- Email: [email protected]
Sliding Mode Control for Detumbling Rigid Spacecraft with Underactuated Configuration
SLIDING MODE CONTROL FOR DETUMBLING RIGID SPACECRA. ..
14th World Congress ofIFAC
P-8a-03-5
Copyright © 1999 IFAC 14th Triennial World Congress. Bcijing, P.R. China
SLIDING MODE CONTROL FOR DETUMBLING RIGID SPACECRAFT WITH UNDERACTUATED CONFIGURA nON
Bing Zbang and Hongxin Wu Beijing Institute a/Control Engineering, P.OBox 2729, Beijing 100080, P.R.c. And
JunZbang High Technology R&D Center,Ministry a/Science and Techn%gy,Beijing l00862,P.R.C.
Abstract: This paper discusses the problem of detumbling a rigid spacecraft with underactuated configuration of one-sided thruster. First, some new definitions on the configuration of one-sided thruster are introduced, such as completeness and incompleteness, and the incomplete configuration is divided into two cases: disturbancefree two-dimensional complete configuration for three-dimensional control and twodimensional complete configuration with disturbance for three-dimensional control. For the latter one a special case as disturbance-coplaner two-dimensional complete configuration is considered to stabilize the angular velocities of a rigid spacecraft, in which a Sliding Mode Control (SMC) law is used. The simulations show the effectiveness ofthe approach. Copyright © 1999 IFAC Keywords: Attitude control, Actuators, Configuration, Angular velocity, Sliding-mode control.
1.INTRODUCTION· Thruster for the attitude and translation control of a spacecraft is a kind of one-sided actuator with multidimensional effects. Take the attitude control problem for instance, a thruster can produce control torque around three principal axes, and the control effect is limited to a fixed direction in threedimensional control space, which is determined by the orientation and position of the thruster. It is onesided means that the control command to individual thrusters can only be positive (Zhang ,1998a; Wiktor, I 989; Wiktor,1994). If the net control effects on the spacecraft by the combined outputs of the individual thrusters can be in any directions, it is called in this paper that the
• The research is supported by Chinese National Science Foundation and 863-2-4 Project
configuration of the thrusters system is complete (Zhang ,1998a). A complete configuration may lose completeness under the failures of some thrusters, and the control system becomes underactuated (Krishnan, et aI, 1994). The control of underactuated spacecraft is a key point of FaultTolerant Control of Reaction Control System (Zhang , 1998b), and attract many attentions( Bymes,and Isidmi, 1991; Coron,1996 Morin, and Samson, I 995;Tsiotras, and Luo, 1997 ). Under the assumption that the thruster systems with incomplete configuration comprise of one or t\vo thrusters' group, in which both positive and negative thruster parallel to one of the three principal axes are in normal condition, then one or two control torque can be generated only along the controllable axes. Almost all researches on the attitude control of underactuated spacecraft are in relation to this case . However,
in
general
case
the
incomplete
8015
Copyright 1999 IF AC
ISBN: 008 0432484
SLIDING MODE CONTROL FOR DETUMBLlNG RIGID SPACECRA ...
configuration of one-sided thruster systems may produce extra effects on the uncontrolled axes. Few have been done in this case to consider the attitude control of underactuated spacecraft. In this paper, the underactuated configuration of one-sided thrusters is considered to detumble the angular velocity of a rigid spacecraft.
14th World Congress ofIFAC
where dimensional
control
Uc
2.1
The rotation motion of a rigid spacecraft under the influence of body-fixed torque is described by Euler's equations of motion of the fonn (1-a) I/j)[ + (02 W 3 (13 - 12 ) =
't
-
13 0 3 + (0,(02(12 where
(0"
0)2'
-
13 )
= '2
I,) =
'3
principal
are
moments
(I-c)
of
inertia"" '2' '] is the body-fixed control torque vector.Defining (2-a) d , = (/ 3 - 1,)11, 2
= (/,
-13)/12
(2-b)
d3
= (J 2
-
1, )/ I 3
(2-c)
d
U i = T"; 1 i Eq. (1) can be rewritten as
i
=
1,2,3
(3)
u,
(4-a)
c:il 1 = - d , (JJ2(03 +
is
m-
negative (one-sided). C E R3 ~m is the configuration of thruster systems, whose columns represent the influence coefficients defining how each thruster affects each component of the control vector. In order to evaluate the control-generating capability of the thruster systems, the following definition is provided. Definition I For configuration matrix C its controllable domain satisfies
{u\u =Cuc,u
c
ER:'
E
R 3xm
}= R3
if
,
(6)
In nonnal condition, if C is complete, then the thruster systems can generate arbitrary control vector in three-dimensional space. When some thrusters failed , the configuration matrix is changed to C f = CF, F = diag(fi) ,i = 1,2,···m (7)
o
1. = {I
(l-b)
are the angular velocities,
(03
ER:'
then C is complete.
EquQtion of motion
1 2 0 2 + (0,(0 3 (1,
Uc
three-
are the burning times, so it can only be non-
Se = 2.PROBLEM DEFINITION
vector,
the
dimensional action vector, m is the total number of thrusters. When the P\VM mode is applied to control thrusters, the physical interpretation of the elements of
The main contributions of this paper are the following: 1) the completeness of the one-sided thruster systems configuration is defined; 2) the incompleteness is analyzed and divided into two cases according to the existence of disturbance on the uncontrolled axis; 3) under a special incomplete configuration with the disturbance on the uncontrolled axis, a sliding mode control law is developed to asymptotically stabilize the angular velocity.
is
where j is configuration
i = j i 0# J
(8)
the failed thruster. The current matrix
Cf
may
loss
completeness(Zhang, 1998a). For n -dimensional control , the controllable domain of incomplete configuration is a convex polyhedral cone on one side of a (n - I)-dimensional hyperplane, and n dimensional incomplete configuration can be considered as an n - r(r < n) -dimensional complete configuration. Now consider a incomplete configuration matrix Cl = (C;, C;2] T for three-dimensional control, where C l' ER"·', Cl' ER"" .
