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Damping-Augmentation Mechanism for Flexible Spacecraft
Copyright © IFA C Automatic Control in Space
:-
DAMPING-AUGMENTATION MECHANISM FOR FLEXIBLE SPACECRAFT M. Inoue and K. Tsuchiya Centra l R esearch Laboratory, Mitsubishi Electric Corporation , Amagasaki, Hyogo , japan
Abs tract . A new design of a tti tude control system for flex i ble space craft was described; The con tro l system has two parts , a rigid body atti tude controller and a hinge con troller. The latter was supposed to suppress the osci lla ti on o f f lex ib le appendages . A rotati onal motion o f spacecraft that has a large flexible solar array was modeled and analysed . An impro v emen t was seen in the frequency responses and digital sim ul ations.
fl ex ible
space c raft,
atti tude
INTRODUCTION
control,
Bode
diagram .
The purpose of this paper is to show an other approach to design attitude control system o f flexible spacecrafts . A space craft is designed such that the flexible appendages are connected loosely with the rigid main body . Two parts, the rigid main body and the flexible appendages , are controlled separately . The mai n body is controlled accurately with high gain a nd the appendages are controlled by hinge controllers with low gain and enough damp ing, so that the hinge control torque may not inter f ere with the attitude motion .
Current designs o f spacecraft empl oy large flexible appendages . The dynamic bandwidth of t hese spacecrafts becomes lower because of their large inertia as well as relative ly weakened strength o f materials. On the other hand, the requ ireme nts o f po inting accuracy make the bandwidth higher. As the results, the elastic oscillat i o n, whose frequencies become c l os e to the bandwidth of the attitude control syste m, sometimes makes the system unstable. To overcome this pro blem, many co ntrol schemes have bee n proposed . Most of them are based on the modal control co ncepts ( Meirovitch, 1979 ; Balas, 1976. etc . ). But this clas s of cont roll ers are far from on -board com putati on on account o f their high order . Mor e ove r, as they make much of the fle xibi I i ty parameters , they often bec om e sensi ti ve to mo del error. These contro ll ers are intended to suppress all oscillational modes , though some of them need not be controlled acti v ely .
1-
\ \
On the othe r hand , Bryson( 1961) investigated the characteristics of the flexible spacecraft by the use of root loci . He indicated that if an actuator and a sens or are co- located flexible modes are stabilized by simple rate feedback. We agreed with his opi ni on that the control system should not be designed only by s o ftware .
Fig. 1 .
347
Sp acecra ft Model
M. Inoue and K. Tsuchiya
34 8
A sirlplified one axis a ttitud e motion of spacecraft whose flex i b l e appendage is connected by a hinge are mode led and ana lysed using classical cont r ol theory . Frequency response diagram (Bo de pl o t ) is s ti 11 a usefull tool when we design a c ontrol system of a fle x ib l e spacecraft. Finally , the usefulness o f thi s c ontroller was derlonstrated by digital s imulations.
Then
r~ODEL
OF SP ACECRAFT
.!. 2 -2 1J Z 2 + J *.ai 11:1 + ~ 1;1 ..
1 N
't
I
I
spacecraft that has a large s o l ar array connected by a hinge is illustrated i n Fig. 1. The coordinte ( y , z) are fi xed t o the main rigid body and the origin 0 coincides '.vi th the r:1ainbody mas s c enter. The atti tude B is the ro t a t ional angle o f the axes l y , z ) from the or b i t a l reference fra r:1e I. Y. Z l . A flexible ap pendage is connected to the rigid main bo dy at 0' which is apart from 0 by r . If no frictional torque ac ts on the hinge . bending moment of th e flexible ap pendage cannot be transferr ed t o the ma in body . Only the shearing f orc e di sturb the atti tude motion via torque arm r. Th e hinge angle lE, is defined as the de ri vative o f the flexible appendage a bo ut y a t 0 '. the tangent line at which i s y '-axis of ( y ', z ' ) frame . Then the bendin g de flection w(y ' . t) has the sam e mo des a s those of the appendage which is fixed at 0 ' like a cantilever . whose parameter s are derived v i a computer program.
