- Email: [email protected]
New Control Schemes for a Magnetic Attitude Control System
CopYTight
©
IFAC Automatic Control in Spac~ 1982
Noordwijk~rhout. Th~ N~th~r1ands
NEW CONTROL SCHEMES FOR A MAGNETIC ATTITUDE CONTROL SYSTEM K. Tsuchiya and M. Inoue Centra l R esearch Laboratory, Mitsubishi Electric Corporation, Amagasaki, Hyogo, 661 Japan
Abstract. In this paper, we propose two new control schemes for a magnetic control system. The first one is a component control scheme, where only an angular momentum error attributable to the secular component of a disturbance torque is controlled . The second one is a P-I control scheme, where an error offset of an angular momentum is reduced to zero . Stability criteria and performances of the control schemes are derived on the basis of the method of averaging. Keywords . Attitude control; artificial satellite; dynamic stability; time - varying system; method of averaging . an error offset of an angular momentum. The uncompensated momentum will results in a large wheel size or degrade a pointing accuracy . In order to overcome the disadvantages of the cross product scheme, two new control schemes are developed in this paper: For the first disadvantage, a component control scheme is developed, which compensates only for a secular component of disturbance torques. For the second disadvantage, a P-I control scheme is developed, which suppresses an error offset of an angular momentum caused by a secular component of disturbance torques . Because of the time varying coefficients associated with using the earth's magnetic field as a torquing source, a linear analysis of the equation of motion is of little value, the analysis of the control schemes is based on the method of averaging (Nayfeh, 1973).
INTRODU CTION For attainment of high pointing accuracy an attitude control system employing a bias momentum wheel and/or reaction wheels is favored: ~~eel torques are used for compensating for disturbance torques. The secular component of external disturbance torques would lead to saturation of the momentum capacity of the wheels . Provision for damp ing the excess wheel angular momentum is needed. A magnetic control system (utilizing interactions of the earth ' s magnetic field with magnets on the spacecraft) is widely used for an angular momentum control of the spacecraft . This type of control system has advantages such as smooth continuous control, unlimited mission life and absence of catastrophic failure mode. A popular control scheme for an angular momentum control is the cross product law (White and others, 1961; Alfriend, 1975, Weiss and others, 1977; Tsuchiya and others, 1981)
M
l:{
EQUATION OF MOTION Consider a spacecraft moving in a low altitude circular orbit. A reference frame (X, Y, 2) is fixed in the spacecraft: The spacecraft roll, pitch and yaw axes correspond to X, Y and 2 axes, respectively . To specify an attitude of the spacecraft, we introduce two reference frames; an orbital reference frame (X o , Yo, 2 0 ) and an inertia reference frame (Xi, Yi, 2i) (Fig. 1). The orbital reference frame is taken so that the Xo axis is in the direction of motion, the -Y o axis is normal to the orbit and the 20 axis points toward the center of the earth. The inertia refer ence frame (Xi, Yi, 2i) is taken to be coin cident with the orbital reference frame when the spacecraft is at the ascending node of the geomagnetic equator. It is assumed that the geornagnetic field can be represented by a tilted dipole: It is expressed in the orbital frame as
6h x B
where M is the dipole moment of the magnet , l:{ is a control gain, 6h is the differences between the wheel angular momenta and their nominal values and B is the earth magnetic field. Although the implementation is relatively simple, the cross product scheme has disadvantages: The control torque compen sates not only for its secular component of disturbance torques but for the entire disturbance torque in such an application. The power will be wasted r emoving i ner t ially periodic angular momentum and this , in t ur n, results in degrad ing the secular momentum damping efficiency and a large size of the torquer. On the o t her hand , this type of control scheme has a proportional characteris tic . A secular disturbance torque will cause 221
222
K. Tsuchiya and M. Inoue For this purpose, an information about an angular momentum error resulting from a secular component is needed. The following moving average provides an approximation to the angular momentum error caused by the secular component
NORTH Pa.E
EARTH
P~i
Wo 2n Wo
P;i
00 BIT
PLANE ~
ASCEf'ONG r-oDE CF
Pxi d t'
(4. a)
Pzi d t'
(4. b)
t-2 n / wo t
f
t-2 n / wo
The pitch magnet is activated by the following law
TrE ('£().IAGl'£TIC EOJATQR My
Fig. 1.
