The study of the dynamics of the Spacecraft Control Laboratory Experiment (SCOLE) is extended to emphasize the synthesis of control laws for both the linearized system as well as the large amplitude slewing maneuvers required to rapidly reorient the antenna line of sight. For control of the system through small amplitude displacements from the nominal equilibrium position LQR techniques are used to develop the control laws. Pontryagin's maximum principle is applied to minimize the time required for the slewing of a general rigid spacecraft system. The minimum slewing time is calculated based on a quasi-linearization algorithm for the resulting two point boundary value problem. The effect of delay in the control input on the stability of a continuously acting controller (designed without considering the delay) is studied analytically for a second order plant. System instability can result even for delays which are only a small fraction of the natural period of motion

Bainum, Peter M.

Diarra, Cheick Modibo

Li, Feiyue

Reddy, A. S. S. R.

English

NASA Technical Reports Server

OR_U_t pAGE 15
e xi  AurY
, y,r T_I ,Ca,A!,.D UNI\FERS ITY
N89-13467
PRINCI PAL IbWESTICATOR :
CO- I NVESTI GATOR:
,q'I1IDENTS •
TE_!NICAL HCNITOR:
Dr. Peter M. Baintm_
Dr. A.S.S.R. Reddy
Cheick bfodibo Diarra and
Fe lyric Li
Mr. Jolm Young (baRC)
The Dyqlamics and Control of the In-Orbit
SCOLE Configuration - NASA- NSG 1414
ABSTRACJr
]]_e study of the dymamics of the Spacecraft Control Laboratory Emperi-
meat (SCOI,E) 1 is extended to emphasize the synthesis of control laws for both
the linearized system as well as the large amplitude slewing maneuvers required
to rapidly reorient the antenna line of sight. For control of the system
through small amplitude displacements from the nominal equilibrium position
IP,R teclmiques are used to develop the control laws. Pontryagin's ma×imt_n
principle is applied to minimize the time required for the slewing of a
general rigid spacecraft system. The minimt_n slewing time is calculated
based on a quasi-Iinearization algorithm for the resulting two point boundary
value problem. 2 _he effect of delay in the control input on the stability of
a coni-inuouslv acting controller (desiflmed without considering the delay)
i'._ studied analyti,:ally for a second order plant. System instability can
result even for 6elays which are only a small fraction of the natural period
of mo_ion. 3
T,i'.!or, L.;; _. m_d italakrishmm, A.V., "A Mathematical Problem ;rod a
,<pacecraft Control Laboratory Experin<_nt (SCOLE) used to Evaluate
Cocttt'ol Laws for Flexible Spacecraft ... NASA/IEEE Desigm Challenge,"
J:_. 1984.
"3
_,], !:<,kva_e c..nd i_ain_n, P.M., 'Ninimtm_ Time Attitude Slewing bkmeuvers
of a Rigid Spacecraft," accepted for presentation, AIAA 26th Aerospace
<_:-_,,_q ,\',eerin_. Reno >bvada, ,Jan 11-14, 19,q8
l_eddy, A.S.S.R. and Bainum, P.M., "Stability Analysis of Large Space
:;t:ucture Control Systems with Delayed lnput," 6th VP[ & SU/AIAA
,q>'mposium on Dynamics and Control of I,arge Stn_ctures, Blacksburg,
\,:a., Jtr_e 29 - July i, 1987.
https://ntrs.nasa.gov/search.jsp?R=19890004096 2020-03-20T04:39:55+00:00Z
b,
O_
Reflector
!
\
\
,\
¢%
_c
O Geo6enLer
FIGURE llI_l: TIlE 3-1) GE(_'.lt::'I'[<Y OI _' TIlE SCOLE
CONFICURATION IN ITS DEFORMED STATE
. Equatlom of Cotton
"Derived using a Newton--Euler approach
, Assumptions
- Reflector and Shuttle rigid
-Mast has constant cross-section
- It is assumed to undergo small elastic deformations only
- Its modal shapes in orbit are assumed to be the same as