T
1
= -
2
I
e• - 2l )f p[ ?
B
(\t'. r)
+ -
a
0
. ('iI' ,t) ]2
~ 'It
e.
J 0
2
P ( y '+r ) dy '
J
J*
J + mrl /2
bi
~ f Y' ~i dy '
A.
~ f
1
4\ dY'
= 1 . 2 , . .. . N)
(i
m • 1 a r e mass and l e ngt h o f res pective l y .
rW
1 N
V = 2-
i_I
2
2
(~)
q
ll
where W . are t h e n at ur a l f re quen ci e s of the i t h m6de . Thes e p a r amet ers S., 6 . and W . are derived by th e u s e o flfi n i t ~ elements1metho d.
f-----1~- --~
o f such a T is
t
~~'
1
1
where q . ( t) are mo d al coordinates an d '; are etgenfunctions sat i s fying Q
~ f 'Pi CP; d~' o
( 3)
o
the a ppenda ge
The elastic s train energ y is expressed as
is mome nt of ine r tia o f t h e main where IB body about f (y ' ) is mass per o and un i t length . Us ually th e e l astic de flection of the cantil e ve red array can be approximated as N w( y ' , t) = [ ~ . ( y' ) q . ( t ) (2)
j_,
•
6i
p y' 2 dy '
The attitude motion is s up pos e d to be constrained about X axis , a nd t h e deflection of the array is const rained in the ( y . Z) plane. The cro ss coup li ng effects o f the flexib l ity mo des and rigid body motion between three ax e s are neglect ed for simplicity . To derive the mathemat i cal mod e l spacecraft , the kinetic e n ergy evaluated as follows .
l
+
B
~:
J
A
•
2' ~I (Ji +ZS.~ +Z( bit t'A,)e)
(4)
wh ere r~.ATHD·ATICAL
2'1
T
Fig. 2 .
Bloc k Di a g ram
349
Damp in g-Augme ntati o n Me c hani sm Equations of motion of the spacecraft that has a hinge are derived from (4) and (5) as I
e"
+
*" J
e
J* "
5
..
; bi~
J
e. e" 1
+
")
+
r.i="" ,
e q i i
te
(
+
~
b iqi
t~
Ci)
= 0
($')
" qi
+
2
W i qi
( i =1 .2 •. . • • N) where t and t are control torques acting on the main body and the hinge respectively and e i 0i + r 6 i . In the case where the hinge is fi xed . terms containing the hinge angle celeration can be neglected from Eqs . and ( 3). The equations of motion of sp3cecraft are then reduced to
.
I
e
the ac ( 6)
the
( 'i) ( 10) (i = 1 . 2 • ...• N)
Fig . 2 shows the block diagram corresponding to Eqs . (6) . (7) and (3) . This system has two inputs t e and t ~ and two outputs and ~ . When the inner loop of the block diagram is neglected . it is reduced to the usual system that has a fixed hinge flexible appendage corresponding to Eqs . (9) and ( 10).
e
t~
( 12)
Fig. 3. is the proposed control system where the inner hinge angle control loop is closed by only the hinge angle signal. the outer attitude control loopuy only the atti tude signal. so that the atti tude for attitude sufficient information is controller . The hinge the form
controller
is
.
d ~
supposed
-
to
have
k ~
The fi rs t term ac ts as a damper and the second term as a spring. This controller may be composed of a spring and a damper passively . otherwise it needs an active servo mechanism. In the equation . k must be selected small not to interact with the atti t ude control loop and d should be large enough to absorb oscillating torques of the f le xibi lity modes .
This hinge controller reduces the system to an one input one output system . which is easily designed using a classical control theory .