2n
t
f
- k xi sin wo t cosWo t [ (I-a ) Pxi + a P~i] - kzi (2sin 2 wo t - cos 2 wot) [(I-a )Pzi
Reference Frames. (5 ) (1. a)
Bzo
(1. b)
where Bxo, Bzo are the components of the geomagnetic field along the Xo , Zo axes, respectively, Bo is the coefficient of the geomagnetic field with small diurnal variation, Wo is the orbital rate, and at t=O the spacecraft is supposed to be at the ascending node of the geomagnetic equator. For the development of the control laws, it is also assumed that only small deviations occur in roll and yaw about the equilibrium configuration of the angular momentum perpendicular to the orbit plane. The linearlized equations of motion can be written in the orbital reference frame as
where kxi, kzi are the feedback gains and a is a parameter. The moving average involves a time lag, and will tend t o make the system unstable. The terms which c ontain Pxi and Pzi provide damping characteristics to the system. Substitution of Eq. (5) into Eqs. (3) leads to dP xi dt [(I-a )Pxi + a P~i1 - kzi (2sin 2 wo t - cos 2 wot) [(I-a )Pzi + a P~i1 )
+ Nxi dPzi dt
d dt P xo
(6.a)
Bo(2sin 2 wo t - cos2 w ot){-~i sin wot cos wot [(I-a )Pxi + a P~i] -
kzi (2sin 2 wo t - cos 2 wot) [(I-a )Pzi
+ aP~il ) + Nzi
(6.b)
and in the inertia reference frame as dPxi dt
(3. a)
dPzi dt where P xo , P zo are the components of the angular momentum vector along the Xo , Zo axes, Pxi, Pzi are the components along the Xi, Zi axes, My is the pitch magnet dipole, and Nxo , Nzo are the components of the external disturbance torque along the Xo , Zo axes, Nxi, Nzi are the components along the Xi, Zi axes.
DEVELOPMENT OF CONTROL SCHEMES (l)
Component Control Scheme
It is important to note that magnetic torquers need not to compensate for the entire disturbance torque but only for the secular component.
The torque exerted on the spacecraft by the pitch magnet is usually small; the variable Pxi, Pzi are assumed to change slowly. We can apply the method of averaging to Eqs. (6). Averaging over the period 2e/w o, we obtain the approximation equations for Pxi, P zi in the form dPxi dt dP zi dt where Nxio, Nzio are the secular components of a disturbance torque along Xi, Zi axes. Application of Laplace transformation to Eqs. (7) and (4) derives the characteristic equations 3 l_e-2nS/wo S+SkiB[(I-a)+a( 2nS/wo )]
o
(8)
A
~a gne ti c
Attitude Co ntr o l System
where k i repr esen t s k x i o r k z i ' Using Pad~ approxima ti on, Eq. (8) can be written
N" -- ----- s.D i-----
· - · - · - - - --; HX
H'
o Fro m Eq . derived
223
(9)
(9) , the stability co nditions are
31TBO
1 + 16w (1-2 c<)k i > O . o
(10)
The e rr or offset at tributabl e t o the secular component of a distu r ban ce t orque is obtained from Eqs. (7) .0
1jl"f I SE::
where Pi a nd Nio r ep resent Pxi (or Pzi) and Nxio (or Nzio) . Some t ypical responses of the sys tem a r e calcul ated by use of Eqs . (6). The sys tem parameters are listed in Table 1. The r esul ts are shown in Figs . 2 '" 4 . Fi gu res 2 a nd 3 shows s t eady state responses t o a disturban c e torqu e ;
5.0 ; " -NH' - -- -
. - ---- . ... - . -
7 4. 0
l i .:I
1 .0
(11)
Fig. 3 .
