TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.1, Jan
uary 20
14
, pp. 468 ~ 4
7
6
DOI: http://dx.doi.org/10.11591/telkomni
ka.v12i1.3477
468
Re
cei
v
ed
Jun
e
15, 2013; Revi
sed
Jul
y
2
7
, 2013; Acce
pted Augu
st 7, 2013
Nonlinear Robust Control for Spacecraft Attitude
Lina Wan
g
*, Zhi Li
Dep
a
rtment of Commun
i
cati
o
n
Engi
ne
erin
g, Schoo
l of Com
puter an
d Com
m
unic
a
tion En
gin
eeri
ng,
Univers
i
t
y
of Scienc
e an
d T
e
chno
log
y
B
e
ij
ing
No.30
Xu
e
y
u
a
n
Roa
d
, Hai
d
ia
n District, Beiji
ng, Ph./F
ax: +
86 10 62
33
28
73
*Corres
p
o
ndi
n
g
author, e-ma
i
l
:
w
l
n_
ustb@1
26.com*
A
b
st
r
a
ct
T
h
is pap
er pro
pose
d
a no
nli
near ro
bust co
ntrol for space
c
raft attitude b
a
sed o
n
pass
i
vity an
d
disturb
ance
su
ppress
i
on
vect
or. T
he sp
ace
c
raft mo
del
w
a
s descri
b
e
d
us
ing
qu
aterni
on.
T
he co
ntrol l
a
w
introd
uced
the
suppr
essio
n
v
e
ctor of exter
n
a
l
dist
ur
banc
es
and
ha
d
no
inf
o
rmatio
n
re
late
d to th
e syste
m
para
m
eters. T
he des
ire
d
perf
o
rmanc
e of sp
acecraft atti
tud
e
control co
ul
d
be achi
eve
d
u
s
ing the
desi
g
n
ed
control
law
.
An
d stabi
lity co
nd
itions
of the
no
nlin
ear r
o
b
u
st control for
spa
c
ecraft attitude
w
e
re giv
en. T
h
e
stability
cou
l
d
be
prove
d
by a
pplyi
ng
Ly
apu
n
o
v a
ppro
a
ch.
T
he v
e
rificati
on
of the
pro
pose
d
attitud
e
c
ontr
o
l
meth
od
w
a
s p
e
rformed
thro
u
gh
a s
e
ries
of
simulati
ons. T
h
e n
u
m
eric
al
re
sults sh
ow
ed t
he
effectiven
es
s o
f
the pro
pos
ed c
ontrol
metho
d
i
n
contro
lli
ng th
e spac
ecra
ft at
titude i
n
the
pr
esenc
e of
exte
rnal
disturb
anc
es.
T
he main
be
n
e
fit of the pro
p
o
sed
attitude c
ontrol
met
hod
does
not ne
ed
ang
ular v
e
loc
i
ty meas
ure
m
e
n
t
and h
a
s its rob
u
stness ag
ai
ns
t mod
e
l unc
ertainti
es an
d externa
l
disturb
a
n
c
es.
Ke
y
w
ords
: sp
acecraft attitud
e
, external d
i
sturba
nc
e, rob
u
s
t
control, quate
r
nio
n
, stability
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
Attitude control is a
parti
cu
larly impo
rtan
t comp
onent f
o
r
spa
c
e
c
rafts. A sp
acecra
ft must
maintain a
certain attitud
e
while i
n
orbit. No
waday
s, attitude co
ntrol of spa
c
ecrafts de
ma
nd
better p
e
rfo
r
mance. The
spa
c
e
c
raft attitude ca
n
be exp
r
e
s
se
d by matrix,
Euler
angle
,
or
quaternion. T
he method of
matrix
repre
s
entatio
n is complicated in
calculation; Euler an
gle a
l
so
exist some li
mitations. Fo
r example, the rotati
on
matrix is not interch
ang
e
able, Euler a
ngle
rotation mu
st
be in a part
i
cula
r order,
and eq
uivale
nt to Euler angle chan
ge may not cau
s
e
equal rotatio
n
, which lea
d
s to a rotati
ng uneve
nne
ss.