°
2
= -d 2 w'(03
+ u2
(4-b)
Definition 2
c:il 3
= -d,W'(02
+U3
(4-c)
dimensional control, and for any two-dimensional control
2.2 Completeness of configuration
[u;,
If ui2
C /2
E
R 2 xm is complete for two-
YE R2 , n,i2 E {I ,2,3 }, il ;t:. i2 ,
there exists U c E R~t yields A thruster for attitude control is a kind of one-sided actuator, and can generate control torque around three principal axes. Moreover, the effects on different axes have fixed direction in control space, which depend on the orientation and position of the thruster. Thus in order to generate arbitrary controls, multithrusters are needed to act together. The net control effect on spacecraft by the combined outputs of the individual thrust can be expressed as U
= CUe
(5)
{
C f1 U = 0 C
(9)
C f2uc =
[Ui1
un] T
then Cl IS a disturbance-free two-dimensional complete configuration for three-dimensional controL Although complete three-dimensional control cannot be generated by Cl' complete two-dimensional controls around axis il, i2 are available, and no extra
8016
Copyright 1999 IF AC
ISBN: 0 08 043248 4
SLIDING MODE CONTROL FOR DETUMBLING RIGID SPACECRA. ..
effect on aXIs i3 exists. Therefore the current controllable domain contain the plane U;3 = 0 . Without loss of generality, suppose controls around 2 and 3 axis are complete and that around axis I is restricted to zero. Then according to Eq. (9), Eq. (4) can be changed to
00 1 = -d l ffi 2 ffi 3 cO 2 =-d2 ffi I 0)3 W3 =-d 3 ffijffi2
(10-a)
+U 2
(lO-b)
+U 3
(lO-c)
14th World Congress ofIFAC
= [J;(w) fz(w) J;(O)]T+ U 0) = [ro( ro 2 ffi) JT •
cO where
Multiplying Eq. (11) with P, then
~ = [~(05) ~ (05) ~(05)] T +U and u=PU = PCjU c
Cj
is
a
= CfUe
Definition 3 if en ER'"'' is complete for twodimensional control, and for any two-dimensional control exists
[un u"Y E R2 , il, i2 E {1,2,31 il "" i2 U c E R"; which yields {
Cf1U e
Y1U il
:
C f2 Uc where YI , Y2 E then
R
+ Y2 U ;2
,
there
complete
configuration with coplaner disturbance, according to U c E R~
which
satisfies ~ [U;l
~ U;2
il, i2, i3
JT E R2 ,U;3 ~ =0
i2 ;;c i3 Without Ims of generality, assuming i3 = 1 , then u=[O u 2 UJT (15) where [Uz Defining
P=[l{ where (9)
and
Un]
[U;I
(14)
two-dimensional
Definition 3 there must exist However, in a more general case the incomplete configuration of one-sided thruster systems may produce extra effects on uncontrolled axes. In this paper a special case of incomplete configuration with the disturbance is considered.
(12)
where
Because
Two-dimensional complete configuration with copianer disturbance
2.3
(11)
UJ
Y
E
;j=.
R2 .
1,2,3)
P; (i
,where
{1,2,3}, il
JrT =r
~Trp=[l( Jf
If
Pr (i =
E
=
are
row
vectors
of
(16)
P
1,2,3) are row vectors of
P- 1 •
are constants
C f is a disturbance-eoplaner two-dimensional
complete configuration for three-dimensional control. Complete two-dimensional controls around axis i1, i2 are available, and in spite of the existence of effect on axis i3, the generated controls around three axes are eoplaner. That is
[y I
Y2
-
I]
U =
0
(17)
where
-
~
J; (m) = -
~
~T-T-~
-dlffi
(18-a)
~
Po, (f)
(l8-b)
.~C(5) = -dijjT p/~(jj
(l8-c)
f2 ( m) = -d 2 0)
(10)
P2 P3 m
~T-T--
Theorem There exists a nonsingular linear transformation that transforms 2-dimensional complete configuration with coplaner disturbance to disturbance-free reduction-dimensional complete configuration.
eT
R 3xm
a tw 0dimensional complete configuration with eoplaner disturbance, according to Theorem 1 there exists a f2 ]TE
nonsingular square matrix P
Cf
= PCf
E
be
R~X3, which makes
be a disturbance-free two-dimensional
complete configuration.
2.4 Control problem description First rewrite Eq. (4) as
where PijU,j =
1,2,3) are elements of P.
Therefore the control problem of system (11) under the configuration with coplaner disturbance can be transfonned to that of system (12) without disturbance, and routing control approach can be used to stabilize 00; (i = 1,2,3). Sliding Mode Control has
8017
Copyright 1999 IF AC
ISBN: 008 0432484
SLIDING MODE CONTROL FOR DETUMBLING RIGID SPACECRA. ..
the unique property that the control law can change instantaneously depending upon the particular region of the state space that the system is currently occupying, and has been successfully applied to the control of non linear systems. Here, a sliding mode control law is applied to the stabilization of system (12).