The goal is to control 8 and ;- without sensing or estimating the modal coordinates qi .
e
CONTROL SYSTEM DESIGN Fig. 2. suggests that the flexible modes are dr i ven by and ~ Through the flexibility dynamics the disturbance torques are provided to the main body and to the hinge . After the disturbance torques are di vided into two paths. they are cancelled each other at the adder A. Especially. "hen the length r is zero . the spacecraft ~otion is independent of flexibility d ynamics .
e
e
+
Sp"'C eCfo.ft D~ no. ", ~ cs
S ----I
r---
I
I
I
I
I
I I I
I I I
I I I
This force
condition means that the shearing also doesn ' t disturb the attitude ~o tion as "ell as bending moment . Although such a requi remen t seems unreal izable . the flexibli ty feedback torques decreases when the length r is small compared to the system size . Generally . control torques te and tE should be synth esized wi t h the function of and ~ However . in our case . prec ise contro l is required on attitude motion while the coarse orientat i on is a d e quat e for hinge angle . For thos e r eas on s attit u de and hinge angle can be contro ll e d separate l y as
I I
I I I I
I
I
--------------------------~
D"""'M
.w..ec~
e
B •\
Fig . 3.
Block Di agram with Controllers
M. lnou e a nd K. Tsu c hi ya
3 50
e
The transfer function from to t, derived fror:-, Eqs . 16) , \ 7 ) and ; 5 ) as P G is ) 1
::>
'. .:here
'11
~
s- I P / - ,p 1 2 1 22
I -
N p
and
b;
L j=,
. 1"
"
P22
J +
d s
+
1
-
1
1
k S2
)
- r 6 .,2
~
"
L i-I
11"
22
:;
N
is
,W . S
-/
\
,
1
(S,+r~i)
-
i
's )2
W
&.2
N
L
i'"
1+ I
I
..,
W
' s l-
is a Laplace opel'ator .
s
In the case o f function becomes G ,s ) 2
fi x ed
hinge ,
the
transfer
----- s
2
P
where T 1 and T;:> are the time constants of the phase lead - network , TL is considered as the l ag time con s tant of a sensor or actuator and g is a feedback gain that is selected so that the crossover freq u en cy is abou t o . 15 rad l sec . Val ues o f these con tr olle r paraf'1eters are given in Table 3 . The Bode diagrams of the open l oo p transfer functions G ( sl · G IS) and G I s l . G/ISl are depicted in cFig . ~ and 6 '; ·, !here higher than 3rd f'1ode are neglected because of their less importance . These Bode diagrams indicate that when the hinge is fixec resonance peaks o f the 1st and the 2nd node exceed the 0 dB line . In ot her 'tJords , those modes are unstable because the phase ma rgin at the frequencies are negative . sho"ls On the other hand , however , Fig . 6 that the hinge controller suppress the pe a k heigh t under the 0 dB line . As the result , the flexible modes cannot be driven to unstable os c illation even by the co n ventional attitude controller .
Cl;)
11
It is apparent that G?( s) has zeros and poles upon the imaginary a x is of compl~x plane because G? I s ) is a function of s "" , so that the hinge fix ed spacecl'3ft has resonance frequencies where the the gains are infinite . On the other hand , zeros and poles of G ( s ) are on the left half 1 plane because of the damping coefficient d This means that the hinge controller augments the stability margin .
NUMERICAL EXAMPLES To make the function of the hinge control ler more apparent , let us gi ve a set of practical values to the spacecraft parame ters as listed in Table 1 . The flexibili ty parameters are also given in Table 2 , '~'here ; , are the daf'1pin g ratios of eoch mode . These fle x ibi li ty paraf'1eters '''ere cOf'1puted using well known structure analy si s program NASTRAN . The pa t terns of the eigen functions are depicted in Fig . 4 , where the dashed lines indicate the patterns before trans f omation .
W .)