When Eu
ler angl
e is e
qual to
2
/
,
there will be a
si
ngula
r
poi
nt,
leadin
g
to
the
lo
ss
of d
egre
e
s
of
fre
edom, whi
c
h is calle
d
a
s
t
h
e
phen
omen
on
of gimbal lock. But expressin
g
3D rotation with quaternion
can avoid the
s
e
limitations, a
nd also ha
s
clea
r ge
omet
ric me
anin
g
and si
mple
calcul
ation. In the past
sev
e
ral
decade
s, re
searche
r
s hav
e devoted
to
the pro
b
lem
of spa
c
e
c
raft attitude stabil
i
zation b
a
sed
o
n
quaternion re
pre
s
entatio
n. Some control met
hod
s have been de
veloped to treat this probl
em,
su
ch a
s
rob
u
st co
ntrol a
ppro
a
ch [1, 2], Lyapunov
-ba
s
ed a
p
p
r
oach [3-5], adaptive con
t
rol
approa
ch [6-9], variable st
ructu
r
e
control approa
ch [10-1
4
].
In gen
eral,
angul
ar vel
o
city and
qu
a
t
ernion,
are
use
d
to
deal
with th
e
stability of
feedba
ck con
t
rol. Ho
weve
r, the angul
ar
velocity
mea
s
urem
ent is
n
o
t necessa
ry
in so
me of th
e
previou
s
works. Fo
r exam
ple, in [10], a desi
gn
crit
erion fo
r a cl
ass of pro
p
o
r
tional
-de
r
ivative
(PD)
cont
rolle
rs was firstly prop
osed by us
in
g the Lyapunov-ba
s
ed
approa
ch, an
d then a desi
gn
crite
r
ion
of controlle
r
with
out ang
ular v
e
locity
mea
s
urem
ent was pre
s
ente
d
b
a
se
d on
pa
ssivity.
The
app
roa
c
h p
r
opo
se
d i
n
[10]
wa
s fu
rther
extende
d to the
syste
m
de
scrib
ed
by the
Rod
r
ig
ues
and modifie
d
Rod
r
igu
e
s p
a
r
amete
r
s [11]
.
Ho
wever, th
e
extern
al di
st
urba
nces,
wh
ich i
nevitably
affect the
mot
i
on of th
e
spa
c
e
c
raft
in its
attitude
, are
igno
re
d
in the
ab
ove-me
nti
one
d literatures.
In
this pap
er, we
fo
cu
s on the
stability of
spacecraft atti
tude in
the
pre
s
en
ce
of
the
boun
de
d external
di
sturb
a
n
c
e
s
a
n
d
prop
ose a
n
online
a
r
rob
u
s
t control me
thod for sp
a
c
ecraft attitud
e
. The
space
c
raft
attitude
is
rep
r
e
s
ente
d
by quate
r
nio
n
. The
sup
p
re
ssi
on ve
ct
or of external di
sturba
nce
s
, which
is
indep
ende
nt
of ang
ula
r
ve
locity si
ze,
is
introdu
ce
d
int
o
the
control
law. In
ad
dtition, the
control
law h
a
s
no in
formation
rel
a
ted to the
system pa
ram
e
ters so that
the ro
bu
stne
ss is
gua
ra
nte
ed.
To demo
n
st
rate the perf
o
rma
n
ce of the pro
p
o
s
e
d
attitude co
ntrol metho
d
in supp
re
ssing
disturban
ce
s
and maintai
n
i
ng stabilit
y, the nume
r
ical simulation
s are carrie
d out usin
g MATLA
B
.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 2302-4
046
TELKOM
NIKA
TELKOM
NIKA
Vol. 12, No
. 1, Janua
ry 2014: 468 – 4
7
6
469
2. Proposed
At
titud
e
Con
t
rol Meth
od
2.1. Spacecr
aft Mo
del
The motion
of spa
c
e
c
raf
t
attitude can be de
scri
bed by kin
e
m
atic and
d
y
namic
equatio
ns.
We u
s
e the u
n
it quaterni
on
to repre
s
ent
spa
c
e
c
raft attitude in orde
r to avoid singularity.
Define the u
n
i
t quaternio
n
as in Eq. (1
).
)
2
/
cos(
ˆ
)
2
/
sin(
0
n
q
q
q
(1)
whe
r
e
3
ˆ
R
n
is the
rotation
axis rep
r
e
s
ente
d
by unit vecto
r
,
is the
rota
tion ang
ular,
3
R
q
and
R
q
0
are the compon
ents of
the unit quaternio
n
, whi
c
h
subje
c
t to the followin
g
co
nstrai
nt:
1
2
0
q
q
q
T
(2)
The kin
e
mati
c equ
ation re
pre
s
ente
d
by the unit quate
r
nion i
s
given
by Eq. (3).