14th World Congress ofIFAC
13 = 0
(29-c)
Straightforward computations give
J; (m l )
(30) 05; Defining Lyapunov function V (05 1 ) = O5~ /2 and =
[2
noting that V(m!) is positive definite except when
m = 0, then l
mll; (m l ) 12mt
3. SLIDING MODE CONTROL OF TRANSFORMED SYSTEM
V(05!) = = <0 (31) which indicates that system (12) is asymptotically stable on the surface (24). Hence have the following theorem:
First Eq. (19) can be simplified as
J; (05) = O5T P05
(20-a)
Z(m) = mOm
(20-b)
~ (m)
(20-c)
T
= mT
Rm
where -
-r-
P =-dIPll~ ~
f
J{ - d3PI3l{T~
- d 2P12 J -
Q
(21-a)
Theorem 3.1 On SI = 0 and S 2 = O,if conditions (29) is satisfied, system (12) is asymptotically stable. Further study is about how to determine the coefficients k 21 , k 22 , k 31 , k32' which condition (29) have solution. From Eq. (29-a), then conclude that if
~I = [(P32
-T-
= -dlP21~ ~ -T-
-T-
R=
-d1P31P;.Tl{
-d2 Pn P/J{ -d3P33R,T~
then fOT any k32 one gets
(21-c)
From Eq. (29-c), if
k22
Ll;
Defining switch functions as
k21 wI - k22W~ S2 (05) = m3 - k 31 05 1 - k32W; on the surfaces SI = 0 and S2 = 0 and SI
(m) = W2
-
+ Pn)2 - 4]522]533 J~ 0
(2l-b)
- d2P22~ I; - d3P23~ P2
(22) (23)
=
make
(32)
[-k32 (P32 + P23) ± k32 ~.I / 2]522 (33)
= CP21 + ]512 + (Pn + P23 )k31 )2
(34)
- 4 P22 (P33 k ;r + CP31 + PIJk31 + Pll) ~ 0 then for any k 31
,
one gets
lsl = [-P21 - PI2 -(J531 +Pn)1SI ±.JR;]/ 2P22(35) Rewrite Eq. (34) as (24-a) (24-b)
~ =[1A2 +Ai -4A:A~1 +[2(]J; 1+A~1A2 +AJ
-4AiplI +AJF:lI +CAI +A~2 -4AA I =~~I +[2(P2 I +PI2)@32+P23)
(36)
-4P22 CP31 +Pl 3)YsI +~ ~O
where 6 2 = (P21 + PI2)2 - 4 P22PI1 . If Eq. (32) is satisfied. then there always exists
= 05;(13 where
11
+ '2WI + llw;)
Pil (i. j = 1.2.3) =
are the elements of
(25)
P.
k31 E n c R which make Eq. (36) satisfied. Rewrite Eq. (29-b) as l2 = ~/s 1+ 2A!s.! +Al +A;)ls.2 +[ (2Pn +A;)ls I +(P12 + AJls 1+A3+ A 1.1k.12
1522k~2 + ('fin + 1523 )k22 k 32 + P33k:2 (26)
~- =(2Ajc.,2+AIs2+A:ls-)lsl+m!s~+2AIs2 --- (27)
+'AIsJ/s 1+CA I+Pt -)Is 2 +(JJ 3 +AJk32 '3 = P22 k ;1 +CP21 + PI2)k21 + P33 kil + CP31 (28) + -A Jk3 ! + CP23 + P31 )k2l k,! + Pll If P22 "* 0 ,and
=)'1 ('s l' k21
Ykn +~('s1,ls.1)kJ2 <0
(37)
where
~(k21,1s1)=A.Js1 +2A!s.I+AI +P; ~(ls.I,/sJ =(2A3 +A)/sI +C'A2+AJkn +A3+Al Proposition I If Eq. (32) is satisfied, there exist
(29-a) (29-b)
k 21 , k22 ' k31 ' k32 ' such that the equations set (29) with inequality constrains has solutions. Proof:
8018
Copyright 1999 IF AC
ISBN: 008 0432484
SLIDING MODE CONTROL FOR DETUMBLlNG RIGID SPACECRA ...
i) When Eq. (32) is met, then there always exists k31
14th World Congress ofIFAC
Substitute Eq. (43) and Eq. (44) to Eq. (41), then
Ere c R which make Eq. (36) satisfied, such
that kz, can be obtained according to Eq. (35), which makes Eq. (29-c) be satisfied. ii) Substitute
k21
and
k31
12 =[~ +~(-P32 -P23±~)/2j522Jls2 <0(38) kn
Select k32 according to Eq. (38), and gain
from Eq. (33), hence Eq. (29-a) and Eq. (29-b) are both satisfied.As mentioned above, based on the steps
k 21
described by i) and ii), there exist
,
kn'
k31'
0
0
PI31522
_0 _
0
P12]513
- P12P23
- P12P13
0
[
;P
into Eq. (37), if }., and
A2 are not all zero, then Substituting Eq. (33) into Eq. (37) ,then
(1522P33 - P23]5}2)
L'l 3 =
JCl
=q~lAlAA3-A;p"J(AjJ22-A;A:J~ (45) End of the proof. Consider the control, which can drive the states of the system to the sliding surfaces from any initial states. From Eq. (22), it can be found that SI (co) = ~2 - k2]cll l Considering Eq. (17)-Eq. (24)
-
2kn
co, cll,
(46)
s] = COT(j:n+ 1:4.---kjJIPm- 2lsihlJlPm
k32 ' such that the equations set (29) with inequality constrains has solutions.End of proof. The proof mentioned above is constructive, and also gives a procedure to obtain the coefficients
The control law can be obtained from Eq. (47) and Eq.
k 21 , k22' k31' k32 of the switch surfaces depicted
(48)
by Eq. (22) and (23).
u2
.s·1
= bll sign(s1) + b'2 S,
then Eq. (32) is satisfied, where Pij(i,j = are the elements of
P ,and Pi} (i, j
=
(39)
P22P33)]~O(40)
]523)2 +4(p32P23 -
L'l3 = P32P23
- ]522]533
~0
(4 J)
then Eq. (40) is satisfied. Because
~T--
- d 3 P13 ]
-
~
T
.
JT
P22P33= - - - PlJiJ22P3tiJ33 (43) - - - ~A;P]";P2;P32 d Pl2P2JiJ32PJ> Pl2P13P3tiJ33
l~'!"~~' -
-
p,,p,,p,,A,] _
-
-
-
-
P3.J52] = J7 PI2P23P32 -)-
~T~~
J
0~
+ b 21 sign(s 2) + b 22 S 2 Proposition 3 The system depicted as Eq.(J2) and Eq.(l4) under a disturbance-coplaner twodimensional complete configuration. can be asymptotically stabilized with the control law described by Eq. (49) and Eq. (50). In order to minimize cbattering in the controllers, the saturation function sat(-) given in Eq. (51) has
sat(s)
= {:/L'l
- - - P13P22fJ32fJ33 - - - Pl2fJl3P32P33 - - - PlzPI3P23P32
- -, PUPZtiJ33
~l ~LJ~22~33 PI2PLJPZzP23
] J
(51)
4. SPACECRAFT DETUMBLING SIMULATION Based on the aforementioned control law the simulation on the detumbling a rigid spacecraft under a incomplete configuration is provided. In the
simulation the principal inertia 11
= lOOkg.m 2 ,
= 200kg.m 2 , 13 = 300kg.m 2 , the number of the thrusters m = 3, and the contlguration matrix Iz
Cl (44 )
1::1:: s < -t,.
