feedback concept.However, modern satellites or spacecrafts have a trendtoward using quaternion feedback instead of Euler anglefeedback (Weiss, 1993; Wie & Barba, 1985; Wie, Weiss,& Arapostathis, 1989). As described in the references,quaternion control enables the attitude change along theshortest path by matching the control torque vector tothe eigenaxis which is not possible with Euler anglecontrol because Euler angles are based on the concept ofsequential rotation. Moreover, Wie et al. (1989) showedthat the quaternion feedback control system is globallystable and near-eigenaxis rotation can be achieved evenin the presence of initial body rate and inertia matrixuncertainty.

However, similar research is hardly found in the areaof attitude control for the tactical missiles operated inthe low atmosphere. It seems due to a view that thequaternion feedback will not retain its advantage wherethe aerodynamic effects are not negligible. But Songet al. proposed a control scheme which might beprospective even in this case (Song, Nam, & Kim,

e front matter r 2005 Elsevier Ltd. All rights reserved.

nengprac.2005.04.003

ing author. Tel.: +8242 8214415; fax: +8242 8212224.

ess: ksjletter@hanmail.net (C. Song).

2000). In this paper, the results are extended to the casewhere the plant uncertainties exist.

Several control design methods for the uncertaindynamical systems are introduced in Barmish, Corless,and Leitmann (1983), Corless and Leitmann (1981) andKhalil (1996). Most of them require that the uncertainsystem satisfies the so-called ‘‘matching condition.’’ Theplant uncertainty considered here satisfies this condi-tion. Based on this plant uncertainty model and a robuststabilization method given in Khalil (1996), an attitudecontroller for a vertical launch anti-submarine rocket(VLASR) model is constructed. This controller consistsof two parts: a nominal feedback part and an additionalterm ensuring the robustness to the plant uncertainties.

This paper is organized as follows. First, missiledynamics with uncertainty and kinematics will bedescribed. After that, a nominal controller and robustcontroller will be given with a stability analysis. Finally,a design example will be given with computer simulationresults which show that the robust controller has betterperformance than the nominal one.

2. Missile dynamics with uncertainty

Missile motion is described by six degrees of freedomequations which consist of the translational and rotational

ARTICLE IN PRESSC. Song et al. / Control Engineering Practice 14 (2006) 811–818812

motion equations as follows:

m_vþmðx� vÞ � g ¼ FaðvÞ þ F tðuÞ, (1)

J _x ¼ x� JxþMaðv;xÞ þM tðuÞ, (2)

where v ¼ ½v1; v2; v3�T is the linear velocity vector, x ¼

½o1;o2;o3�T is the angular velocity vector, u ¼

½dr; dp; dy�T is the control input vector, m is the mass of

a missile, J is a inertia matrix, g is gravity, Fa and Ma

are aerodynamic force and moment vectors, respec-tively, and F t and M t are control force and controlmoment vectors, respectively.

Now, in order to simplify Eqs. (1) and (2), thefollowing assumptions are used:

(A1)

Velocity and altitude of the missile are constant. (A2) Gravity is neglected. (A3) v2 and v3 are much smaller than v1. (A4) Missile body has the symmetrical cruciform. (A5) F tðuÞ and M tðuÞ are linear in u and invertible.

Under A1–A4, the translational motion equation (1) canbe written as (Hemsch & Nielsen, 1986):

_a ¼ o2 þ ZoðQ; n; aÞ � Ztdp,

_b ¼ �o3 þ Y oðQ; n;bÞ þ Ztdy, ð3Þ

where a is the angle of attack, b is the sideslip angle, Q isthe dynamic pressure, and n is Mach number. ZoðQ; n; aÞand Y oðQ; n;bÞ are aerodynamic coefficients, and Zt isthe control thrust coefficient. They are given by

ZoðQ; n; aÞ ¼QS

mv1Czðn; aÞ,

Y oðQ; n;bÞ ¼QS

mv1Cyðn;bÞ,

Zt ¼Tc

mv1, ð4Þ

respectively, where Czðn; aÞ and Cyðn;bÞ are nondimen-sional aerodynamic coefficients, S is the reference area,and Tc is the magnitude of the control thrust. Eq. (3)can be rewritten in a compact form:

_z ¼ hðz;xÞ þ F0 þ Eu, (5)

where

z ¼a

b

" #; hðz;xÞ ¼

o2

�o3

" #; F0 ¼

Z0

Y 0

" #,

E ¼

0 0 0

0 �Zt 0

0 0 Zt

2664

3775.