T
0
0
2
1
)
(
2
1
)
(
2
1
q
q
q
I
q
q
E
q
(3)
whe
r
e
T
3
2
1
is the
spa
c
e
c
raft angul
ar velo
city vector
with re
sp
ect
to the inert
i
al
referen
c
e fra
m
e, expre
ssed in the
sp
ace
c
raft bod
y-fixed refe
re
nce f
r
ame,
I
is the
3×3
unit
matrix,
q
is the ske
w symm
e
t
ric matrix which is d
e
fined
by Eq. (4).
0
0
0
1
2
1
3
2
3
q
q
q
q
q
q
q
(4)
The dyn
a
mi
c model
of th
e spa
c
e
c
raft
attitude
con
t
rol sy
stem i
s
d
e
scri
bed
by the
differential eq
uation, as in
Eq. (5).
d
u
J
J
(5)
whe
r
e
3
3
T
R
J
J
is the
inertia matrix whi
c
h is a
sy
mmetric a
nd positive defin
e matrix,
3
R
u
is the ve
ctor of control to
rque,
T
3
2
1
d
d
d
d
is the
vector of ex
ternal di
stu
r
b
ance which i
s
boun
ded a
s
i
i
d
|
|
, where
i
is a positive co
nsta
nt, for
i
=1, 2, 3.
2.2. Passiv
i
ty
and Distur
bance S
upp
r
ession Base
d Attitude
Control
First, we
co
nsid
er attitud
e
control wit
h
angula
r
ve
locity measurment. The n
online
a
r
control law i
s
given in Eq. (6).
q
E
k
k
u
T
2
1
(6)
whe
r
e
k
1
a
nd
k
2
are p
o
sitiv
e
con
s
tant
s.
Con
s
id
er the
Lyapun
ov function candi
da
te
q
q
k
J
V
T
2
T
2
1
(7)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 2302-4
046
Nonli
nea
r Ro
bust Control f
o
r Spa
c
e
c
raft
Attitude (Lina Wan
g
)
470
Usi
ng Eqs. (3
), (5) a
nd (6
), the time deriv
ate of
V
can be com
puted
to
d
k
q
E
k
d
u
J
E
q
k
d
u
J
q
q
k
J
V
T
T
1
T
2
T
T
2
T
T
2
T
)
(
)
(
2
(8)
Whe
n
d
=
0
, Eq
. (
8
)
c
a
n be s
i
mp
lifie
d
a
s
0
T
1
k
V
. Since the
Lyapun
ov fu
nction
can
d
idate
V
is po
sitive definite and radially unbo
u
nded.
By LaSalle invaria
n
ce p
r
in
ciple,
all
trajecto
rie
s
converg
e
to th
e larg
est inva
riant set
}
0
:
)
,
{(
}
0
:
)
,
{(
q
V
q
, whi
c
h impli
e
s
that
0
. Sinc
e
0
, then
0
q
,
0
0
q
from
Eq.(3). From
Eq. (5
),
we have that
u
J
J
(
d
=0
) o
r
0
J
J
u
. From Eq.
(6),
we obtai
n that
0
1
T
2
k
u
q
E
k
.
So
0
q
. The la
rg
est inva
riant
set is
}
0
,
0
:
)
,
{(
q
q
, which
co
rre
sp
ond
s to the
stable
equilibrium.
Whe
n
d
≠
0,
we will present a controller for the system.
First of all, we introdu
ce a
resu
lt about Input-to
-
State Stability [15].
Lemma 1
Let
R
R
V
n
,
0
:
be a conti
nuou
sly differentiable fun
c
t
i
on that sati
sfies
the followin
g
prop
ertie
s
:
)
(
)
,
(
)
(
2
1
x
x
t
V
x
(9)
)
(
)
,
,
(
3
x
W
u
x
t
f
x
V
t
V
,
0
)
(
u
x
(10
)
m
n
R
R
u
x
t
,
0
)
,
,
(
, where
1
an
d
2
are
class
func
tions
,
is a class
functio
n
,
and
)
(
3
x
W
is a con
t
inuou
s po
sitive definite function o
n
n
R
. Then, the syst
em is input-t
o-state
stable with
2
1
1
.