-1
Pl2P22P2:J133 Pl2PlJiJnP33 Pl.jjlJ;2J;23
l~~',AJ>" PI2Pz3PnP33
~
Rro+k3l ro Pro+2k 32 ro 1ro Pro
=
= [- d,p],
-
-T-~
been introduced in stead of sign(·).
[P2JP33 Pl3P33 Pl3P23] J (42-a) ]532 = [PnP32 Pl3P32 1513Pn] d (42-b) 15,21522] d (42-c) ]522 = [PnP32 PI2]521] Cl (42-d) /523 = [PnP33
-
~
U-=-(J)
1,2,3) are
If
J
(49)
Similarly,
1,2,3)
elements of P. Proof: Eq. (32) can be rewritten as
where Then
(48)
+ bllsign(sl) + bl2 s ,
d.d:A pJA.A3 - AAJ(JJ..P12 - AAJ ~O
]533
bl " bl 2 < 0
_WT (205 + k2]05 T Pro + 2k 22 ro,ro T Pro
=
Proposition 2 If
~] = [(D32 -
(47)
Let
=[:
o
~ ~~l I
is a disturbance-coplaner two-dimen sional complete configuration. The parameters
bZ2 and
~
b,t' b12
,
b 21
,
are 0.01.
8019
Copyright 1999 IF AC
ISBN: 0 08 043248 4
SLIDING MODE CONTROL FOR DETUMBLING RIGID SPACECRA. ..
14th World Congress ofIFAC
5.CONCLUSION
Table I Simulation condition Initial states
Parameter of controller
ill 10
-11A
-5.7
-5.4
5.71
5
1.1
10
The simulation condition is shown in Table 1.
..
..!!!
Cl
o
~ u
.. :. ---~-~-~-----~--:
..2
. .. -.~,,", . -:-~~-~'~'~.~'----~--~------..... ,
·,
~
,." ,,, ,,
L
III
~ ~
· ··
,
..
·
..
,
REFERENCE
-,.-. - - - - - ----
- - - -- ---:- - -- - - - - - --:- --- - - - - -
-10
..
B)-Tiles, c.l. and Isidori(l99 I), A. On the attitude stabilization of a rigid spacecraft. Automatica, VoI.27, pp.87-95. Coron, J.M. (1996). Explicit feedback stabilizing the attitude control of a rigid spacecraft with two control torque. Automatica, Vo1.32, pp.669-677. Krishnan, H.C 1994). Attitude stabilization of a rigid spacecraft using two control torques: a non-linear control approach based on the spacecraft attitude dynamics. Automatica, Vo1.30, pp.1023-1027 Morin,P. And C. Samson(l995). Time-varying exponential stabilization of the attitude of a rigid spacecraft with two control. In: Proceedings of the 34th Conference on Decision & Control, New Orleans,LA,Dec. Tsiotras, P. and J. Luo (1997). Reduced-effort control laws for underactuated rigid spacecraft. AIAA 97-
-150L--~2~O---4~0----:6"-O----:8:'-O--....,.-J100
t(Sec)
25.---~--~---------,
20 ..... - .. -.~ ......•...;. - ........ ~ ......... -; - ... - .. -.• •
, •
, ,
I I
•
I
I
I
E;
15
i
10
t----·~-..L~----L ..
-5
~-
-- -- --. -.~ -- -- -- --. ~-- -- -----. ~-- -- -- -- -- ~ -- -- -- ----
f :I\'T·.'I·~.·. • .I.··.··.··I·····•. · • ..
----~.-
.
0113.
..
.. ------:.- .. ----.-~ .. --------~---.---.-,
Wikor,P.J.(l989). Optimal thruster configurations for the GP-B spacecraft. In: Proceedings of IFAC Automatic Control in Aerospace. Tsukuba, Janpan Wiktor ,Po ].(1994). Minimum control authority plot: a tool for designing thruster systems. Journal of Guidance,Control and Dynamics, VoI.17, pp.9981006 Zhang Bing and Hongxin Wu, (1998a). Completeness of configuration of one-sided actuators system. In: Proceedings of the Chinese Control Conference '98.Ningbo Univ., Zhejiang P.R.China Zhang Bing(l998b). Fault-Tolerant Control of Reaction Control System and its applications. In: PHD dissertation of Chinese Academy of Space Technology. Beijing Institute of Control Technology.
-10~--~--~--~--~--~
o
20
40
60
80
100
t(Secl
Fig.2. Control torque vector u
1'
u
2'
u
3
15,---~--~--~--~---,
E c
'J 10 C'"
o
-
,
:
1\:
:
- - - or.\ ---. - -~ ------ - --.:., --- - ----- ~-, - -- --- -->' : ;
~ : Il i
\..
,
~ 5 ~'.-.-I" '\" .~-- ... ---.~ 13
This paper introduces some new definitions on the configuration of one-sided thruster, such as completeness and incompleteness, and divides the incomplete configuration into two cases: disturbancefree two-dimensional complete configuration for three-dimensional control and two-dimensional complete configuration with disturbance for threedimensional control. For the latter one a special case as disturbance-coplaner two-dimensional complete configuration for three-dimensional control is considered to stabilize the angular velocities of a rigid spacecraft, in which a Sliding Mode Control law is used. The simulations show the effectiveness of the approach .
.