1 . l7 rad "sec
Though appendage actual flexible 'tJas analysed in 3- di ension space , 've selected only the r:1o des which produce the dis turbance torque to the considering axis . The attitude contro l ler can be designed in a conventional mann er, where the spacecraft is s u pposed to be rigid . The transfer funct i on o f the cont r o ll er has the form as 3rd mode G (s)
c
g
1
+
T
L
s
3 . 9 7 rad / sec
(16' Fig . 4 .
Eigen f unctions of the flexible appendage
Damping-Augmentation Mechanism
GAIN
GAIN
dB
40
40
-40
-40
- BO
-BO
-120
-12 0
- 16 0
35 1
10 ·-
10'
d3
-~':"--- ----i
10"
10 '
10'
10'
10'
10 '
,'HAS[ de~ . , - - - - , - -- - - ,- - - -.-------,
o
!
-IBO
-IBO
-36 0
-3 60 10"
Fig .
10'
l.) .
10'
G
L'ocie Diagram o f l
c
f i x ed :1ing e)
i
I
i
I
10 '
10"
10"
10 '
1 0'
IS)
Fig . 6 .
'-----~-----L.__:_----'I-
! O'
Bode Diagrwn o f G, 's) Gc i s ) ( controlled hinge ) "
::r=------ J""" :IS---------=j" " -=__
,.:~''''-__ ..........................,J ""
.I.DLI_~ __. _--_~-__
v=~:J
'".DC :r _-----I~.!"" L..._ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _
':t:=
-1. 1
.D
l. D
I. D
' .D
11MEISEC)
Fig. 7 .
12 . 0
I S.D
11 . 0
.0
l.'
*10 '
Responce to Initial Rate ( fixed hinge)
1. 0
' .0
11MEISE C)
Fig. S .
12 . 0
1S.D
11 . 0
*1 0'
Response to Initial Rate (control led Hinge )
M. Inoue and K. Tsuchiya
352
To demonstrate the desirable characteristics , so me digital simulations were done . The results are seen in Figs . 7 and 5 . Fig . 7 shows the cas e where the hinge is f i xed and initial attitude rate o f 0 . 001 rad / sec is given . Fig . 5 shows the case ~ he re the hinge cont r oller is er.lployed and the initial condition is the same as Fig . 7 . As be i ng expe c ted , the flexib""i:y n odes diverge when the hinge is fixed.
TA5L::
Spacecraft Paramete rs
J
1 ')6'.'
kgr.
760
kgr.l
SO
'"
kg
S . .2
r.
i
r ____ ~~5
m________~1
! ______
TABLE 2
mode
2
flexibility Paramete rs
GU i
~i
Ll i
23 . 3
6 . 30
0 . 000 1
4 . 51
1. 46
0 . 0001
1. 63
1 . 94
0 . 0001
C, i
-----------------------------------0 . 77 s
2
1 . 37 s
3
3 . 97 s
TA BLE 3
-1 -1 -1
Controller Parameters
Tl
2 . 5 sec
T2
12 . 5 sec
TL
2 . 0 sec
g
k d
- 10 . 0 Nm / rad
10 . 0 150. 0
Nm/ r a d Nmsec/rad
CO NCLUSION This investigation showed that f or some types of flexible spacecraft the attitude controllers designed in a convent iona l manner worked well . A hinge controller was introduced to co nne ct the rigid main body and the flex i ble appendage for the purpose tha t i t absorbs the 0 S C i lla ti ng ac As the c eleration o f flexible appendage. results , the attit ude motion didn' t interfer with the flexibility dynamics and vice
versa . The stability was analized by the he l p o f classical control theo ry ·which provided us useful suggestions.
REFERENCES Bryson , A. and Wie , B. ( 1951 ) . Attitude Control of a Flexible Triangular Truss i n Space . preprint o f IFAC World Confference,vol.16 . Meirovitch , L. and Oz , H. (1979) . Obse rver Modal Control of Dual - Spin Flexible Spacecraft. J . Guidance and Control , vol . 2 , no . 2 . Ba la~~ ( 1975 ) . Feedback Control of Flexible Systems. IEEE T. Automatic Control , vol. 23 , no. 4 .