For the missile with cruciform configuration, Ma inEq. (2) can be described as

Ma ¼

Lo

Mo

No

2664

3775þ

Lpo1

Mqo2

Mqo3

2664

3775 ¼

QSDClðn; aÞ sin 4g

QSDCmðn; aÞ

QSDCmðn;bÞ

2664

3775

þQSD2

2v1

ClpðnÞo1

CmqðnÞo2

CmqðnÞo3

2664

3775 ð6Þ

where D is the reference length, Cl , Cm, Clp, and Cmq arethe nondimensional moment coefficients, and g is thebank angle defined by

g ¼v2

v3¼

ba.

Moreover, without loss of generality we assume

J ¼

J1 0 0

0 J2 0

0 0 J2

264

375

for some constants J1, J2. The control moment M t isgiven by

M t ¼

Ltdr

Mtdp

Mtdy

264

375 ¼

Tclydr

Tclxdp

Tclxdy

264

375, (7)

where Tc is the magnitude of the control thrust, lx and ly

are the moment arms defined by the distance from thecenter of gravity to the location of the control thrustvector, and dr, dp, and dy are control inputs.

Now, consider the uncertainty terms by DF0, DE, DJ,DMa, and DB corresponding to F0, E, J, Ma, and B,respectively. Then, the motion Eqs. (2) and (5) can bemodified as

ðJ þ DJÞ _x ¼ XðJ þ DJÞxþMa þ DMa þ ðB þ DBÞu,

(8)

_z ¼ hðz;xÞ þ F0 þ DF0 þ ðE þ DEÞu (9)

with the skew symmetric matrix X and the input matrixB defined by

X ¼

0 o3 �o2

�o3 0 o1

o2 �o1 0

264

375; B ¼

Lt 0 0

0 Mt 0

0 0 Mt

264

375.

From the matrix inversion lemma

ðJ þ DJÞ�1 ¼ J�1 � DX (10)

with DX defined by

DX ¼ J�1 DJðI þ J�1 DJÞ�1J�1.

Eqs. (8) and (10) lead to

_x ¼ f ðx; tÞ þ DH þ ðG þ DGÞu, (11)

ARTICLE IN PRESSC. Song et al. / Control Engineering Practice 14 (2006) 811–818 813

where

f ðx; tÞ ¼ J�1XJx,

G ¼ J�1B, (12)

and

DH ¼ ðJ�1 � DXÞðMa þ DMaÞ

� DX XJxþ ðJ�1 � DXÞXDJ x,

DG ¼ J�1 DB � DXðB þ DBÞ. ð13Þ

Since G is an invertible square matrix, DH and DG canbe written as

DH ¼ GDh,

DG ¼ GDg

with some matrices Dh and Dg.Then, Eq. (11) can be rewritten as

_x ¼ f ðx; tÞ þ GðDh þ ðI þ DgÞuÞ. (14)

Now, impose the following assumption on Dg:

ðA6Þ kDgko1.

With assumption A6, ðI þ DgÞ in Eq. (14) is invertible,which implies that the uncertainty terms satisfy thematching condition because all the uncertainties can beexactly cancelled by taking u ¼ �ðI þ DgÞ

�1Dh. Definethe control input u by

u ¼ wn þ m, (15)

where wn is the nominal one and m is an additional termfor guaranteeing robustness to the plant uncertainties.Inserting Eq. (15) into Eq. (14) gives

_x ¼ f ðx; tÞ þ Gwn þ Gðdþ mÞ, (16)

where

d ¼ Dh þ Dgwn þ Dgm (17)

with Dh and Dg satisfying

Dh ¼ G�1 DH ,

Dg ¼ G�1 DG . (18)

3. Kinematics

Euler’s rotational theorem states that the attitude of arigid body can change from one orientation to anotherby rotating the body about an axis called Euler axis oreigenaxis. The definition of quaternion comes from suchan eigenaxis rotation. The vector part of the quaternionindicates the direction of the eigenaxis and the scalarpart indicates the rotation angle about the eigenaxis, i.e.,a quaternion is defined by

qi ¼ ei sinj2

� �; i ¼ 1; 2; 3,

q4 ¼ cosj2

� �,

where j is the magnitude of the eigenaxis rotation angleand a vector ðe1; e2; e3Þ is the direction cosine of theeigenaxis with respect to the reference frame. Thekinematic equation in terms of quaternion is describedby

_v ¼ 12Xvþ 1

2q4x,

_q4 ¼ �12xTv, (19)

where v ¼ ½q1 q2 q3�T.