'"
:,
o
o
i
20
40
60
100
80
t(Sec)
Fig.3. Thruster action torque vector u
c1 '
u
e2 '
u
c3
8020
Copyright 1999 IF AC
ISBN: 008 0432484
14th World Congress ofIFAC
P-8a-03-5
Copyright © 1999 IFAC 14th Triennial World Congress. Bcijing, P.R. China
SLIDING MODE CONTROL FOR DETUMBLING RIGID SPACECRAFT WITH UNDERACTUATED CONFIGURA nON
Bing Zbang and Hongxin Wu Beijing Institute a/Control Engineering, P.OBox 2729, Beijing 100080, P.R.c. And
JunZbang High Technology R&D Center,Ministry a/Science and Techn%gy,Beijing l00862,P.R.C.
Abstract: This paper discusses the problem of detumbling a rigid spacecraft with underactuated configuration of one-sided thruster. First, some new definitions on the configuration of one-sided thruster are introduced, such as completeness and incompleteness, and the incomplete configuration is divided into two cases: disturbancefree two-dimensional complete configuration for three-dimensional control and twodimensional complete configuration with disturbance for three-dimensional control. For the latter one a special case as disturbance-coplaner two-dimensional complete configuration is considered to stabilize the angular velocities of a rigid spacecraft, in which a Sliding Mode Control (SMC) law is used. The simulations show the effectiveness ofthe approach. Copyright © 1999 IFAC Keywords: Attitude control, Actuators, Configuration, Angular velocity, Sliding-mode control.
1.INTRODUCTION· Thruster for the attitude and translation control of a spacecraft is a kind of one-sided actuator with multidimensional effects. Take the attitude control problem for instance, a thruster can produce control torque around three principal axes, and the control effect is limited to a fixed direction in threedimensional control space, which is determined by the orientation and position of the thruster. It is onesided means that the control command to individual thrusters can only be positive (Zhang ,1998a; Wiktor, I 989; Wiktor,1994). If the net control effects on the spacecraft by the combined outputs of the individual thrusters can be in any directions, it is called in this paper that the
• The research is supported by Chinese National Science Foundation and 863-2-4 Project
configuration of the thrusters system is complete (Zhang ,1998a). A complete configuration may lose completeness under the failures of some thrusters, and the control system becomes underactuated (Krishnan, et aI, 1994). The control of underactuated spacecraft is a key point of FaultTolerant Control of Reaction Control System (Zhang , 1998b), and attract many attentions( Bymes,and Isidmi, 1991; Coron,1996 Morin, and Samson, I 995;Tsiotras, and Luo, 1997 ). Under the assumption that the thruster systems with incomplete configuration comprise of one or t\vo thrusters' group, in which both positive and negative thruster parallel to one of the three principal axes are in normal condition, then one or two control torque can be generated only along the controllable axes. Almost all researches on the attitude control of underactuated spacecraft are in relation to this case . However,
in
general
case
the
incomplete
8015
Copyright 1999 IF AC
ISBN: 008 0432484
SLIDING MODE CONTROL FOR DETUMBLlNG RIGID SPACECRA ...
configuration of one-sided thruster systems may produce extra effects on the uncontrolled axes. Few have been done in this case to consider the attitude control of underactuated spacecraft. In this paper, the underactuated configuration of one-sided thrusters is considered to detumble the angular velocity of a rigid spacecraft.
14th World Congress ofIFAC
where dimensional
control
Uc
2.1
The rotation motion of a rigid spacecraft under the influence of body-fixed torque is described by Euler's equations of motion of the fonn (1-a) I/j)[ + (02 W 3 (13 - 12 ) =
't
-
13 0 3 + (0,(02(12 where
(0"
0)2'
-
13 )
= '2
I,) =
'3
principal
are
moments
(I-c)
of
inertia"" '2' '] is the body-fixed control torque vector.Defining (2-a) d , = (/ 3 - 1,)11, 2
= (/,
-13)/12
(2-b)
d3
= (J 2
-
1, )/ I 3
(2-c)
d
U i = T"; 1 i Eq. (1) can be rewritten as
i
=
1,2,3
(3)
u,
(4-a)
c:il 1 = - d , (JJ2(03 +
is
m-
negative (one-sided). C E R3 ~m is the configuration of thruster systems, whose columns represent the influence coefficients defining how each thruster affects each component of the control vector. In order to evaluate the control-generating capability of the thruster systems, the following definition is provided. Definition I For configuration matrix C its controllable domain satisfies
{u\u =Cuc,u
c
ER:'
E
R 3xm
}= R3
if
,
(6)
In nonnal condition, if C is complete, then the thruster systems can generate arbitrary control vector in three-dimensional space. When some thrusters failed , the configuration matrix is changed to C f = CF, F = diag(fi) ,i = 1,2,···m (7)
o
1. = {I
(l-b)
are the angular velocities,
(03
ER:'
then C is complete.
EquQtion of motion
1 2 0 2 + (0,(0 3 (1,
Uc
three-
are the burning times, so it can only be non-
Se = 2.PROBLEM DEFINITION
vector,
the
dimensional action vector, m is the total number of thrusters. When the P\VM mode is applied to control thrusters, the physical interpretation of the elements of
The main contributions of this paper are the following: 1) the completeness of the one-sided thruster systems configuration is defined; 2) the incompleteness is analyzed and divided into two cases according to the existence of disturbance on the uncontrolled axis; 3) under a special incomplete configuration with the disturbance on the uncontrolled axis, a sliding mode control law is developed to asymptotically stabilize the angular velocity.
is
where j is configuration
i = j i 0# J
(8)
the failed thruster. The current matrix
Cf
may
loss
completeness(Zhang, 1998a). For n -dimensional control , the controllable domain of incomplete configuration is a convex polyhedral cone on one side of a (n - I)-dimensional hyperplane, and n dimensional incomplete configuration can be considered as an n - r(r < n) -dimensional complete configuration. Now consider a incomplete configuration matrix Cl = (C;, C;2] T for three-dimensional control, where C l' ER"·', Cl' ER"" .