31.0
( ~.
0
H .C
*J0 '
Steady State Response (Cross Product Law) . k x i=k z i =15(Am2 /Nms) , c< =0 Upper Pxo , . .. P zo Lower My
._------,,,
5.' ~~"~''---------
01
.. -5 . ' L _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _-'
------------:H'
I Q1 FlH,ul
1.1 1A
·'Vvvv~------1
'+l 1W i
-1. '
1.1,l:1D!!.:·c..·!!.!T!!.··!!·,,'- - - -- - - - - - - - -- - - - , .T
L ___ ._____________
....J
.- -- - - - - , ",IX
-1.1 L _ __ _ __ _ _ _ _ _ _ _ _ _ _ _ _-'
, .• (' ,!.!l.:./',!!"!1..'~\--------------, 0'"
1/
1"'11 .,
,
Fig. 2.
'.1
1'.0
H.D
TIM£ IS£CI
11 . 0
H.D
H.D
----------:::~---------__1
..
'.1
* 10 •
Steady State Response (Component Control Scheme). k x i=k x i=15(Am 2 / Nms), a =0.8 Upper P xo , Pzo Middle M~ Lower Pxi, P~i
... ,::: 92 _ ....
offl
-1.' L _ _ _ _ _ _ _ _ _ _ _ __ _ _ __
-1.' L I _ __
..
~
Fig. 4.
" .0
24 . 0
TIMEISEC I
n .D
* 10 •
'D . a
.J
H .D
Transient Response (Compone nt Control Scheme) . k x i=k z i=15(Am2 /Nms), a =0. 8 Upper Pxo , P zo Middle M~ Lower Pxi. P~ i
K. Tsuchiya and M. Inoue
224
d
4.1 sinwo t - 0 . 46 sin2wo t
Nx i
(xlO - 3
- 0 . 84 s in 3wo t
dt Pxi
Nm)
1 Txi
t
f
(14 . a)
+ Nx i o ~ P .
(xlO - 3 Nm)
+ 0 . 84 cos3wo t
Pxi d t' )
0
0.33 - 1. 4 coswot + 0 . 46 cos2wo t
Nzi
dt
Figure 2 corresponds t o t he componen t con tr ol scheme and Fig . 3 co rre sponds t o the c ro ss product law, i. e . , 0.=0 . It should be n o ted that a requir ed t o r que r s i ze fo r the component control scheme is considerable smal l in comparison wi t h t hat f o r the c r oss pr o du c t law . Fi gure 4 shows a transient r esponse of th e compo nen t contr ol scheme . (2 )
3 - SBokxi(Pxi +
z~
3 - S Bokzi (P z i +
1 Tzi
t
f
P zi d t' )
0
(14. b)
+ NZio
..,
S.,
"1
"I
P-I Cont r ol Scheme
One of th e desriable characteristics of a magnetic cont r ol system is t o suppress the error offse t a ttri butable to the secul ar compone nt of a disturbance t o rqu e . For this purpose, consider t he following co ntrol scheme . Ny
- kxi[Pxi + s inwot
1 Txi
j
-5 , 0
IC
1
RH''''-Z
.T
1.1
t Pxi d t' ] cos wo t
0
- kzdPzi + 1 Tzi
-1.2
t
J
Pzi dt ' ]
0
.,
I.'
1' . 0
1' . 0
1IHE I SEC )
(2sin 2wo t - cos 2wot)
1
Txi
Fi g . S .
