1. INTRODUCTION
Recently, satellite lifetimes have increased, and the pointing
requirements have become more stringent for observation missions.
By applying a magnetic bearing wheel (MBW) as the attitude
control actuator of a satellite, instead of an ordinary ball bearing
wheel, it is expected that the lifetime of the bearing will increase and
the disturbance of the wheel will decrease ; therefore, we have
developed a MBW with inclined magnetic poles ; this enables the
5DOF magnetic bearing to be composed of six electromagnets and
six displacement sensorsP
[1]`[4]
PD
In order to equip a satellite with the MBW, the equations of
motion of the MBW are accurately obtained through various
examination results and then based on these equations, the equations
of motion of the MBWsatellite system are formulated and the
motion property of the system is considered.
With regard to the former equations of motion of the MBW and
by modeling the asymmetric magnetic bearing stiffness caused by
the difference in the electromagnet property as well as disturbance
factors of the magnetic bearing, we derived the equations of motion
that accurately describe the actual motion propertyP
[5]
P.
On the other hand, with regard to the motion property of the
MBWsatellite system, it is known that the crossfeedback control
of the MBW rotor gimbal angle in the magnetic bearing controller
makes the satellite motion unstable. Inoue and NinomiyaP
[6]
P
proposed a method in which the integral of the satellite angular rate
estimated from the MBW rotor gimbal angle is fed back to the
magnetic bearing controller for the stabilization of both the MBW
and the satellite motion. However, the equations of motion derived
2
by them were not entirely accurate; therefore, a detailed analysis is
required for practical applicationDMoreover, if the disturbance
feedback controller proposed in our reportP
[3]
P is applied, the motion
property of the system must be considered.
This paper deals with the dynamic interactions between the
satellite and the MBW. First, based on the equations of motion of
the MBWP
[5]
P, the equations of motion of the MBWsatellite system
in which the MBW rotor is coupled with the satellite by the
magnetic bearing forces and torques are formulated. Second, a
stability analysis based on these equations is considered.
2. EQUATIONS OF MOTION OF THE
MBWSATELLITE SYSTEM
2E1 Reference Frames and Symbols The reference frames
and symbols in the following discussions are defined as shown in
Figure 1 and as follows.
W FMBW reference frame
( , , W W W X Y Z = principal axes of inertia of the MBW)
B Fsatellite reference frame
( , , B B B X Y Z principal axes of inertia of the satellite)
subscript on the upperleft sideFreference frame to describe a
variable
B M , B I Fsatellite mass, tensor of inertia
W M , W I FMBW rotor mass, tensor of inertia
=MBMW(MB+MW)Fconversion mass
B v Fvelocity of the center of mass of the satellite
[ ]T
B Bx By Bz = Fsatellite angular rate
W v Fvelocity of the center of mass of the MBW rotor
W FMBW rotor angular rate
[ ]T
c cx cy cz v = v v v Fvelocity of the center of mass of the
MBWsatellite system
zw Fangular rate of the MBW rotor spin
[ ]T
w w w xw yw d = x y z F5DOF MBW rotor displacement
[ ]T
x y z f = f f f Fmagnetic bearing forces
Fig. 1 Definition of reference frames
[ ]T
x y = Fmagnetic bearing torques
0 0 0 0 [ ]T
WB w w w r = x y z Fposition of the center of mass of the
MBW rotor relative to that of the satellite under the condition
d = 0
2E2 Formulation of Equations of motion In this study,
the equations of motion of the MBWsatellite system are
formulated by Kanefs methodP
[7],[8]P. For convenience of
formulation, the following assumptions are applied in the
following discussions.
EThe MBW and the satellite are rigid bodies, and the
MBW rotor is coupled with the satellite by magnetic
bearing forces and torques.
EThe gravity and gravitygradient torque are negligible.
Degrees of Freedom and Reference Frame The
degrees of freedom of the MBWsatellite system are the
6DOF of the system and 5DOF of the relative motion.