°
2
= -d 2 w'(03
+ u2
(4-b)
Definition 2
c:il 3
= -d,W'(02
+U3
(4-c)
dimensional control, and for any two-dimensional control
2.2 Completeness of configuration
[u;,
If ui2
C /2
E
R 2 xm is complete for two-
YE R2 , n,i2 E {I ,2,3 }, il ;t:. i2 ,
there exists U c E R~t yields A thruster for attitude control is a kind of one-sided actuator, and can generate control torque around three principal axes. Moreover, the effects on different axes have fixed direction in control space, which depend on the orientation and position of the thruster. Thus in order to generate arbitrary controls, multithrusters are needed to act together. The net control effect on spacecraft by the combined outputs of the individual thrust can be expressed as U
= CUe
(5)
{
C f1 U = 0 C
(9)
C f2uc =
[Ui1
un] T
then Cl IS a disturbance-free two-dimensional complete configuration for three-dimensional controL Although complete three-dimensional control cannot be generated by Cl' complete two-dimensional controls around axis il, i2 are available, and no extra
8016
Copyright 1999 IF AC
ISBN: 0 08 043248 4
SLIDING MODE CONTROL FOR DETUMBLING RIGID SPACECRA. ..
effect on aXIs i3 exists. Therefore the current controllable domain contain the plane U;3 = 0 . Without loss of generality, suppose controls around 2 and 3 axis are complete and that around axis I is restricted to zero. Then according to Eq. (9), Eq. (4) can be changed to
00 1 = -d l ffi 2 ffi 3 cO 2 =-d2 ffi I 0)3 W3 =-d 3 ffijffi2
(10-a)
+U 2
(lO-b)
+U 3
(lO-c)
14th World Congress ofIFAC
= [J;(w) fz(w) J;(O)]T+ U 0) = [ro( ro 2 ffi) JT •
cO where
Multiplying Eq. (11) with P, then
~ = [~(05) ~ (05) ~(05)] T +U and u=PU = PCjU c
Cj
is
a
= CfUe
Definition 3 if en ER'"'' is complete for twodimensional control, and for any two-dimensional control exists
[un u"Y E R2 , il, i2 E {1,2,31 il "" i2 U c E R"; which yields {
Cf1U e
Y1U il
:
C f2 Uc where YI , Y2 E then
R
+ Y2 U ;2
,
there
complete
configuration with coplaner disturbance, according to U c E R~
which
satisfies ~ [U;l
~ U;2
il, i2, i3
JT E R2 ,U;3 ~ =0
i2 ;;c i3 Without Ims of generality, assuming i3 = 1 , then u=[O u 2 UJT (15) where [Uz Defining
P=[l{ where (9)
and
Un]
[U;I
(14)
two-dimensional
Definition 3 there must exist However, in a more general case the incomplete configuration of one-sided thruster systems may produce extra effects on uncontrolled axes. In this paper a special case of incomplete configuration with the disturbance is considered.
(12)
where
Because
Two-dimensional complete configuration with copianer disturbance
2.3
(11)
UJ
Y
E
;j=.
R2 .
1,2,3)
P; (i
,where
{1,2,3}, il
JrT =r
~Trp=[l( Jf
If
Pr (i =
E
=
are
row
vectors
of
(16)
P
1,2,3) are row vectors of
P- 1 •
are constants
C f is a disturbance-eoplaner two-dimensional
complete configuration for three-dimensional control. Complete two-dimensional controls around axis i1, i2 are available, and in spite of the existence of effect on axis i3, the generated controls around three axes are eoplaner. That is
[y I
Y2
-
I]
U =
0
(17)
where
-
~
J; (m) = -
~
~T-T-~
-dlffi
(18-a)
~
Po, (f)
(l8-b)
.~C(5) = -dijjT p/~(jj
(l8-c)
f2 ( m) = -d 2 0)
(10)
P2 P3 m
~T-T--
Theorem There exists a nonsingular linear transformation that transforms 2-dimensional complete configuration with coplaner disturbance to disturbance-free reduction-dimensional complete configuration.
eT
R 3xm
a tw 0dimensional complete configuration with eoplaner disturbance, according to Theorem 1 there exists a f2 ]TE
nonsingular square matrix P
Cf
= PCf
E
be
R~X3, which makes
be a disturbance-free two-dimensional
complete configuration.
2.4 Control problem description First rewrite Eq. (4) as
where PijU,j =
1,2,3) are elements of P.
Therefore the control problem of system (11) under the configuration with coplaner disturbance can be transfonned to that of system (12) without disturbance, and routing control approach can be used to stabilize 00; (i = 1,2,3). Sliding Mode Control has
8017
Copyright 1999 IF AC
ISBN: 008 0432484
SLIDING MODE CONTROL FOR DETUMBLING RIGID SPACECRA. ..
the unique property that the control law can change instantaneously depending upon the particular region of the state space that the system is currently occupying, and has been successfully applied to the control of non linear systems. Here, a sliding mode control law is applied to the stabilization of system (12).