Steady State Response (P-I Control Scheme) . k x i=k z i=lS( Am 2 /Nms) , Txi=Tzi= 3(sec) Upper -- Pxo ' . . . P zo Lowe r -- My
t
J Pxi 0
" .0
40 . 0
*I 0 '
(12)
Subs titution of Eq . (12) int o Eqs . (3) leads to
+
12.0
dt ' ]coswo t sinwo t
s . 0 I,!.!IO'-"...c·"'·=-.'- - - -- - - - - - - - - - - - - - - - , "' "I
(13.a)
d
1.1 r'-IO,--'-,"",1""',-,,'-'-.1- - -- - - - -- ---------, or
dt P z i
1
- kzi[P z i + T zi
f 0
~
t
-1. 1 L -_ _ _ _ _ __ _ _ __ _ _ _ _ _
P zi dt']
.0
I.'
11.0
1' ,0
1lHE ISEC)
32.0
I
~
40.0
".D
* 10 '
(13. b)
As mentioned above, the method of averaging
Fig. 6.
Ste ady State Response (Cross Product Law).
is su cc essfull y applied to Eqs. (13): Averag ing over the period 2n/w o , we obtain an approximation equations t o Eqs. (13) in the fonn
kxi=kzi=lS (Am2 /Nms) Upper -- Pxo , ... P zo Lower -- My
A Magnetic Attitude Control System where Nxio, Nzio are the secular components of a disturbance torque along the Xi, Zi axes. From Eqs. (14), it is straightforward to derive the characteristic equations
o
(15)
where ki and Ti represent kxi (or kzi) and Txi (or Tzi)' From Eq. (15), the following formulas can be derived Stability conditions (16) Damping factor 0 i (17)
Some t ypical responses are calculated by use of Eqs . (13). The system parameters are listed in Tab le 1. The steady state responses t o a secular component of a disturbance torque are shown in Figs . 5, 6:
225
Figure 5 corresponds to the proposed P-I control scheme and Fig. 6 corresponds to the cross product scheme. The e rror offse t attributable to a secular component of a disturbance torque is shown to be suppressed for the case of the P-I control scheme. Figure 7 shows a transient response of the P-I control scheme.
CONCLUSION In this paper, two new contro l schemes for a magnetic control system have been proposed; component control scheme and P-I control scheme. The first one is suitable for a spacecraft in a high disturbance torque, where magnitude of inertial periodic compo nents of a disturbance torque are also large, for which a magnetic t orquer needs not to compensate. The second one is suitable for a bias momentum spacecraft in a high secular disturbance torque where the error offset directly degrade the pointing accuracy .
REFERENCES
o ,
Nzi
~
0.8 x 10
-3
Nm
5.' r----"'"·'-"'- - - - - - - -- - - -
- - - ' HX HI
- -- - -- - - ---,01
., -1.1 L-_ _ __ _ _ _ _ _ _ _ __
...
., Fig. 7.
1"a
_ _ _ _ _-'
31,0
14 .0
lIMEISEC)
40. D
41.0
*10'
Transient Response (P-I Control Scheme).
kxi~kzi~15(Am2/Nms), Txi~Tzi~3(sec) Upper Lower
TABLE 1.
-
P xo ' My
... P zo
System Parameters
Parameter
Value
Unit
Bo
0.3
x
10-4
Wb/m 2
Wo
1
x
10- 3
rad/S
White, J.S., Shigemoto, F.H., and Bo urquin, K. (1961), Satellite Attitude Control Utilizing the Earth's Magnetic Field, NASA TN-D1068, Aug. Alfriend, K.T. (1975), Magnetic Attitude Control System for Dual-Spin Sate llites, AlAA J. 13, 817-822. Weiss, R., Rodden, J.J., and Hendricks, R.J. (1978), SEASAT-A Attitude Control System, J. Guidance and Control, 1, 6-13. Tsuchiya, K. Inoue, M., Wakasugi, N., and Suzuki, T. (1981), Attitude Control System with Wheels and Magnetic Torquers, 76.2 8th IFAC Wo rld Congress, Kyoto , Japan. Nayfey, A.H. (1973), Perturbation Hethods, Wiley, New York .