Here, these degrees of freedom are defined as
E6DOF of the systemFTranslation of the center of mass of
the MBWsatellite system (3DOF), rotation of the
satellite (3DOF)
E5DOF of the relative motionFTranslation (3DOF) and
rotation(2DOF) of the MBW rotor relative to the
satellite
In particular, if the 6DOF of the system are selected as
stated above, the translational motion can be separated from
the rotational motion in the equations of motion.
In addition, to delete the satellite attitude angle from the
equations of motion, the reference frame for describing the
equations of motion is determined with regard to the satellite
reference frame B .
Formulation by Kanefs Method The generalized
speeds ( 1 11) j u j= ? and generalized forces ( 1 j Q j= ?
11) of the MBWsatellite system are given as
Generalized Speeds :[ B
j cx u v B
cy v B
cz v B
Bx
B
By B
Bz B
w x?? B
w y?? B
w z?? B
xw
??
B ]
yw
??
(1)
Generalized Forces :[0 j Q 0 0 0 0 0
B
x f B
y f B
z f B
x B ]
y (2)
The equations of motion of the MBWsatellite system can be
obtained by Kanefs method as follows:
( )
( )0
CB B
CW
B B j B B j
j B CB
B B j B B j
W CW W
Q + ? + ?
+ ? + ? =
f v n
f v n
?? ??
@@?? ??
(j =1?11) (3)
where
B
B f?? , B
CB n?? Finertia force, inertia torque around the center of mass
of the satellite
B
W f?? , B
CW n?? Finertia force, inertia torque around the center of mass
B
XBBB
YBBB
ZBBB
W X BWB
YBWB
ZBWB
MBW Satellite
rotor
3
of the MBW rotor
CB
B j B
v = vB uj , B
B j B
B j = u Fpartial velocity of
B
B v Cpartial angular velocity of B
B with respect to
the generalized speed j u
CW
B j B
W j v = v u , B j B
W W j = u F partial velocity
of B
W v Cpartial angular velocity of B
W with respect
to the generalized speed j u
Moreover, the tensors of inertia are defined as
B [ ]
B Bij I = J (i, j ={1,2,3}={x,y,z}) (4)
W [ , , ]
W W W PW I = diag J J J (5)
by substituting the components of inertia forces, inertia torques,
partial velocities, and partial angular velocities into equation (3); by
neglecting the 2ndorder terms or more, the linearized equations of
motion of the MBWsatellite system are given as follows.
0 0
2 2
0 0 0 0 0 0 { ( )} { } { } ( ) 0 Bxx W w w Bx Bxy w w By Bzx w w Bz w w w w W xw PW zw By yw J +J + y +z ?? + J ?x y ?? + J ?z x ?? ?z ??y?? +y ??z?? +J???? +J +?? =
c(61)
0 0
2 2
0 0 0 0 0 0 { } { ( )} { } ( ) 0 Bxy w w Bx Byy W w w By Byz w w Bz w w w w W yw PW zw Bx xw J ?xy ?? +J +J +z +x ?? +J ?yz ?? +z??x?? ?x??z?? +J???? ?J +?? =
c(62)
0 0
2 2
0 0 0 0 0 0 { } { } { ( )} 0 Bzx w w Bx Byz w w By Bzz PW w w Bz w w w w J ?z x ?? + J ?y z ?? + J +J + x + y ?? ?y ??x?? +x ??y?? = (63)
0 0 ( ) w By w Bz w x ??x?? +??z ???y =f (64)
0 0 ( ) w Bz w Bx w y ??y?? +??x ???z =f (65)
0 0 ( ) w Bx w By w z ??z?? +??y ???x =f (66)
( ) ( ) W Bx xw PW zw By yw x J ?? +???? +J +?? = (67)
( ) ( ) W By yw PW zw Bx xw y J ?? +???? ?J +?? = (68)
Here, for brevity of expression, the subscript at the upperleft side is
deleted. Note that equation (6) and the following discussions are
described within the satellite reference frame B .