No
w Eq. (8)
can be re
written as follo
ws
)
(
)
(
)
(
)
(
3
1
T
1
3
1
2
3
1
T
1
T
T
T
1
i
i
i
i
i
i
i
i
i
d
k
d
k
d
k
V
(11
)
Therefore, when
sat
i
sf
ie
s
/
i
i
d
for
i
=
1
,2,
3
, we have
that
T
1
)
(
k
V
whe
r
e
1
0
k
. By
Lemma 1, the prop
osed controlle
r ca
n make the
clo
s
ed
-loo
p system
achi
eve input
-to-
state stabl
e.
Now we will
provide an improved controller. In
order to suppress the effect of external
disturban
ce
s,
we i
n
trod
uce the
sup
p
re
ssi
on ve
ctor
v
of external
d
i
sturb
a
n
c
e
s
i
n
to the
control
law. Let
q
E
k
k
u
T
2
1
, where
T
3
2
1
]
[
v
v
v
v
,
)
sgn(
i
i
i
v
for
i
=1, 2, 3. The
symbolic
function
)
sgn(
x
is defined by
0
1
0
0
0
1
)
sgn(
x
x
x
x
(12
)
Then the time
derivative of
Lyapun
ov function candi
da
te
V
can be computed to
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 2302-4
046
TELKOM
NIKA
TELKOM
NIKA
Vol. 12, No
. 1, Janua
ry 2014: 468 – 4
7
6
471
i
i
i
i
i
i
i
d
k
d
k
d
k
q
E
k
d
u
J
V
3
1
3
1
T
1
T
3
1
T
1
1
T
T
2
T
)
(
)
(
(13
)
Therefore, when
sat
i
sf
ie
s
/
i
i
d
for
i
=
1
,2,3,
i
i
i
V
3
1
. Acco
rdin
g to Lemm
a
1, we
kno
w
t
hat the p
r
op
o
s
ed
attitude controlle
r
can
make
the
clo
s
ed
-loo
p sy
st
em inp
u
t-to-st
a
te
stable.
Secon
d
, we consi
der nonli
near attitude
contro
l
witho
u
t angul
ar vel
o
city mea
s
u
r
ment. In
[10], a co
ntroller
witho
u
t angul
ar vel
o
city me
a
s
u
r
e
m
ent was propo
sed
usi
n
g pa
ssivity-base
d
approa
ch. Along the line o
f
[10], we con
s
tru
c
t a co
ntroller a
s
follows.
q
E
k
y
E
k
u
x
P
B
Bq
Ax
P
B
y
Bq
Ax
x
T
2
T
1
T
T
)
(
(14
)
whe
r
e
B
is
a full rank mat
r
ix. There
ex
is
t pos
i
tive definite matrices
P
and
Q
which can ma
ke
matrix
A
satisfy the following Lyapun
ov equation.
Q
PA
P
A
T
(15
)
It can be se
en that the distur
ban
ce sup
p
re
ssion vector
v
is rel
a
ted to the angul
ar
velocity in th
e co
ntrol l
a
w. The di
sturb
ance
supp
re
ssion ve
cto
r
can be
dete
r
mined
only if the
dire
ction of the angul
ar vel
o
city is kn
own. While t
he size of the ang
ular velo
city is not ne
ce
ssary.
Con
s
id
er the
followin
g
Lyapunov fun
c
tio
n
can
d
idate
)
(
)
(
2
1
T
1
T
2
T
Bq
Ax
P
Bq
Ax
k
q
q
k
J
V
(16
)
Usi
ng Eqs. (3
), (5), (1
0) a
n
d
(14
)
, the time derivate o
f
V
is can be
comp
uted to
d
x
Q
x
k
q
B
x
A
P
x
k
x
P
q
B
x
A
k
q
q
k
J
V
i
i
i
T
3
1
T
1
T
1
T
1
T
2
T
)
(
)
(
2
(17
)
Whe
n
d
=0, Eq. (17) ca
n be simplifi
ed as
0
3
1
T
1
i
i
i
x
Q
x
k
V
. Sin
c
e the
Lyapun
ov function
ca
ndi
date
V
is p
o
sitive defini
t
e and radia
lly unboun
de
d. By LaSalle
invarian
ce
prin
ciple, a
ll trajecto
ri
es conve
r
g
e
to the large
s
t invariant
set
}
0
,
0
:
)
,
,
{(
}
0
:
)
,
,
{(
x
x
q
V
x
q
whi
c
h im
plie
s that
0
. Since
0
, then
0
q
,
0
0
q
from Eq.(3). It is
easy to have
x
P
B
y
T
. At the sa
me time, we have that
u
J
J
(
d
=0
) or
0
J
J
u
from Eq. (5).