:-
DAMPING-AUGMENTATION MECHANISM FOR FLEXIBLE SPACECRAFT M. Inoue and K. Tsuchiya Centra l R esearch Laboratory, Mitsubishi Electric Corporation , Amagasaki, Hyogo , japan
Abs tract . A new design of a tti tude control system for flex i ble space craft was described; The con tro l system has two parts , a rigid body atti tude controller and a hinge con troller. The latter was supposed to suppress the osci lla ti on o f f lex ib le appendages . A rotati onal motion o f spacecraft that has a large flexible solar array was modeled and analysed . An impro v emen t was seen in the frequency responses and digital sim ul ations.
fl ex ible
space c raft,
atti tude
INTRODUCTION
control,
Bode
diagram .
The purpose of this paper is to show an other approach to design attitude control system o f flexible spacecrafts . A space craft is designed such that the flexible appendages are connected loosely with the rigid main body . Two parts, the rigid main body and the flexible appendages , are controlled separately . The mai n body is controlled accurately with high gain a nd the appendages are controlled by hinge controllers with low gain and enough damp ing, so that the hinge control torque may not inter f ere with the attitude motion .
Current designs o f spacecraft empl oy large flexible appendages . The dynamic bandwidth of t hese spacecrafts becomes lower because of their large inertia as well as relative ly weakened strength o f materials. On the other hand, the requ ireme nts o f po inting accuracy make the bandwidth higher. As the results, the elastic oscillat i o n, whose frequencies become c l os e to the bandwidth of the attitude control syste m, sometimes makes the system unstable. To overcome this pro blem, many co ntrol schemes have bee n proposed . Most of them are based on the modal control co ncepts ( Meirovitch, 1979 ; Balas, 1976. etc . ). But this clas s of cont roll ers are far from on -board com putati on on account o f their high order . Mor e ove r, as they make much of the fle xibi I i ty parameters , they often bec om e sensi ti ve to mo del error. These contro ll ers are intended to suppress all oscillational modes , though some of them need not be controlled acti v ely .
1-
\ \
On the othe r hand , Bryson( 1961) investigated the characteristics of the flexible spacecraft by the use of root loci . He indicated that if an actuator and a sens or are co- located flexible modes are stabilized by simple rate feedback. We agreed with his opi ni on that the control system should not be designed only by s o ftware .
Fig. 1 .
347
Sp acecra ft Model
M. Inoue and K. Tsuchiya
34 8
A sirlplified one axis a ttitud e motion of spacecraft whose flex i b l e appendage is connected by a hinge are mode led and ana lysed using classical cont r ol theory . Frequency response diagram (Bo de pl o t ) is s ti 11 a usefull tool when we design a c ontrol system of a fle x ib l e spacecraft. Finally , the usefulness o f thi s c ontroller was derlonstrated by digital s imulations.
Then
r~ODEL
OF SP ACECRAFT
.!. 2 -2 1J Z 2 + J *.ai 11:1 + ~ 1;1 ..
1 N
't
I
I
spacecraft that has a large s o l ar array connected by a hinge is illustrated i n Fig. 1. The coordinte ( y , z) are fi xed t o the main rigid body and the origin 0 coincides '.vi th the r:1ainbody mas s c enter. The atti tude B is the ro t a t ional angle o f the axes l y , z ) from the or b i t a l reference fra r:1e I. Y. Z l . A flexible ap pendage is connected to the rigid main bo dy at 0' which is apart from 0 by r . If no frictional torque ac ts on the hinge . bending moment of th e flexible ap pendage cannot be transferr ed t o the ma in body . Only the shearing f orc e di sturb the atti tude motion via torque arm r. Th e hinge angle lE, is defined as the de ri vative o f the flexible appendage a bo ut y a t 0 '. the tangent line at which i s y '-axis of ( y ', z ' ) frame . Then the bendin g de flection w(y ' . t) has the sam e mo des a s those of the appendage which is fixed at 0 ' like a cantilever . whose parameter s are derived v i a computer program.