©
IFAC Automatic Control in Spac~ 1982
Noordwijk~rhout. Th~ N~th~r1ands
NEW CONTROL SCHEMES FOR A MAGNETIC ATTITUDE CONTROL SYSTEM K. Tsuchiya and M. Inoue Centra l R esearch Laboratory, Mitsubishi Electric Corporation, Amagasaki, Hyogo, 661 Japan
Abstract. In this paper, we propose two new control schemes for a magnetic control system. The first one is a component control scheme, where only an angular momentum error attributable to the secular component of a disturbance torque is controlled . The second one is a P-I control scheme, where an error offset of an angular momentum is reduced to zero . Stability criteria and performances of the control schemes are derived on the basis of the method of averaging. Keywords . Attitude control; artificial satellite; dynamic stability; time - varying system; method of averaging . an error offset of an angular momentum. The uncompensated momentum will results in a large wheel size or degrade a pointing accuracy . In order to overcome the disadvantages of the cross product scheme, two new control schemes are developed in this paper: For the first disadvantage, a component control scheme is developed, which compensates only for a secular component of disturbance torques. For the second disadvantage, a P-I control scheme is developed, which suppresses an error offset of an angular momentum caused by a secular component of disturbance torques . Because of the time varying coefficients associated with using the earth's magnetic field as a torquing source, a linear analysis of the equation of motion is of little value, the analysis of the control schemes is based on the method of averaging (Nayfeh, 1973).
INTRODU CTION For attainment of high pointing accuracy an attitude control system employing a bias momentum wheel and/or reaction wheels is favored: ~~eel torques are used for compensating for disturbance torques. The secular component of external disturbance torques would lead to saturation of the momentum capacity of the wheels . Provision for damp ing the excess wheel angular momentum is needed. A magnetic control system (utilizing interactions of the earth ' s magnetic field with magnets on the spacecraft) is widely used for an angular momentum control of the spacecraft . This type of control system has advantages such as smooth continuous control, unlimited mission life and absence of catastrophic failure mode. A popular control scheme for an angular momentum control is the cross product law (White and others, 1961; Alfriend, 1975, Weiss and others, 1977; Tsuchiya and others, 1981)
M
l:{
EQUATION OF MOTION Consider a spacecraft moving in a low altitude circular orbit. A reference frame (X, Y, 2) is fixed in the spacecraft: The spacecraft roll, pitch and yaw axes correspond to X, Y and 2 axes, respectively . To specify an attitude of the spacecraft, we introduce two reference frames; an orbital reference frame (X o , Yo, 2 0 ) and an inertia reference frame (Xi, Yi, 2i) (Fig. 1). The orbital reference frame is taken so that the Xo axis is in the direction of motion, the -Y o axis is normal to the orbit and the 20 axis points toward the center of the earth. The inertia refer ence frame (Xi, Yi, 2i) is taken to be coin cident with the orbital reference frame when the spacecraft is at the ascending node of the geomagnetic equator. It is assumed that the geornagnetic field can be represented by a tilted dipole: It is expressed in the orbital frame as
6h x B
where M is the dipole moment of the magnet , l:{ is a control gain, 6h is the differences between the wheel angular momenta and their nominal values and B is the earth magnetic field. Although the implementation is relatively simple, the cross product scheme has disadvantages: The control torque compen sates not only for its secular component of disturbance torques but for the entire disturbance torque in such an application. The power will be wasted r emoving i ner t ially periodic angular momentum and this , in t ur n, results in degrad ing the secular momentum damping efficiency and a large size of the torquer. On the o t her hand , this type of control scheme has a proportional characteris tic . A secular disturbance torque will cause 221
222
K. Tsuchiya and M. Inoue For this purpose, an information about an angular momentum error resulting from a secular component is needed. The following moving average provides an approximation to the angular momentum error caused by the secular component
NORTH Pa.E
EARTH
P~i
Wo 2n Wo
P;i
00 BIT
PLANE ~
ASCEf'ONG r-oDE CF
Pxi d t'
(4. a)
Pzi d t'
(4. b)
t-2 n / wo t
f
t-2 n / wo
The pitch magnet is activated by the following law
TrE ('£().IAGl'£TIC EOJATQR My
Fig. 1.