From Eq. (14), we h
a
ve that
0
T
1
T
2
v
y
E
k
u
q
E
k
. So
0
q
. Consequ
ently, the large
s
t invariant
set is
}
0
,
0
:
)
,
{(
q
q
, which
corresponds to
the stable equilibrium.
Whe
n
d
≠
0, E
q
. (17)
can b
e
rewritten a
s
follows.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 2302-4
046
Nonli
nea
r Ro
bust Control f
o
r Spa
c
e
c
raft
Attitude (Lina Wan
g
)
472
)
(
3
1
3
1
2
3
1
T
min
1
3
1
3
1
3
1
2
2
3
1
T
min
1
3
1
3
1
T
1
T
3
1
T
1
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
d
x
x
k
d
x
x
k
d
x
Q
x
k
d
x
Q
x
k
V
(18
)
Therefore, when
sat
i
sf
ies
i
i
d
and
min
1
k
x
i
for
i
=1,
2
,3, we
have
that
0
V
,
whe
r
e
min
is the minimum
eigenvalu
e
o
f
Q
. Accordi
ng to Lemm
a
1, we
kno
w
that the
prop
osed
co
ntrol m
e
thod
can
ma
ke
th
e cl
osed-l
oop
syste
m
d
e
scribed
by Eq
s.
(3),
(5
) a
nd
(14)
input-to
-
state stable.
3. Rese
arch
Metho
d
In orde
r to d
e
mon
s
trate a
nd verify the
effectiveness and
rob
u
st
nes
s of the
prop
osed
attitude control method
for spacecraft, several numeri
cal
simul
a
tions are carri
ed
out
using
MATLAB.
A s
p
ac
ec
raft
with the following inertia matrix
)
m
(kg
10
0
0
0
20
0
0
0
15
2
J
is consi
dered.
The oth
e
r
m
a
in pa
ram
e
te
rs
are
k
1
=8,
k
2
=4,
-
A
=
P
=
B
=I. And the initial states are
8018
.
0
2673
.
0
5345
.
0
ˆ
n
,
6
/
11
,
0
)
0
(
. Then
the
corre
s
p
ondin
g
qu
aterni
on
rep
r
e
s
entatio
ns
are
2075
.
0
0692
.
0
1383
.
0
)
0
(
q
,
9659
.
0
)
0
(
0
q
. At this time, yaw angle, roll angle a
nd pitch an
gl
e are 23.56
º, 17.21º and
4.98º, re
spe
c
tively.
Con
s
id
er the
followin
g
four cases.
Ca
se 1: Ch
oose the e
x
ternal distu
r
ban
ce
m)
(N
)
4
.
0
sin(
014
.
0
)
3
.
0
sin(
006
.
0
)
2
.
0
sin(
01
.
0
1
t
t
t
d
, and the
comp
one
nts
of the distur
b
ance su
ppression ve
ctor
0
3
2
1
.
Ca
se 2: Ch
o
o
se the exte
rnal distu
r
ba
n
c
e
m)
(N
)
4
.
0
sin(
014
.
0
)
3
.
0
sin(
006
.
0
)
2
.
0
sin(
01
.
0
1
t
t
t
d
, and
01
.
0
1
,
006
.
0
2
,
014
.
0
3
.
Ca
se 3:
Cho
o
se
the external distu
r
b
a
n
ce
m)
(N
)
4
.
0
sin(
14
.
0
)
3
.
0
sin(
06
.
0
)
2
.
0
sin(
1
.
0
2
t
t
t
d
, and
1
.
0
1
,
06
.
0
2
,
14
.
0
3
.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 2302-4
046
TELKOM
NIKA
TELKOM
NIKA
Vol. 12, No
. 1, Janua
ry 2014: 468 – 4
7
6
473
Ca
se
4:
Cho
o
se
the
external di
stu
r
ban
ce
m)
(N
)
4
.
0
sin(
014
.
0
)
3
.
0
sin(
006
.