T
1
= -
2
I
e• - 2l )f p[ ?
B
(\t'. r)
+ -
a
0
. ('iI' ,t) ]2
~ 'It
e.
J 0
2
P ( y '+r ) dy '
J
J*
J + mrl /2
bi
~ f Y' ~i dy '
A.
~ f
1
4\ dY'
= 1 . 2 , . .. . N)
(i
m • 1 a r e mass and l e ngt h o f res pective l y .
rW
1 N
V = 2-
i_I
2
2
(~)
q
ll
where W . are t h e n at ur a l f re quen ci e s of the i t h m6de . Thes e p a r amet ers S., 6 . and W . are derived by th e u s e o flfi n i t ~ elements1metho d.
f-----1~- --~
o f such a T is
t
~~'
1
1
where q . ( t) are mo d al coordinates an d '; are etgenfunctions sat i s fying Q
~ f 'Pi CP; d~' o
( 3)
o
the a ppenda ge
The elastic s train energ y is expressed as
is mome nt of ine r tia o f t h e main where IB body about f (y ' ) is mass per o and un i t length . Us ually th e e l astic de flection of the cantil e ve red array can be approximated as N w( y ' , t) = [ ~ . ( y' ) q . ( t ) (2)
j_,
•
6i
p y' 2 dy '
The attitude motion is s up pos e d to be constrained about X axis , a nd t h e deflection of the array is const rained in the ( y . Z) plane. The cro ss coup li ng effects o f the flexib l ity mo des and rigid body motion between three ax e s are neglect ed for simplicity . To derive the mathemat i cal mod e l spacecraft , the kinetic e n ergy evaluated as follows .
l
+
B
~:
J
A
•
2' ~I (Ji +ZS.~ +Z( bit t'A,)e)
(4)
wh ere r~.ATHD·ATICAL
2'1
T
Fig. 2 .
Bloc k Di a g ram
349
Damp in g-Augme ntati o n Me c hani sm Equations of motion of the spacecraft that has a hinge are derived from (4) and (5) as I
e"
+
*" J
e
J* "
5
..
; bi~
J
e. e" 1
+
")
+
r.i="" ,
e q i i
te
(
+
~
b iqi
t~
Ci)
= 0
($')
" qi
+
2
W i qi
( i =1 .2 •. . • • N) where t and t are control torques acting on the main body and the hinge respectively and e i 0i + r 6 i . In the case where the hinge is fi xed . terms containing the hinge angle celeration can be neglected from Eqs . and ( 3). The equations of motion of sp3cecraft are then reduced to
.
I
e
the ac ( 6)
the
( 'i) ( 10) (i = 1 . 2 • ...• N)
Fig . 2 shows the block diagram corresponding to Eqs . (6) . (7) and (3) . This system has two inputs t e and t ~ and two outputs and ~ . When the inner loop of the block diagram is neglected . it is reduced to the usual system that has a fixed hinge flexible appendage corresponding to Eqs . (9) and ( 10).
e
t~
( 12)
Fig. 3. is the proposed control system where the inner hinge angle control loop is closed by only the hinge angle signal. the outer attitude control loopuy only the atti tude signal. so that the atti tude for attitude sufficient information is controller . The hinge the form
controller
is
.
d ~
supposed
-
to
have
k ~
The fi rs t term ac ts as a damper and the second term as a spring. This controller may be composed of a spring and a damper passively . otherwise it needs an active servo mechanism. In the equation . k must be selected small not to interact with the atti t ude control loop and d should be large enough to absorb oscillating torques of the f le xibi lity modes .
This hinge controller reduces the system to an one input one output system . which is easily designed using a classical control theory .