2n
t
f
- k xi sin wo t cosWo t [ (I-a ) Pxi + a P~i] - kzi (2sin 2 wo t - cos 2 wot) [(I-a )Pzi
Reference Frames. (5 ) (1. a)
Bzo
(1. b)
where Bxo, Bzo are the components of the geomagnetic field along the Xo , Zo axes, respectively, Bo is the coefficient of the geomagnetic field with small diurnal variation, Wo is the orbital rate, and at t=O the spacecraft is supposed to be at the ascending node of the geomagnetic equator. For the development of the control laws, it is also assumed that only small deviations occur in roll and yaw about the equilibrium configuration of the angular momentum perpendicular to the orbit plane. The linearlized equations of motion can be written in the orbital reference frame as
where kxi, kzi are the feedback gains and a is a parameter. The moving average involves a time lag, and will tend t o make the system unstable. The terms which c ontain Pxi and Pzi provide damping characteristics to the system. Substitution of Eq. (5) into Eqs. (3) leads to dP xi dt [(I-a )Pxi + a P~i1 - kzi (2sin 2 wo t - cos 2 wot) [(I-a )Pzi + a P~i1 )
+ Nxi dPzi dt
d dt P xo
(6.a)
Bo(2sin 2 wo t - cos2 w ot){-~i sin wot cos wot [(I-a )Pxi + a P~i] -
kzi (2sin 2 wo t - cos 2 wot) [(I-a )Pzi
+ aP~il ) + Nzi
(6.b)
and in the inertia reference frame as dPxi dt
(3. a)
dPzi dt where P xo , P zo are the components of the angular momentum vector along the Xo , Zo axes, Pxi, Pzi are the components along the Xi, Zi axes, My is the pitch magnet dipole, and Nxo , Nzo are the components of the external disturbance torque along the Xo , Zo axes, Nxi, Nzi are the components along the Xi, Zi axes.
DEVELOPMENT OF CONTROL SCHEMES (l)
Component Control Scheme
It is important to note that magnetic torquers need not to compensate for the entire disturbance torque but only for the secular component.
The torque exerted on the spacecraft by the pitch magnet is usually small; the variable Pxi, Pzi are assumed to change slowly. We can apply the method of averaging to Eqs. (6). Averaging over the period 2e/w o, we obtain the approximation equations for Pxi, P zi in the form dPxi dt dP zi dt where Nxio, Nzio are the secular components of a disturbance torque along Xi, Zi axes. Application of Laplace transformation to Eqs. (7) and (4) derives the characteristic equations 3 l_e-2nS/wo S+SkiB[(I-a)+a( 2nS/wo )]
o
(8)
A
~a gne ti c
Attitude Co ntr o l System
where k i repr esen t s k x i o r k z i ' Using Pad~ approxima ti on, Eq. (8) can be written
N" -- ----- s.D i-----
· - · - · - - - --; HX
H'
o Fro m Eq . derived
223
(9)
(9) , the stability co nditions are
31TBO
1 + 16w (1-2 c<)k i > O . o
(10)
The e rr or offset at tributabl e t o the secular component of a distu r ban ce t orque is obtained from Eqs. (7) .0
1jl"f I SE::
where Pi a nd Nio r ep resent Pxi (or Pzi) and Nxio (or Nzio) . Some t ypical responses of the sys tem a r e calcul ated by use of Eqs . (6). The sys tem parameters are listed in Table 1. The r esul ts are shown in Figs . 2 '" 4 . Fi gu res 2 a nd 3 shows s t eady state responses t o a disturban c e torqu e ;
5.0 ; " -NH' - -- -
. - ---- . ... - . -
7 4. 0
l i .:I
1 .0
(11)
Fig. 3 .