0
)
2
.
0
sin(
01
.
0
1
t
t
t
d
, an
d
01
.
0
1
,
006
.
0
2
,
014
.
0
3
. And assume that th
ere exi
s
t model e
rro
r a
nd mod
e
l p
a
ram
e
ter
uncertainty. That is
,
)
m
(kg
6
.
0
10
0
0
0
4
.
0
20
0
0
0
5
.
0
15
2
J
.
4. Results a
nd Analy
s
is
Whe
n
there e
x
ists the exte
rnal di
stu
r
ban
ce
d
1
an
d the
control la
w d
oes
not incl
u
de the
disturban
ce suppr
essio
n
vector
v
(Co
ndit
i
on: Ca
se 1),
the perfo
rm
a
n
ce of the attitude co
ntrolle
r
without an
gul
ar velocity m
easure
m
ent i
s
sh
own in
Figure 1. Whe
n
the con
d
ito
n
is ch
ang
ed
to
Ca
se 2, th
e perfo
rma
n
c
e of the
prop
osed at
t
i
tude co
ntrol
l
er with
out angul
ar velo
city
measurement
is given
in F
i
gure
2. By compa
r
ing
Fig
u
re
1 with
Fi
gure
2, we can see that t
he
controlle
r
without the
di
sturba
nce
sup
p
re
ssi
on ve
ct
or
v
can not converge
to the
equilibri
um
point and b
e
not any more
stable.
While t
he
prop
osed co
ntrolle
r with
the disturb
ance
s
u
pp
r
e
ss
io
n ve
c
t
o
r
v
can m
a
ke
the clo
s
e
d
-loo
p syste
m
whi
c
h
is
d
e
s
c
rib
ed
by
Eqs. (
3
),
(5
) a
n
d
(14) achi
eve the input
state stability. It proves
that the di
sturbance
suppressi
on vector
can
sup
p
re
ss the effect whi
c
h e
x
ternal di
stu
r
ban
ce
s have
on the clo
s
e
d
-
loop
system.
Figure 1. The
angula
r
velo
city curve
without
angul
ar velo
city measurem
ent (Co
nditio
n
:
Ca
se 1)
Figure 2. The
angula
r
velo
city curve
without
angul
ar velo
city measurem
ent
(Conditio
n
:
Ca
se 2)
Figure 3 sho
w
s the
conv
erge
nce of the pro
p
o
s
ed
controll
er wi
thout angul
ar velocity
measurement
unde
r the
co
nditon of
Ca
se 2. Co
mpa
r
ed with th
e
controlle
r
with
angul
ar velo
city
measurement
unde
r the
same
con
d
itio
n, who
s
e
qu
aternio
n
curv
e is
sh
own i
n
Figu
re
4, the
prop
osed con
t
roller in thi
s
pape
r ca
n co
nverge m
o
re
fastly to the equilibri
um poi
nt. It il
lustrate
s
the effectiven
ess of the pro
posed attitud
e
control met
hod.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 2302-4
046
Nonli
nea
r Ro
bust Control f
o
r Spa
c
e
c
raft
Attitude (Lina Wan
g
)
474
Figure 3. The
quaterni
on curve witho
u
t
angul
ar velo
city measurem
ent (Co
nditio
n
:
Ca
se 2)
Figure 4. The
quaterni
on curve with an
g
u
lar
velocity measurem
ent (Con
dition: Ca
se 2
)
Figure 5. The
angula
r
velo
city curve
without
angul
ar velo
city measurem
ent (Co
nditio
n
:
Ca
se 3)
Figure 6. The
enlarg
ed figu
re of
2
in Figure 2
and Figu
re 5
for
]
100
,
35
[
s
s
t
Whe
n
the external di
stub
a
n
ce in
crea
se
s from
d
1
to
d
2
, that is the conditio
n
of Ca
se 3,
the re
sult i
s
d
epicte
d
in
Fig
u
re
5. And Fi
gure
6
gives t
he enl
arged
part of
ang
ul
ar velo
city cu
rve
s
h
ow
n in
F
i
gu
r
e
2
a
n
d
F
i
gu
r
e
5
for
]
100
,
35
[
s
s
t
. From Fig
u
re 2,
Figure 5
an
d
Figure 6,
it can
be observe
d that the area
which the a
ngula
r
veloci
t
y
converg
e
s
to is related
to the external
disturban
ce
s;
that is to
sa
y, the bigge
r the am
p
litud
e of external
distu
r
ba
nce, the la
rge
r
t
h
e
conve
r
ge
nce
area of ang
ular velo
city and the le
ss effective the pro
p
o
s
ed
attitude con
t
rol
method.