14th World Congress ofIFAC
13 = 0
(29-c)
Straightforward computations give
J; (m l )
(30) 05; Defining Lyapunov function V (05 1 ) = O5~ /2 and =
[2
noting that V(m!) is positive definite except when
m = 0, then l
mll; (m l ) 12mt
3. SLIDING MODE CONTROL OF TRANSFORMED SYSTEM
V(05!) = = <0 (31) which indicates that system (12) is asymptotically stable on the surface (24). Hence have the following theorem:
First Eq. (19) can be simplified as
J; (05) = O5T P05
(20-a)
Z(m) = mOm
(20-b)
~ (m)
(20-c)
T
= mT
Rm
where -
-r-
P =-dIPll~ ~
f
J{ - d3PI3l{T~
- d 2P12 J -
Q
(21-a)
Theorem 3.1 On SI = 0 and S 2 = O,if conditions (29) is satisfied, system (12) is asymptotically stable. Further study is about how to determine the coefficients k 21 , k 22 , k 31 , k32' which condition (29) have solution. From Eq. (29-a), then conclude that if
~I = [(P32
-T-
= -dlP21~ ~ -T-
-T-
R=
-d1P31P;.Tl{
-d2 Pn P/J{ -d3P33R,T~
then fOT any k32 one gets
(21-c)
From Eq. (29-c), if
k22
Ll;
Defining switch functions as
k21 wI - k22W~ S2 (05) = m3 - k 31 05 1 - k32W; on the surfaces SI = 0 and S2 = 0 and SI
(m) = W2
-
+ Pn)2 - 4]522]533 J~ 0
(2l-b)
- d2P22~ I; - d3P23~ P2
(22) (23)
=
make
(32)
[-k32 (P32 + P23) ± k32 ~.I / 2]522 (33)
= CP21 + ]512 + (Pn + P23 )k31 )2
(34)
- 4 P22 (P33 k ;r + CP31 + PIJk31 + Pll) ~ 0 then for any k 31
,
one gets
lsl = [-P21 - PI2 -(J531 +Pn)1SI ±.JR;]/ 2P22(35) Rewrite Eq. (34) as (24-a) (24-b)
~ =[1A2 +Ai -4A:A~1 +[2(]J; 1+A~1A2 +AJ
-4AiplI +AJF:lI +CAI +A~2 -4AA I =~~I +[2(P2 I +PI2)@32+P23)
(36)
-4P22 CP31 +Pl 3)YsI +~ ~O
where 6 2 = (P21 + PI2)2 - 4 P22PI1 . If Eq. (32) is satisfied. then there always exists
= 05;(13 where
11
+ '2WI + llw;)
Pil (i. j = 1.2.3) =
are the elements of
(25)
P.
k31 E n c R which make Eq. (36) satisfied. Rewrite Eq. (29-b) as l2 = ~/s 1+ 2A!s.! +Al +A;)ls.2 +[ (2Pn +A;)ls I +(P12 + AJls 1+A3+ A 1.1k.12
1522k~2 + ('fin + 1523 )k22 k 32 + P33k:2 (26)
~- =(2Ajc.,2+AIs2+A:ls-)lsl+m!s~+2AIs2 --- (27)
+'AIsJ/s 1+CA I+Pt -)Is 2 +(JJ 3 +AJk32 '3 = P22 k ;1 +CP21 + PI2)k21 + P33 kil + CP31 (28) + -A Jk3 ! + CP23 + P31 )k2l k,! + Pll If P22 "* 0 ,and
=)'1 ('s l' k21
Ykn +~('s1,ls.1)kJ2 <0
(37)
where
~(k21,1s1)=A.Js1 +2A!s.I+AI +P; ~(ls.I,/sJ =(2A3 +A)/sI +C'A2+AJkn +A3+Al Proposition I If Eq. (32) is satisfied, there exist
(29-a) (29-b)
k 21 , k22 ' k31 ' k32 ' such that the equations set (29) with inequality constrains has solutions. Proof:
8018
Copyright 1999 IF AC
ISBN: 008 0432484
SLIDING MODE CONTROL FOR DETUMBLlNG RIGID SPACECRA ...
i) When Eq. (32) is met, then there always exists k31
14th World Congress ofIFAC
Substitute Eq. (43) and Eq. (44) to Eq. (41), then
Ere c R which make Eq. (36) satisfied, such
that kz, can be obtained according to Eq. (35), which makes Eq. (29-c) be satisfied. ii) Substitute
k21
and
k31
12 =[~ +~(-P32 -P23±~)/2j522Jls2 <0(38) kn
Select k32 according to Eq. (38), and gain
from Eq. (33), hence Eq. (29-a) and Eq. (29-b) are both satisfied.As mentioned above, based on the steps
k 21
described by i) and ii), there exist
,
kn'
k31'
0
0
PI31522
_0 _
0
P12]513
- P12P23
- P12P13
0
[
;P
into Eq. (37), if }., and
A2 are not all zero, then Substituting Eq. (33) into Eq. (37) ,then
(1522P33 - P23]5}2)
L'l 3 =
JCl
=q~lAlAA3-A;p"J(AjJ22-A;A:J~ (45) End of the proof. Consider the control, which can drive the states of the system to the sliding surfaces from any initial states. From Eq. (22), it can be found that SI (co) = ~2 - k2]cll l Considering Eq. (17)-Eq. (24)
-
2kn
co, cll,
(46)
s] = COT(j:n+ 1:4.---kjJIPm- 2lsihlJlPm
k32 ' such that the equations set (29) with inequality constrains has solutions.End of proof. The proof mentioned above is constructive, and also gives a procedure to obtain the coefficients
The control law can be obtained from Eq. (47) and Eq.
k 21 , k22' k31' k32 of the switch surfaces depicted
(48)
by Eq. (22) and (23).
u2
.s·1
= bll sign(s1) + b'2 S,
then Eq. (32) is satisfied, where Pij(i,j = are the elements of
P ,and Pi} (i, j
=
(39)
P22P33)]~O(40)
]523)2 +4(p32P23 -
L'l3 = P32P23
- ]522]533
~0
(4 J)
then Eq. (40) is satisfied. Because
~T--
- d 3 P13 ]
-
~
T
.
JT
P22P33= - - - PlJiJ22P3tiJ33 (43) - - - ~A;P]";P2;P32 d Pl2P2JiJ32PJ> Pl2P13P3tiJ33
l~'!"~~' -
-
p,,p,,p,,A,] _
-
-
-
-
P3.J52] = J7 PI2P23P32 -)-
~T~~
J
0~
+ b 21 sign(s 2) + b 22 S 2 Proposition 3 The system depicted as Eq.(J2) and Eq.(l4) under a disturbance-coplaner twodimensional complete configuration. can be asymptotically stabilized with the control law described by Eq. (49) and Eq. (50). In order to minimize cbattering in the controllers, the saturation function sat(-) given in Eq. (51) has
sat(s)
= {:/L'l
- - - P13P22fJ32fJ33 - - - Pl2fJl3P32P33 - - - PlzPI3P23P32
- -, PUPZtiJ33
~l ~LJ~22~33 PI2PLJPZzP23
] J
(51)
4. SPACECRAFT DETUMBLING SIMULATION Based on the aforementioned control law the simulation on the detumbling a rigid spacecraft under a incomplete configuration is provided. In the
simulation the principal inertia 11
= lOOkg.m 2 ,
= 200kg.m 2 , 13 = 300kg.m 2 , the number of the thrusters m = 3, and the contlguration matrix Iz
Cl (44 )
1::1:: s < -t,.