31.0
( ~.
0
H .C
*J0 '
Steady State Response (Cross Product Law) . k x i=k z i =15(Am2 /Nms) , c< =0 Upper Pxo , . .. P zo Lower My
._------,,,
5.' ~~"~''---------
01
.. -5 . ' L _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _-'
------------:H'
I Q1 FlH,ul
1.1 1A
·'Vvvv~------1
'+l 1W i
-1. '
1.1,l:1D!!.:·c..·!!.!T!!.··!!·,,'- - - -- - - - - - - - -- - - - , .T
L ___ ._____________
....J
.- -- - - - - , ",IX
-1.1 L _ __ _ __ _ _ _ _ _ _ _ _ _ _ _ _-'
, .• (' ,!.!l.:./',!!"!1..'~\--------------, 0'"
1/
1"'11 .,
,
Fig. 2.
'.1
1'.0
H.D
TIM£ IS£CI
11 . 0
H.D
H.D
----------:::~---------__1
..
'.1
* 10 •
Steady State Response (Component Control Scheme). k x i=k x i=15(Am 2 / Nms), a =0.8 Upper P xo , Pzo Middle M~ Lower Pxi, P~i
... ,::: 92 _ ....
offl
-1.' L _ _ _ _ _ _ _ _ _ _ _ __ _ _ __
-1.' L I _ __
..
~
Fig. 4.
" .0
24 . 0
TIMEISEC I
n .D
* 10 •
'D . a
.J
H .D
Transient Response (Compone nt Control Scheme) . k x i=k z i=15(Am2 /Nms), a =0. 8 Upper Pxo , P zo Middle M~ Lower Pxi. P~ i
K. Tsuchiya and M. Inoue
224
d
4.1 sinwo t - 0 . 46 sin2wo t
Nx i
(xlO - 3
- 0 . 84 s in 3wo t
dt Pxi
Nm)
1 Txi
t
f
(14 . a)
+ Nx i o ~ P .
(xlO - 3 Nm)
+ 0 . 84 cos3wo t
Pxi d t' )
0
0.33 - 1. 4 coswot + 0 . 46 cos2wo t
Nzi
dt
Figure 2 corresponds t o t he componen t con tr ol scheme and Fig . 3 co rre sponds t o the c ro ss product law, i. e . , 0.=0 . It should be n o ted that a requir ed t o r que r s i ze fo r the component control scheme is considerable smal l in comparison wi t h t hat f o r the c r oss pr o du c t law . Fi gure 4 shows a transient r esponse of th e compo nen t contr ol scheme . (2 )
3 - SBokxi(Pxi +
z~
3 - S Bokzi (P z i +
1 Tzi
t
f
P zi d t' )
0
(14. b)
+ NZio
..,
S.,
"1
"I
P-I Cont r ol Scheme
One of th e desriable characteristics of a magnetic cont r ol system is t o suppress the error offse t a ttri butable to the secul ar compone nt of a disturbance t o rqu e . For this purpose, consider t he following co ntrol scheme . Ny
- kxi[Pxi + s inwot
1 Txi
j
-5 , 0
IC
1
RH''''-Z
.T
1.1
t Pxi d t' ] cos wo t
0
- kzdPzi + 1 Tzi
-1.2
t
J
Pzi dt ' ]
0
.,
I.'
1' . 0
1' . 0
1IHE I SEC )
(2sin 2wo t - cos 2wot)
1
Txi
Fi g . S .