Whe
n
the
r
e
e
x
ist model
error a
nd
model
param
ete
r
u
n
ce
rtainty, the pe
rform
a
n
c
e of the
clo
s
ed
-loo
p system und
er
control torq
u
e
is foun
d
in
Figure 7. Obv
i
ously,
the sy
stem can be
still
stable at equi
librium poi
nt. It shows that the prop
os
ed
attitude control method is robu
st to model
error an
d mo
del paramete
r
uncertai
n
ty.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 2302-4
046
TELKOM
NIKA
TELKOM
NIKA
Vol. 12, No
. 1, Janua
ry 2014: 468 – 4
7
6
475
Figure 7. The
angula
r
velo
city curve
with the
existen
c
e of model e
rro
r and m
o
d
e
l para
m
eter
uncertainty (Con
dition: Ca
se 4)
5. Conclusio
n
We con
s
ide
r
ed the stab
ility of spacecraft
attitude in the presn
e
ce of external
disturban
ce
s
and mod
e
l un
certai
nties in
this pap
er
. A nonlin
ear
rob
u
st co
ntrolle
r
is pro
p
o
s
ed b
y
usin
g passivity-based ap
proach and
introdu
cing the suppressio
n
vect
or of external distu
r
ba
n
c
e
into the cont
rol law. The p
r
opo
se
d co
ntrolle
r doe
s n
o
t need the a
ngula
r
velocit
y
measu
r
em
ent
and
can
su
p
p
re
ss the eff
e
ct of ex
tern
al distu
r
ba
nce to a
certai
n
extent. In ad
dition, the
co
ntrol
law do
esn’t contain inform
ation relate
d to the
system
param
eters, whi
c
h ma
ke
s the spa
c
e
c
raft
attitude cont
rol system
rob
u
st to model
error a
nd mo
del pa
ramete
r un
certai
nty. The stability
of
the proposed cont
roller is
proved
theoretically and the num
eri
c
al
simulation result
s illustrated
the effectiven
ess and robu
stne
ss of t
he
spa
c
e
c
raft attitude co
ntrol
method.
Ackn
o
w
l
e
dg
ements
This work was supp
orte
d by the National Natu
ral Scien
c
e
Found
ation
of China
(608
720
46),
Funda
mental
Re
sea
r
ch F
und
s for th
e
Central
Univ
ersitie
s
(F
RF
-TP-1
2
-0
88A) and
Scientific Research Fu
nd
s.
Referen
ces
[1]
Joshi
SM, Ke
l
k
ar AG, W
en J
T
. Robust Attit
ude
Stabi
liz
atio
n of S
pacecr
a
ft us
in
g Non
lin
e
a
r
Quater
nio
n
F
eedb
ack.
IEEE Transactions
on Autom
a
tic
Control
. 199
5; 40(1
0
): 180
0-1
803.
[2]
T
ang Q, W
ang Y, Chen
XL, Lei YJ.
N
onl
in
ear
Attitude C
ontrol of
Rig
id
Body w
i
th Bou
nde
d Co
ntro
l
Input a
n
d
Ve
lo
city-F
ree
. ROB
IO’09 Proc
ee
di
ngs
of the
2
0
09 In
ter
nati
o
n
a
l
Confer
enc
e
on
Ro
botics
and Bi
omimeti
cs. Guilin. 200
9: 2221-
22
26.
[3]
Lin YY, L
i
n
GL. Nonl
ine
a
r
Contro
l
w
i
th
L
y
ap
un
ov Stabil
i
t
y
App
lie
d
to Spacecr
a
ft
w
i
th F
l
e
x
i
b
l
e
Structures.
Jou
r
nal of Syste
m
s and Co
ntrol
Engi
neer
in
g
. 2001; 21
5(1
2
): 131-1
41.
[4]
Ba
yat F
,
Bola
ndi H, Ja
lal
i
AA. A Heuristi
c De
sig
n
Met
hod for Attitud
e
Stabil
i
zati
on
of Magneti
c
Actuated Sate
ll
ites.