The goal is to control 8 and ;- without sensing or estimating the modal coordinates qi .
e
CONTROL SYSTEM DESIGN Fig. 2. suggests that the flexible modes are dr i ven by and ~ Through the flexibility dynamics the disturbance torques are provided to the main body and to the hinge . After the disturbance torques are di vided into two paths. they are cancelled each other at the adder A. Especially. "hen the length r is zero . the spacecraft ~otion is independent of flexibility d ynamics .
e
e
+
Sp"'C eCfo.ft D~ no. ", ~ cs
S ----I
r---
I
I
I
I
I
I I I
I I I
I I I
This force
condition means that the shearing also doesn ' t disturb the attitude ~o tion as "ell as bending moment . Although such a requi remen t seems unreal izable . the flexibli ty feedback torques decreases when the length r is small compared to the system size . Generally . control torques te and tE should be synth esized wi t h the function of and ~ However . in our case . prec ise contro l is required on attitude motion while the coarse orientat i on is a d e quat e for hinge angle . For thos e r eas on s attit u de and hinge angle can be contro ll e d separate l y as
I I
I I I I
I
I
--------------------------~
D"""'M
.w..ec~
e
B •\
Fig . 3.
Block Di agram with Controllers
M. lnou e a nd K. Tsu c hi ya
3 50
e
The transfer function from to t, derived fror:-, Eqs . 16) , \ 7 ) and ; 5 ) as P G is ) 1
::>
'. .:here
'11
~
s- I P / - ,p 1 2 1 22
I -
N p
and
b;
L j=,
. 1"
"
P22
J +
d s
+
1
-
1
1
k S2
)
- r 6 .,2
~
"
L i-I
11"
22
:;
N
is
,W . S
-/
\
,
1
(S,+r~i)
-
i
's )2
W
&.2
N
L
i'"
1+ I
I
..,
W
' s l-
is a Laplace opel'ator .
s
In the case o f function becomes G ,s ) 2
fi x ed
hinge ,
the
transfer
----- s
2
P
where T 1 and T;:> are the time constants of the phase lead - network , TL is considered as the l ag time con s tant of a sensor or actuator and g is a feedback gain that is selected so that the crossover freq u en cy is abou t o . 15 rad l sec . Val ues o f these con tr olle r paraf'1eters are given in Table 3 . The Bode diagrams of the open l oo p transfer functions G ( sl · G IS) and G I s l . G/ISl are depicted in cFig . ~ and 6 '; ·, !here higher than 3rd f'1ode are neglected because of their less importance . These Bode diagrams indicate that when the hinge is fixec resonance peaks o f the 1st and the 2nd node exceed the 0 dB line . In ot her 'tJords , those modes are unstable because the phase ma rgin at the frequencies are negative . sho"ls On the other hand , however , Fig . 6 that the hinge controller suppress the pe a k heigh t under the 0 dB line . As the result , the flexible modes cannot be driven to unstable os c illation even by the co n ventional attitude controller .
Cl;)
11
It is apparent that G?( s) has zeros and poles upon the imaginary a x is of compl~x plane because G? I s ) is a function of s "" , so that the hinge fix ed spacecl'3ft has resonance frequencies where the the gains are infinite . On the other hand , zeros and poles of G ( s ) are on the left half 1 plane because of the damping coefficient d This means that the hinge controller augments the stability margin .
NUMERICAL EXAMPLES To make the function of the hinge control ler more apparent , let us gi ve a set of practical values to the spacecraft parame ters as listed in Table 1 . The flexibili ty parameters are also given in Table 2 , '~'here ; , are the daf'1pin g ratios of eoch mode . These fle x ibi li ty paraf'1eters '''ere cOf'1puted using well known structure analy si s program NASTRAN . The pa t terns of the eigen functions are depicted in Fig . 4 , where the dashed lines indicate the patterns before trans f omation .
W .)
1 . l7 rad "sec
Though appendage actual flexible 'tJas analysed in 3- di ension space , 've selected only the r:1o des which produce the dis turbance torque to the considering axis . The attitude contro l ler can be designed in a conventional mann er, where the spacecraft is s u pposed to be rigid . The transfer funct i on o f the cont r o ll er has the form as 3rd mode G (s)
c
g
1
+
T
L
s
3 . 9 7 rad / sec
(16' Fig . 4 .