Magnetic Bearing Forces and Torques The magnetic
bearing forces and torques, which are shown at the righthand side
of equations (64)`(68), depend on the MBW controller. When
only the magnetic bearing controller is applied as the MBW
controller, the magnetic bearing forces and torques are given asP
[5]
P
_ [ ( ) ( ) ] x X DR w PR w x dist f = ?K K D x?? +K K x + f (71)
_ [ ( ) ( ) ] y Y DR w PR w y dist f = ?K K D y?? +K K y + f (72)
_ [ ( ) ( ) ] z Z DZ w PZ w z dist f = ?K K D z?? +K K z + f (73)
_
[ ( ) ( ) ]
( ) ( )
D xw P xw
x Y
r zw yw a zw c yw
x dist
K D K K
K
K D K W
= ? ??? + ???
?? ? + ??
+
??
@@??
@@@
(74)
_
[ ( ) ( ) ]
( ) ( )
D yw P yw
y X
r zw xw a zw c xw
y dist
K D K K
K
K D K W
= ? ??? + ???
?? + ? ??
+
??
@@??
@@@
(75)
where
_ _ , dist dist f ? ? Fexcitation force, torque caused by the disturbance
factors
, , X Y Z K K K Fmagnetic bearing stiffness considering the
difference in the electromagnet property
, / , / , DR PR DZ PZ D P K K K K K K Fcontrol gain of the radial
translation/axial translation/radial rotation
,r a K K Fcrossfeedback gain for the stabilization of the
gyroscopic motion
( ), ( ), ( ) c D K W Ffrequency transfer function of the
components of the magnetic bearing controller
irefer to appendix for these internal structuresj
Equations (6) and (7) are the equations of motion of the
MBWsatellite system with only the magnetic bearing controller.
On the other hand, when the disturbance feedback controller is
added to the magnetic bearing controller, the magnetic bearing
forces and torques are given as P
[5]
_
1 2 ( )[ ( ) ( ) ]
x xmag X W DR PR
Ru w Ru w
f f KM K K
K C x C y
= + ~
@@???? ? ????
(81)
_
2 1 ( )[ ( ) ( ) ]
y ymag Y W DR PR
Ru w Ru w
f f KM K K
K C x C y
= + ~
@@???? + ????
(82)
_ ( ) ( ) z z mag Z W DZ PZ Zu w f =f +KMK K@KC ??z?? (83)
_
1 2
2 1
[ ( ) ( ) ]
( )
[ ( ) ( ) ]
x xmag Y D P
W u xw u yw
PW zw u xw u yw
K K K
J C C
K
J C C
Ƀ Ƀ
= + ~
?? ? + ??
? ?
??? + ??
???? ????
?? ??
@@
(84)
_
2 1
1 2
[ () () ]
( )
[ ( ) ( ) ]
y ymag X D P
W u xw u yw
PW zw u xw u yw
K K K
J C C
K
J C C
Ƀ Ƀ
= + ~
?? ? ? ??
? ?
??? ? + ??
???? ????
?? ??
@
@@
(85)
where
_ _ , mag mag f ? ? Fmagnetic bearing force, torque with only the
magnetic bearing controlleriequation (7)j
( ), ( ), ( ) Rui Zu ui C C C Ffrequency transfer function of
components of the disturbance feedback controller
irefer to appendix for these internal structuresj
Equations (6) and (8) are the equations of motion of the MBWu
u
u
u
u
4
satellite system with both the magnetic bearing controller and
disturbance feedback controller.
3. STABILITY ANALYSIS OF THE MBWSATELLITE
SYSTEM
Based on the equations of motion of the MBWsatellite system
formulated in chapter 2, the stability of the MBWsatellite system is
considered for two cases: (1) with only the magnetic bearing
controller and (2) with both the magnetic bearing controller and
disturbance feedback controller.
3E1 Without The Disturbance Feedback Controller
With regard to the equations of motion (6) and (7)Cthe physical and
control parameters of the MBW are shown in Table 1. On the other
hand, the physical parameters of the satellite are assumed as shown
in Table 2.