Let the commanded attitude be denoted by aquaternion ½q1c; q2c; q3c; q4c�. Then the error quaternion½q1e; q2e; q3e; q4e� between the current attitude and thecommanded one is given by

q1e

q2e

q3e

q4e

266664

377775 ¼

q4c q3c �q2c �q1c

�q3c q4c q1c �q2c

q2c �q1c q4c �q3c

q1c q2c q3c q4c

266664

377775

q1

q2

q3

q4

266664

377775.

4. Nominal controller

Since the kinematic differential equation of the errorquaternion has the same form as Eq. (19), from Eqs.(16) and (19), we get

_ve ¼12Xve þ

12

q4ex,

_q4e ¼ �12xTve,

_x ¼ f ðx; tÞ þ Gwn þ Gðdþ mÞ, ð20Þ

where vTe ¼ ½q1e; q2e; q3e� is the vector part of the errorquaternion. Let us define x to be

xT ¼ ½q1e; q2e; q3e; q4e;o1;o2;o3�.

Then, Eq. (20) can be rewritten as

_x ¼ f aðx; tÞ þ Gawn þ Gaðdþ mÞ, (21)

where

f aðx; tÞ ¼

12Xve þ

12

q4ex

�12xTve

f ðx; tÞ

264

375; Ga ¼

04�3

G3�3

� �.

Let the nominal controller be in the form

wn ¼ �Kpve � Kdx, (22)

where

Kp ¼

Kp1 0 0

0 Kp2 0

0 0 Kp3

264

375; Kd ¼

Kd1 0 0

0 Kd2 0

0 0 Kd3

264

375.

ARTICLE IN PRESSC. Song et al. / Control Engineering Practice 14 (2006) 811–818814

For the nominal system where there is no plantuncertainity, i.e. d and m are zero in Eq. (20), theequilibrium states are

o1 ¼ o2 ¼ o3 ¼ 0; q1e ¼ q2e ¼ q3e ¼ 0; q4e ¼ 1.

Now, consider the following Lyapunov function candi-date V :

V ¼ 12xTK�1p G�1xþ vTe ve þ ð1� q4eÞ

2 (23)

where Kp is a matrix to be determined so that GKp ispositive definite. Some manipulations give

_V ¼ xTK�1p B�1XJx� xTK�1p Kdx.

Select Kp so that it satisfies

ðBKpÞ�1¼ c1J þ c2I (24)

for some constants c1; c2. Then, using the facts thatXx ¼ 0 and xTXx ¼ 0 for any vector x which comefrom the properties of the skew symmetric matrix X

(Wie et al., 1989), we get

_V ¼ �xTK�1p Kdx. (25)

V is a decrescent and radially unbounded function. So,if Kd is chosen so that K�1p Kd is positive definite, then_Vp0. Therefore, the equilibrium state is globallyuniformly stable.

5. Robust controller

In this section, an additional term m is designedthrough the process in Khalil (1996). In the case that theplant uncertainities exist, i.e. d is not zero in Eq. (20), theLyapunov derivative becomes

_V u ¼ _V þ lTðdþ mÞ, (26)

where

lT ¼qV

qxG . (27)

If we choose m so that

lTðdþ mÞp0, (28)

then, since _Vp0, _Vup0. Therefore, the closed-loopsystem is globally uniformly stable. From Eq. (17),

kdðx; t;wn þ mÞkprþ kDgk kmk, (29)

where

r ¼ kDh þ Dgwnk. (30)

From inequalities (28) and (29),

lTðdþ mÞplTm þ klk kdk

plTm þ klkðrþ kDgk kmkÞ. ð31Þ

Now, take the following assumption:

(A7)

A function Z with ZXr and kDgkmax withkDgkmaxXkDgk are known, and kDgkmaxo1.