those of an Identical:non-rotating beam,
Stability Analysis (Rigidized SCOLE)
A stability analysis of the rigldzed SCOLE was conducted for
the following configurations:
a) Rigid - no offset, Pitch motion decouples from roll
and yaw in the linear ranges, System not stable
b) Rigid - with offset parallel to roll axis, Pitch motion
still decouples from roll and yaw in the linear range,
System unstable,
c) Rigid - With both offsets (parallel to roll and pitch
axes), The motions in all 3 degrees of freedom are
coupled, System found to be unstable,
,Control Laws
Assumption: All the states of the system are available.
It was suggested by J.G. Lin that an intuitively appealing
practical approach to achieve the LOS pointing objective
is a two-stage procedure. (a) Slew as if riaid then,
(b) damp-out flexible dynamics.
.The linear regulator theorv used here to control
-the linear model of the rigidized SCOLE,
-The linear model of the actual SCOLE configuration including
the first fo,,;flexible modes of the mast.
Next
Preliminary slew maneuvers s+ rigtdized SCOLE.
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NININUN TINE nTTITUDE SLEI4ING
N_NEtlVERS OF n RIGID SPnCECRAFT
OBJECTIUE
. DEVELOP CONPUTnTIONAL TECHNIQUES TO SLEH
A GENERAL RIGID SPACECRAFT ( INCLHDING
RICIDIZED SCOLE ) FROMAN ARBITRARY INITIAL
ATTITUDE TO A FINAL REQUIREDATTITUDE
PRECISELY, AND SATISFYING THE FOLLO_4INC
CONDI TIONS:
. IN NININUN TINE
. THE CONTROLS HAVE _ATURATION LEVELS
METHODOLOGY
• THE MAXIMUM PRINCIPLE FROM OPTINNL CONTROL THEORY IS
nPPLIED TO THE EULER _ S DYNAMICAL EQUATIONS riND THE
QUATERNION EINEMATICAL EQUATIONS OF THE SYSTEM TO
DERIUE THE NECESSARY CONDITIONS FOR THE CONTROLS.
THIS LEADS TO THE TWO--POINT BOUNDARY--UALflE PROBLEM.
• AN INTEGRAL OF n QUADRATIC FUNCTION OF THE CONTROLS
IS USED AS A COST FUNCTION. BUT THE INTECRATION
PERIOD OF THIS INTEGRAL,, CALLED THE SLEWING TIME.
IS TO BE CHANGED UNTIL IT REACHES ITS MINIMUM U_LUE.
• THE RESULTING TPBUP IS SOLUED BY _ QUASILINEARIZATION
AL(_-ORITHM ( METHOD OF PARTICULAR SOLUTIONS ).
• EULER" S EICENA_IS ROTATION THEOREM IS USED TO
APPROXIMATELY DETERMINE THE INITIAL U_LUES OF THE
COSTATES _ND THE SLEMING TIME _S MELL AS THE
NOMINRL SOLUTIONS MHICH ARE UESD TO START THE
QUAS I LINEARIZ_T ION AL('-ORI THM.
• THE MINIMUM SLEMIN( _" TIME IS DETERMINED BY SHORTENIN( _-
THE TOTAL SLEMINC- TIME UNTIL AT L_"ST ONE OF THE
CONTROLS BECOMES A BAN,G--BANG TYPE.
.-_.J
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TORQUESxlO,O00 FT-LB
U X
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Uy, Uz
I Ux
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0.0
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FORCESxSO0 LBS
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i I L
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4.0
20.0
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5.0
0.0
ANGLES (DEG.)
!..............................ii: ....
0.0
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i .0 2.0 3.0 4.0
TIME (SEC.) /69
1.0
0.0
-I .0
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T 0 R_Q_U..E_Sxl O_,_Q.(3(3..._F_T_--L_B.......
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/
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0.0 3.0 6.0 9.0
30.0
20.0
I0.0
/ 70 TIME (SEC.)
COHCLUDI NG BENARES
• THE SLEWING MOTION NEED NOT BE RESTRICTED TO A
SINGLE A)_IS HNNEUVER.
• THE GUESSED STARTING U_LUE OF THE SLEWING TINE IS
UEBY CLOSE TO THE CONUERCED VALUE FOR THE SCOLE
EXANPLES AND SUBROUTINE USED HERE•
• THE GUESSED INITIAL UALUES OF THE COSTATES ABE