Acta Astronautic
a
. 200
9; 65(1
1
): 181
3-1
825.
[5]
Xi
a YQ, Z
hu Z, F
u
MY, W
a
n
g
S. Attitude
T
r
ackin
g
of Rigi
d
Spacecr
a
ft
w
i
t
h
Bou
nde
d Disturb
ance
s
.
IEEE Transactions on Industrial Electronics
. 201
1; 58(2): 64
7-65
9.
[6]
Dong SH, Li SH. Stabi
l
i
zati
on
of the Attitud
e
of a R
i
gi
d Sp
acecraft
w
i
th
E
x
tern
al
Disturb
ances
usi
n
g
F
i
nite-T
ime Control T
e
chniqu
es.
Aerospac
e
Scienc
e an
d T
e
chn
o
lo
gy
. 200
9; 13(4-5): 25
6
-
265.
[7]
Alon
ge F
,
Di
p
polit
o F
,
Raim
ond
i F
M
. Glob
all
y
Conv
erg
e
n
t
Adaptiv
e a
n
d
Rob
u
st Co
ntrol of
Rob
o
ti
c
Mani
pul
ators for T
r
ajector
y
T
r
ackin
g
.
Contro
l Engi
neer
in
g Practice
. 200
4; 12(9): 10
91-
11
00.
[8]
Niu L, Li J s
h
. Adaptiv
e N
eura
l
Net
w
ork
G
eneral
ize
d
Predictiv
e Co
n
t
rol for Unkn
o
w
n
No
nli
n
e
a
r
Sy
s
t
e
m
.
TEL
K
OMNIKA
Indon
esia
n Journ
a
l o
f
Electrical Eng
i
ne
erin
g
. 201
3; 11(7): 361
1-3
617.
[9]
W
ang Q, Gao
T
,
He H. An
Adaptiv
e F
u
zz
y
Contro
l Met
hod
for Sp
ace
c
rafts Based
o
n
T
-
S Model
.
T
E
LKOMNIKA Indon
esi
an Jou
r
nal of Electric
al Eng
i
ne
eri
n
g
.
2013; 1
1
(11):
687
9-68
88.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 2302-4
046
Nonli
nea
r Ro
bust Control f
o
r Spa
c
e
c
raft
Attitude (Lina Wan
g
)
476
[10]
Lizarra
ld
e F
,
W
en JT
. Attitu
de Co
ntrol
w
i
t
h
out
Ang
u
lar V
e
locit
y
Meas
ure
m
ent: A Passiv
i
t
y
Ap
proac
h.
IEEE Transactions on Aut
o
m
a
tic Control
. 19
9
6
; 41(3): 46
8-4
72.
[11]
T
s
iotras P. F
u
rther P
a
ssivit
y
Re
sults for the Attitude
Control Problem.
IEEE Transactions
on
Autom
a
tic Control
. 1998; 3
9
(1
1): 1597-
16
00.
[12]
D
w
yer T
A
, Si
ra RH.
Vari
a
b
le-Structur
e
Contro
l of
Sp
acecraft Attitu
de M
a
n
euvers
.
Journ
a
l
of
Guida
n
ce, Co
n
t
rol and Dy
na
mics
. 1988; 1
1
(3
): 262-27
0.
[13]
Hu Q, F
r
is
w
e
l
l
MI. Robust Var
i
abl
e Structur
e
Atti
tude Co
ntro
l
w
i
th
L 2-
gai
n
Performanc
e for A F
l
e
x
i
b
l
e
Spacecr
a
ft Includi
ng In
put S
a
turatio
n
.
Jour
nal
of Systems and
Contro
l
Engi
neer
in
g
. 201
0; 22
4(2)
:
153-
167.
[14]
Sun Z
W
,
W
u
SN, Li H. Varia
b
le Struct
ur
e Attitude Contro
l of Starin
g Mode S
pacecr
a
ft
w
i
th
Disturb
ance O
b
server.
Jour
n
a
l of Harb
in Ins
t
itute of T
e
chnolo
g
y
. 201
0; (9): 1374-1
3
7
8
.
[15]
Hassa
n KK. Nonli
n
e
a
r S
y
ste
m
s.
T
h
ird Edition. Beij
ing: Pu
blish
i
n
g
Hous
e
of Electronics
Industr
y
.
2
0
0
7
.
Evaluation Warning : The document was created with Spire.PDF for Python.