Under assumption A7, select m by

m ¼ �Z

1� kDgkmax

l

klk. (32)

Then, (31) becomes

lTðdþ mÞ

p�Z

1� kDgkmax

klk þ rklk þkDgkZ

1� kDgkmax

klk

¼ �Z1

1� kDgkmax

�kDgk

1� kDgkmax

� �klk þ rklk

pðr� ZÞklkp0,

which satisfies the condition (28). But this control law isnot defined at klk ¼ 0. Practically, such discontinuitycauses chattering phenomena. A method to avoid thistrouble is to modify the control law as follows:

m ¼

�Z

1� kDgkmax

l

klk; if ZklkX�;

�Z2

1� kDgkmax

l

�; if Zklko�:

8>>><>>>:

(33)

Now, consider the following structure of the uncertaintyterms in Dg and Dh:

DJ ¼ KJJ ,

DMa ¼ KMMa,

DB ¼ KBB, ð34Þ

where

KJ ¼

k1 0 0

0 k2 0

0 0 k2

2664

3775; KM ¼

k3 0 0

0 k4 0

0 0 k4

2664

3775,

KB ¼

k5 0 0

0 k6 0

0 0 k6

2664

3775

with unknown constants ki, i ¼ 1; . . . ; 6. SubstitutingEq. (34) into Eq. (13) and calculating Dg and Dh fromEq. (18), we get

ARTICLE IN PRESS

Dh ¼

1

Ltð1þ k1Þ½Lo þ Lpo1 þ k3ðLo þ Lpo1Þ þ ðk2 � k1ÞðJ2 � J3Þo2o3�

1

Mtð1þ k2Þ½Mo þMqo2 þ k4ðMo þMqo2Þ þ ðk2 � k1ÞJ1o1o3�

1

Mtð1þ k2Þ½No þMqo3 þ k4ðNo þMqo3Þ þ ðk1 � k2ÞJ1o1o2�

266666664

377777775, (35)

C. Song et al. / Control Engineering Practice 14 (2006) 811–818 815

Dg ¼

k5 � k1

1þ k10 0

0k6 � k2

1þ k20

0 0k6 � k2

1þ k2

266666664

377777775. (36)

kDgkmax and Z are to be determined so thatkDgkmaxXkDgk and ZXkDh þ Dgwnk.

6. Design example

To confirm the performance of the proposed robustcontroller, the VLASR model was taken. The VLASRhas jet vanes for thrust vector control (TVC) whichenables pushover maneuver aligning the missile velocityvector to the desired ballistic flyout vector. During thisinitial pushover maneuver, the pitch attitude is de-creased from the initial 90� to 50�. And when theVLASR achieves the required velocity at the demandedrange, the thrust is cut off. The details related with

Kp

QuaternionComputation

Error QuaternionComputation

QuaternionComputation

φc, θc, ψc

Fig. 1. Block diagram of th

Table 1

Simulation cases

Case True values of parameter ki Co

1 All zero No

2 k1 ¼ 0:1, k2 ¼ �0:1, k3 ¼ 1:0, k4 ¼ �1:0, k5 ¼ 0:4, k6 ¼ �0:4 No

3 k1 ¼ 0:1, k2 ¼ �0:1, k3 ¼ 1:0, k4 ¼ �1:0, k5 ¼ 0:4, k6 ¼ �0:4 Ro

4 All zero Ro

VLASR dynamics are described in Dunne, Black,Schmidt, and Lewis (1990). For this purpose, an attitudecontroller was designed based on the proposed controlscheme. For the actuator dynamics, the followingsecond-order system was used:

GactðsÞ ¼o2

n

s2 þ 2zonsþ o2n

; on ¼ 15Hz; z ¼ 0:53.

The limit of jet vanes is �20� and control commandsshould be restricted to this range.

Fig. 1 shows the block diagram of the nominalcontroller. It was designed to have the fast time-domainresponse. The result was

Kp ¼

6:5621 0 0

0 6:3662 0

0 0 6:3662

2664

3775,

Kd ¼

0:1959 0 0

0 0:1670 0

0 0 0:1670

2664

3775.

Kd

InertialMeasurement

Unit

ActuatorFlight

Dynamics+

-

ω1, ω2, ω3

φ, θ, ψ

e nominal controller.

ntroller used Remark

minal Nominal case without uncertainties

minal Uncertainties considered, but nominal controller used

bust Uncertainties considered and robust controller used

bust Robust controller used without uncertainties

ARTICLE IN PRESSC. Song et al. / Control Engineering Practice 14 (2006) 811–818816

The uncertainties considered here were

�0:15pk1p0:15; �0:15pk2p0:15,

�1:0pk3p1:0; �1:0pk4p1:0,

�0:15pk5p0:15; �0:15pk6p0:15.