ADEQUATE FOR THE ALGORITHN TO CONUERGE.
• THE HETHODS USED HERE NnY BE IHPLEHENTED FOB
PRACTICAL CONTROL SOURCES WHICH NAY HAUE NOHE
CONS T RA I NT S .
• AN EXTENSION TO THE NININUN TINE SLEMING NOTION
OF THE SCOLE NODEL CONTAINING BOTH RIGID _ND
FLEXIBLE CONPONENTS IS PLANNED.
171
Appendix - Chapter II
Stability Analysis of Second Order System
with Delayed State Feedback
As a second order differential equation describes the dynamics of
a single mode of any large space structure, the stability analysis of such
a system with delayed state feedback is analyzed and the amount of delay
that can be tolerated by the system without becoming unstable is arrived
at analytically.
The differential equation of second order with state feedback can be
written as:
0. T " 2
Xi + 2_imiXi + mi Xi
where
Xi =
_i' =
h
= -k i(t-h) - kp;% t-h)
ith modal coordinate
ith natural frequency
ith mode inherent damping ratio
= rate feedback gain
= position feedback gain
= time delay
(1)
The feedback gains kr, kp are designed for the required stability and
transient response specifications without taking the delay into consideration.
!
The inherent damping ratio, Ei and the feedback gains, kr and _; will
give rise to five possible combinations as shown in Table 1 and are thus
analyzed separately for mathematical convenience and easy understanding.
172
' = 0, _ = 0 and kr > 0Case I: _i
The differential equation of the system can be written as"
"" 2
Xi + _i Xi -- - krXi(t-h) (3:)
!
Case _ i kr kP
I =0 >0 =0
II >0 >0 =0
III = 0 > 0 > 0
IV >0 =0 #0
V >0 >0 _0
Note: The remaining three combinations are
neither feasible nor of interest.
Table I: Feasible Combinations of _'i' kr'
for Stability Analysis
and the corresponding characterstic equation is given by:
s2"+ m_ + 2_imise'Sh = 0
where kr = 2_i_ i.
(4a)
The value of h for which the roots of equation (3) cross the imaginary
axis can be evaluated by substituting s = jm.
Thus
2 2
-_ + -x_iZ;'_-'_sir_h+ 2;i_i_cos_h = 0W i (4b)
For equation (4b) to be satisfied
and
sin_h = 0
2 2
-_ + 2t;_._-_cosoh = 0
I..k
(s)
Thus _h = _/2
and h =
mi [{i + / I+_i2 ]
(6)
' > 0 kr = 2{i_i and _ = 0Case II : {i '
The characteristic equation of the system described by equation (I)
is given by
C_-_2+zq_i_s_h)+ j(2q_i_+Zq_i_co_h)--o (7)
Thus co_h : -q/q
!
and h = cos'l(_il_i
For the case where {i<{i thesystem will always be stable since no value
of h exists for which the roots of (7) cross the imaginary axis.
V
of _ih versus _i for various values of ¢i is shown in Figure 2.1.
A plot
Case III:
or
Thus
!
_i=0, kp=kr> 0
The characteristic equation is given by
2+_2+k%.se" h kpe "s
S S+ h=0
(il-12+_krsinlh+kpco_h) + j{ _krcos_h-kpsin_h ) = 0
tan _h = _kr
(9)
(.10)
and 2 1 2r £[kJ+4JkZ+4k2] ] (119= _-[(2_+k ).+. x i r p
Plots of h_ i versus kr/_ i for various values o£ kp/_ 2 are shcwn in
Figure 2.2. It,can be seen here that these are many combinations o£ kp and
kr for which the roots of Eq. (I0) can cross the ImagJmary axis - i.e.
value of hm i which leads to instability.
tCase IV: _i >o, kr = 0, _ _ 0
The characteristic equation is given by
(_i2-_2+kpCOS_h) + j (2gi'_i_-_sir_h)=. 0 (12)
I
2g i _i _
Thus sir_h = (13)
+ + 2 2
and m2 = mi2(1.2_'i 2) mi2/[(1_2_[2 )2 (kp/m i ) ] (14)
I
The plots of h_ i versus kp/_ 2 for various values of _i are shown in
Figure 2.3
C_ev: q ,o,kr,o,b _p
The characteristic equation is given by
(_£2._2+_kr sir_h+kpcos_h)
+ j(2_ _i_+_krcos_h-kpsir_h) = 0
By equating the imaginary part to zero, _h can be evaluated as
(15)
"_h = sin "I
!
2g i _i_
{ "'_ kp2 +_ kr 2 .
_k r
) - tan -i (--_ ) (16)