-1
Pl2P22P2:J133 Pl2PlJiJnP33 Pl.jjlJ;2J;23
l~~',AJ>" PI2Pz3PnP33
~
Rro+k3l ro Pro+2k 32 ro 1ro Pro
=
= [- d,p],
-
-T-~
been introduced in stead of sign(·).
[P2JP33 Pl3P33 Pl3P23] J (42-a) ]532 = [PnP32 Pl3P32 1513Pn] d (42-b) 15,21522] d (42-c) ]522 = [PnP32 PI2]521] Cl (42-d) /523 = [PnP33
-
~
U-=-(J)
1,2,3) are
If
J
(49)
Similarly,
1,2,3)
elements of P. Proof: Eq. (32) can be rewritten as
where Then
(48)
+ bllsign(sl) + bl2 s ,
d.d:A pJA.A3 - AAJ(JJ..P12 - AAJ ~O
]533
bl " bl 2 < 0
_WT (205 + k2]05 T Pro + 2k 22 ro,ro T Pro
=
Proposition 2 If
~] = [(D32 -
(47)
Let
=[:
o
~ ~~l I
is a disturbance-coplaner two-dimen sional complete configuration. The parameters
bZ2 and
~
b,t' b12
,
b 21
,
are 0.01.
8019
Copyright 1999 IF AC
ISBN: 0 08 043248 4
SLIDING MODE CONTROL FOR DETUMBLING RIGID SPACECRA. ..
14th World Congress ofIFAC
5.CONCLUSION
Table I Simulation condition Initial states
Parameter of controller
ill 10
-11A
-5.7
-5.4
5.71
5
1.1
10
The simulation condition is shown in Table 1.
..
..!!!
Cl
o
~ u
.. :. ---~-~-~-----~--:
..2
. .. -.~,,", . -:-~~-~'~'~.~'----~--~------..... ,
·,
~
,." ,,, ,,
L
III
~ ~
· ··
,
..
·
..
,
REFERENCE
-,.-. - - - - - ----
- - - -- ---:- - -- - - - - - --:- --- - - - - -
-10
..
B)-Tiles, c.l. and Isidori(l99 I), A. On the attitude stabilization of a rigid spacecraft. Automatica, VoI.27, pp.87-95. Coron, J.M. (1996). Explicit feedback stabilizing the attitude control of a rigid spacecraft with two control torque. Automatica, Vo1.32, pp.669-677. Krishnan, H.C 1994). Attitude stabilization of a rigid spacecraft using two control torques: a non-linear control approach based on the spacecraft attitude dynamics. Automatica, Vo1.30, pp.1023-1027 Morin,P. And C. Samson(l995). Time-varying exponential stabilization of the attitude of a rigid spacecraft with two control. In: Proceedings of the 34th Conference on Decision & Control, New Orleans,LA,Dec. Tsiotras, P. and J. Luo (1997). Reduced-effort control laws for underactuated rigid spacecraft. AIAA 97-
-150L--~2~O---4~0----:6"-O----:8:'-O--....,.-J100
t(Sec)
25.---~--~---------,
20 ..... - .. -.~ ......•...;. - ........ ~ ......... -; - ... - .. -.• •
, •
, ,
I I
•
I
I
I
E;
15
i
10
t----·~-..L~----L ..
-5
~-
-- -- --. -.~ -- -- -- --. ~-- -- -----. ~-- -- -- -- -- ~ -- -- -- ----
f :I\'T·.'I·~.·. • .I.··.··.··I·····•. · • ..
----~.-
.
0113.
..
.. ------:.- .. ----.-~ .. --------~---.---.-,
Wikor,P.J.(l989). Optimal thruster configurations for the GP-B spacecraft. In: Proceedings of IFAC Automatic Control in Aerospace. Tsukuba, Janpan Wiktor ,Po ].(1994). Minimum control authority plot: a tool for designing thruster systems. Journal of Guidance,Control and Dynamics, VoI.17, pp.9981006 Zhang Bing and Hongxin Wu, (1998a). Completeness of configuration of one-sided actuators system. In: Proceedings of the Chinese Control Conference '98.Ningbo Univ., Zhejiang P.R.China Zhang Bing(l998b). Fault-Tolerant Control of Reaction Control System and its applications. In: PHD dissertation of Chinese Academy of Space Technology. Beijing Institute of Control Technology.
-10~--~--~--~--~--~
o
20
40
60
80
100
t(Secl
Fig.2. Control torque vector u
1'
u
2'
u
3
15,---~--~--~--~---,
E c
'J 10 C'"
o
-
,
:
1\:
:
- - - or.\ ---. - -~ ------ - --.:., --- - ----- ~-, - -- --- -->' : ;
~ : Il i
\..
,
~ 5 ~'.-.-I" '\" .~-- ... ---.~ 13
This paper introduces some new definitions on the configuration of one-sided thruster, such as completeness and incompleteness, and divides the incomplete configuration into two cases: disturbancefree two-dimensional complete configuration for three-dimensional control and two-dimensional complete configuration with disturbance for threedimensional control. For the latter one a special case as disturbance-coplaner two-dimensional complete configuration for three-dimensional control is considered to stabilize the angular velocities of a rigid spacecraft, in which a Sliding Mode Control law is used. The simulations show the effectiveness of the approach .
.
'"
:,
o
o
i
20
40
60
100
80
t(Sec)
Fig.3. Thruster action torque vector u
c1 '
u
e2 '
u
c3
8020
Copyright 1999 IF AC
ISBN: 008 0432484