Steady State Response (P-I Control Scheme) . k x i=k z i=lS( Am 2 /Nms) , Txi=Tzi= 3(sec) Upper -- Pxo ' . . . P zo Lowe r -- My
t
J Pxi 0
" .0
40 . 0
*I 0 '
(12)
Subs titution of Eq . (12) int o Eqs . (3) leads to
+
12.0
dt ' ]coswo t sinwo t
s . 0 I,!.!IO'-"...c·"'·=-.'- - - -- - - - - - - - - - - - - - - - , "' "I
(13.a)
d
1.1 r'-IO,--'-,"",1""',-,,'-'-.1- - -- - - - -- ---------, or
dt P z i
1
- kzi[P z i + T zi
f 0
~
t
-1. 1 L -_ _ _ _ _ __ _ _ __ _ _ _ _ _
P zi dt']
.0
I.'
11.0
1' ,0
1lHE ISEC)
32.0
I
~
40.0
".D
* 10 '
(13. b)
As mentioned above, the method of averaging
Fig. 6.
Ste ady State Response (Cross Product Law).
is su cc essfull y applied to Eqs. (13): Averag ing over the period 2n/w o , we obtain an approximation equations t o Eqs. (13) in the fonn
kxi=kzi=lS (Am2 /Nms) Upper -- Pxo , ... P zo Lower -- My
A Magnetic Attitude Control System where Nxio, Nzio are the secular components of a disturbance torque along the Xi, Zi axes. From Eqs. (14), it is straightforward to derive the characteristic equations
o
(15)
where ki and Ti represent kxi (or kzi) and Txi (or Tzi)' From Eq. (15), the following formulas can be derived Stability conditions (16) Damping factor 0 i (17)
Some t ypical responses are calculated by use of Eqs . (13). The system parameters are listed in Tab le 1. The steady state responses t o a secular component of a disturbance torque are shown in Figs . 5, 6:
225
Figure 5 corresponds to the proposed P-I control scheme and Fig. 6 corresponds to the cross product scheme. The e rror offse t attributable to a secular component of a disturbance torque is shown to be suppressed for the case of the P-I control scheme. Figure 7 shows a transient response of the P-I control scheme.
CONCLUSION In this paper, two new contro l schemes for a magnetic control system have been proposed; component control scheme and P-I control scheme. The first one is suitable for a spacecraft in a high disturbance torque, where magnitude of inertial periodic compo nents of a disturbance torque are also large, for which a magnetic t orquer needs not to compensate. The second one is suitable for a bias momentum spacecraft in a high secular disturbance torque where the error offset directly degrade the pointing accuracy .
REFERENCES
o ,
Nzi
~
0.8 x 10
-3
Nm
5.' r----"'"·'-"'- - - - - - - -- - - -
- - - ' HX HI
- -- - -- - - ---,01
., -1.1 L-_ _ __ _ _ _ _ _ _ _ __
...
., Fig. 7.
1"a
_ _ _ _ _-'
31,0
14 .0
lIMEISEC)
40. D
41.0
*10'
Transient Response (P-I Control Scheme).
kxi~kzi~15(Am2/Nms), Txi~Tzi~3(sec) Upper Lower
TABLE 1.
-
P xo ' My
... P zo
System Parameters
Parameter
Value
Unit
Bo
0.3
x
10-4
Wb/m 2
Wo
1
x
10- 3
rad/S
White, J.S., Shigemoto, F.H., and Bo urquin, K. (1961), Satellite Attitude Control Utilizing the Earth's Magnetic Field, NASA TN-D1068, Aug. Alfriend, K.T. (1975), Magnetic Attitude Control System for Dual-Spin Sate llites, AlAA J. 13, 817-822. Weiss, R., Rodden, J.J., and Hendricks, R.J. (1978), SEASAT-A Attitude Control System, J. Guidance and Control, 1, 6-13. Tsuchiya, K. Inoue, M., Wakasugi, N., and Suzuki, T. (1981), Attitude Control System with Wheels and Magnetic Torquers, 76.2 8th IFAC Wo rld Congress, Kyoto , Japan. Nayfey, A.H. (1973), Perturbation Hethods, Wiley, New York .