Y
after substituting _h in the real part of equation (i_)
the following equation in the single unknown variable _ can be obtained
2
_i "m2+t_krSin (sin'ly-tan-l( _k''/'r))
kp
_kr
+ kp cos (sin'ly-tan'l(--_ )):o
_p
(17)
Using equations (17) and (16), the limiting value for given values of Ci'
kr, kp and _i can be determined. As the equation (17) is nonlinear,
numerical procedures may have to be used and thus the generalized plots
similar to the other cases may be obtained,
2-14
h_ i
'............ t° .............. ".............. .,,. ........ ,,,4. ......
Figure 2.1:
..................i ..J.... ...T .....
.3eo .4sa .600 ._so .gee ,.ese
active damping ratio
Plots of h_. vs { correspondence to Case
as a parameter, i
= inherent damping ,
0.6 0.8 1.0
....... .,............ ;.,! ............... ..;............................ :.
• J
,.'............................3,..................................................
•_ • \
, \ \
•_,:...........................\ !,i'................._,,
•_ ! \. : ,
.......... i ............ , ...... %. ...... _ .......... ._,.,:" : -- :4 ? .......................
\ i N ' ',i
N : %i " :" :
•k : "...
: . i \ \
: ........ i........ _.._ ........... ._i .......... ::-,.
• . ..
: 4......--...-.-. .....•':'_..._."
: : ,-.,.
1.200 1.360 l.GaO
!
II with _i
177
2-15
,gae ....... ............... ............... .............. i............. "
.,,,, f.......I..._-_ ......./¢....o...6....._..............._ ............_ .............:
I:.... /..... __._'_',.-'._'_,_. ........ -.............................. _..............
• SIlO ...................] _'." _ ...."*':.....................................................................! : : _ i ""_"_: "-"-- .............................:j .i ! i
I:I ':,,." _ ...: :",_ _ ! _ i
' .:,," i "_ _"_ _0.8 ! _ _
.41111 .......
'300 _"' ..... /" : _'.0 :
.zoo [I,-/_i_ ............................... i............................................: i:............................................................i
,_.,/':" ........ ! ............... ; ....... : : :
• I00 ....... ' ............... "............... :".............. i .............................. ............... i ..............
0.800" .............................. ' ...................
O,OeO .3so .?eo 1 .oso 1.480 1.750 z.loe 2.450 2.800 3.150 3.580
o
Figure 2.2 Plot of h_ vs kr/_ corresponding to Case III kr/_i
with kp/__i ias a parameter
2-16
b._ i
1 ,$t*
/
1/ 1.620
| .,448
t .2_0
1.080
.900
• ?20
.640
.368
.100
0.000
0.000 .400
Figure 2.3:
"............i ....................................................._ _i _ : .o..._i ........................]
: _ _ : i
: _ _. i '.._-"'-'- i ! 0,7 _ ]
,"............i.........' ....)..........X! ............._ .........'"__ ......... S ...........! .............:.............',1
f _ i ',i !\ ) - ) ",,.'./ _ 0.3 _ _
' "i i _ : i -./...__......._../.......'2..:" : : J
............................- ,.-.._ ........ .............. _ ............ _...... :..-,,..4 .............. ; ......
............ _k.............. i .............. 4. ........... ".".............. :'_ ......... :...-./. ..... _.-_..A ...... _ ............. 2_..,:1
.i....X.i _ -i.__ .-/---.i_/ :---.]
: . : " ""."" _'.W;,,. . " '..............":'""_'_".k_£,k.............; ..........
' .----".............I_-,--'--4
"__:::T_ -":I
.aee 1.2ee i.6n 2.aeo z.,ee 2.aee ,._e_ ,.,ee ,.**e
• kp/_
Plots of h_ i _ kp/_ 2 correspondence to Case IV with
¢_ as a parameter
/7_
2-17
180


The dynamics and control of the in-orbit SCOLE configuration

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