0.0 0.5 1.0 1.5 2.0-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

η an

d ρ

ηρ

Time (s)(a) (b

Fig. 2. Z with r and kDgkma

0.0 0.2 0.4 0.6 0.8 1.0-20

-10

0

10

20

30

40

Rol

l Atti

tude

Ang

le (

deg)

Case 1

Case 2

Case 3

Case 4

Time (s)

0.0 0.2 0.4-10

0

10

20

30

Yaw

Atti

tude

Ang

le (

deg)

Tim

(a) (b

(c)

Fig. 3. Time-domain responses: (a) roll angle response, (b

True values of parameter ki; i ¼ 1; . . . ; 6, are not knownbut within these bounds. For this type of uncertainties,upper bounds of r and kDgk can be easily found so thattheir magnitude becomes as small as possible. kDgkmax

was chosen as

kDgkmax ¼ 0:611.

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

||∆g|

| max

and

||∆ g

||

||∆g||max

||∆g||

Time (s))

x with kDgk in Case 3.

0.0 0.5 1.0 1.5 2.020

30

40

50

60

70

80

90

100

Case 1

Case 2

Case 3

Case 4

Pitc

h A

ttitu

de A

ngle

(de

g)

Time (s)

0.6 0.8 1.0

Case 1

Case 2

Case 3

Case 4

e (s)

)

) pitch angle response and (c) Yaw angle response.

ARTICLE IN PRESS

0.0 0.5 1.0 1.5 2.0-30

-20

-10

0

10

20

30

Case 1

Case 2

Case 3

Case 4

Case 1

Case 2

Case 3

Case 4

Case 1

Case 2

Case 3

Case 4

Rol

l Con

trol

Com

man

d (d

eg)

Time (s)

Pitc

h C

ontr

ol C

omm

and

(deg

)

Yaw

Con

trol

Com

man

d (d

eg)

0.0 0.5 1.0 1.5 2.0Time (s)

0.0 0.5 1.0 1.5 2.0Time (s)

-30

-20

-10

0

10

20

30

-30

-20

-10

0

10

20

30

(a) (b)

(c)

Fig. 4. Control commands: (a) roll control command, (b) pitch control command and (c) Yaw control command.

C. Song et al. / Control Engineering Practice 14 (2006) 811–818 817

Z was calculated at each time in the following way. First,values of kDh þ Dgwnk were calculated at the cornerpoints of the hypercube

Q6i¼1 ½kimin kimax� where kimin

and ki max are minimum and maximum values of ki.Next, the maximum value was taken among these valuesand assigned to Z. Other design parameter � in Eq. (33)was taken to be

� ¼ 0:005.

Simulations for four cases given in Table 1 were carriedout. In these simulations, desired and initial attitudes ofroll, pitch, and yaw were assumed to be ½0�; 50�; 0�� and½30�; 90�; 20��. Fig. 2 shows Z and kDgkmax which satisfyZXr and kDgkmaxXkDgk all the time. The time-domainresponses are given in Figs. 3 and 4. Fig. 3 shows thatwhen the plant uncertainties were given and the nominalcontroller was used (Case 2) the responses of pitch, yawand roll became worse (larger settling time and moreovershoot), but settling time and overshoot wereconsiderably reduced when the robust controller wasused (Case 3). Note that responses with the robustcontroller have smaller overshoot than those with thenominal controller. It is due to the additional term m

which acts like a damper. It is also notable that theresponse with the robust controller shows good perfor-mance even in the case of no uncertainties (Case 4).

7. Conclusion

In this paper, a robust quaternion feedback controlscheme which can be used efficiently in the attitudecontrol of the tactical missiles employing thrust vectorcontrol was proposed. It was shown that the plantuncertainty can be modeled so that the matchingcondition is satisfied. Furthermore, it was shown thateven in the presence of uncertainties the proposedcontrol law makes the closed-loop system globallyuniformly stable. Finally, it was shown via computersimulations that the proposed control scheme providesbetter performance than the nominal controller. In thisstudy, a simple structure of uncertainties were consid-ered so that Z and kDgkmax used in the robust controllercan be easily found, but more general form ofuncertainties will need more complex optimization

ARTICLE IN PRESSC. Song et al. / Control Engineering Practice 14 (2006) 811–818818

techniques to find kDgkmax and Z as small as possible.We leave it to further study.

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