Eigen f unctions of the flexible appendage
Damping-Augmentation Mechanism
GAIN
GAIN
dB
40
40
-40
-40
- BO
-BO
-120
-12 0
- 16 0
35 1
10 ·-
10'
d3
-~':"--- ----i
10"
10 '
10'
10'
10'
10 '
,'HAS[ de~ . , - - - - , - -- - - ,- - - -.-------,
o
!
-IBO
-IBO
-36 0
-3 60 10"
Fig .
10'
l.) .
10'
G
L'ocie Diagram o f l
c
f i x ed :1ing e)
i
I
i
I
10 '
10"
10"
10 '
1 0'
IS)
Fig . 6 .
'-----~-----L.__:_----'I-
! O'
Bode Diagrwn o f G, 's) Gc i s ) ( controlled hinge ) "
::r=------ J""" :IS---------=j" " -=__
,.:~''''-__ ..........................,J ""
.I.DLI_~ __. _--_~-__
v=~:J
'".DC :r _-----I~.!"" L..._ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _
':t:=
-1. 1
.D
l. D
I. D
' .D
11MEISEC)
Fig. 7 .
12 . 0
I S.D
11 . 0
.0
l.'
*10 '
Responce to Initial Rate ( fixed hinge)
1. 0
' .0
11MEISE C)
Fig. S .
12 . 0
1S.D
11 . 0
*1 0'
Response to Initial Rate (control led Hinge )
M. Inoue and K. Tsuchiya
352
To demonstrate the desirable characteristics , so me digital simulations were done . The results are seen in Figs . 7 and 5 . Fig . 7 shows the cas e where the hinge is f i xed and initial attitude rate o f 0 . 001 rad / sec is given . Fig . 5 shows the case ~ he re the hinge cont r oller is er.lployed and the initial condition is the same as Fig . 7 . As be i ng expe c ted , the flexib""i:y n odes diverge when the hinge is fixed.
TA5L::
Spacecraft Paramete rs
J
1 ')6'.'
kgr.
760
kgr.l
SO
'"
kg
S . .2
r.
i
r ____ ~~5
m________~1
! ______
TABLE 2
mode
2
flexibility Paramete rs
GU i
~i
Ll i
23 . 3
6 . 30
0 . 000 1
4 . 51
1. 46
0 . 0001
1. 63
1 . 94
0 . 0001
C, i
-----------------------------------0 . 77 s
2
1 . 37 s
3
3 . 97 s
TA BLE 3
-1 -1 -1
Controller Parameters
Tl
2 . 5 sec
T2
12 . 5 sec
TL
2 . 0 sec
g
k d
- 10 . 0 Nm / rad
10 . 0 150. 0
Nm/ r a d Nmsec/rad
CO NCLUSION This investigation showed that f or some types of flexible spacecraft the attitude controllers designed in a convent iona l manner worked well . A hinge controller was introduced to co nne ct the rigid main body and the flex i ble appendage for the purpose tha t i t absorbs the 0 S C i lla ti ng ac As the c eleration o f flexible appendage. results , the attit ude motion didn' t interfer with the flexibility dynamics and vice
versa . The stability was analized by the he l p o f classical control theo ry ·which provided us useful suggestions.
REFERENCES Bryson , A. and Wie , B. ( 1951 ) . Attitude Control of a Flexible Triangular Truss i n Space . preprint o f IFAC World Confference,vol.16 . Meirovitch , L. and Oz , H. (1979) . Obse rver Modal Control of Dual - Spin Flexible Spacecraft. J . Guidance and Control , vol . 2 , no . 2 . Ba la~~ ( 1975 ) . Feedback Control of Flexible Systems. IEEE T. Automatic Control , vol. 23 , no. 4 .