Table 1 Physical and control parameters of the MBW
MBWB 7.58 kg KBXB 3.413 ~ 10 P
? 1
P
JBWB 4.24 ~ 10 P
? 2
PkgmP
2
P
KBYB 3.238 ~ 10 P
? 1
P
JBPWB 7.70 ~ 10 P
? 2
PkgmP
2
P
KBZB 3.325 ~ 10 P
? 1
P
KBPRB 42.0 rad/s BiB 1.257 rad/s
KBPZB 38.0 rad/s BdB 1.257 ~ 10P
3
P rad/s
KBPB 38.0 rad/s ւB B 502.7 rad/s
KBDRB 2.50 ~ 10 P
3
P Ns/m KBIRB 1.0 ~ 10 P
? 2
P m/N
KBDZB 2.75 ~ 10 P
3
P Ns/m KBIZB 1.0 ~ 10 P
? 2
P m/N
KBDB 14.4 Nms/rad KBIB 5.0 rad/Nm
KBrB 4.78 ~ 10 P
? 2
P NmsP
2
P/radP
2
P
BlpfB 12.57 rad/s
KBaB 3.82 Nms/radP
2
P
BdisB 6.283 ~ 10 P
? 4
P rad/s
Table 2 Physical parameters of the satellite
MBBB 1000 kg JBByzB ?20 kgmP
2
P
JBBxxB 1200 kgmP
2
P
JBBzxB 50 kgmP
2
P
JBByyB 900 kgmP
2
P
xBw0B ?0.20 m
JBBzzB 1000 kgmP
2
P
yBw0B 0.30 m
JBBxyB 30 kgmP
2
P
zBw0B ?1.00 m
With regard to equations (6) and (7), by substituting the solution
given as
[? ? ? ? ? ? ? ?]T t
Bx By Bz w w w xw yw x = x y z e (9)
into the homogeneous equation in which the excitation force and
torque caused by the disturbance factors are neglected, the following
equation is derived.
E()x= 0 (10)
where E() R8~8 is the coefficient matrix of the solution
obtained using x , and the characteristic equation of the
MBWsatellite system is given as
det E( ) = 0 (11)
In general, the integrator of the magnetic bearing controller makes
the MBW precession unstable; however, the crossfeedback control
of the MBW rotor gimbal angle can stabilize this precessionP
[9]
P. On
the other hand, the satellite nutation is affected by the crossfeedback
control of the MBW rotor gimbal angleP
[6]
P. The root loci of these
(a) MBW precession
(b) Satellite nutation
Fig. 2 Root loci of the MBWsatellite system with an ordinary
magnetic bearing control
motions based on equation (11) are shown in Figure 2. Here, all the
motions of the MBWsatellite system, except the motions shown in
Figure 2, are stable; therefore, the ordinary magnetic bearing
controlleriequation (7)jmakes only the satellite nutation unstable.
In order to clarify the instability factors responsible for satellite
nutation, approximate characteristic roots of the MBWsatellite
system are derived. Approximations to the equations of motion
(61,2),(67,8), and (74,5) of the MBW rotor gimbal angle
, xw yw and the satellite angular rate , Bx By are applied as
E, , , , w w w Bz x y z ; the product of inertia of the satellite are
negligible.
Ethe principal moment of inertia of the MBW rotor W J is
negligibleD
Ein the magnetic bearing controller, the derivative controlC
crossfeedback control gain r K for the stabilization of the
MBW nutation, and the phase delay of the controller are
negligiblei ( ), ( ) 1 c W D = jD
E, B Bxx Byy XY X Y J ?J ?J K ?K ?K
For an approximate characteristic equation of motion of the satellite
nutationCequation (11) is simplified as
2
2
( ) 0
det 1 0
0 1
B B
C B P P i
B
j h J h J
K jh JK jK
h Jh h
?? ? ?
? ?
?? + + + ??=
? ?
?? ? ??
c(12)
6000 rpm
0 rpm
0 rpm
0 rpm
6000 rpm
6000 rpm
~10P
5
P
5
where h=JPW zw gives the MBW rotor angular momentum;
P XY D P K K K K = , the proportional control gain; C XY K =K ~
a zw K , the crossfeedback gain of the gimbal angle; and =
, Bx By xw yw +j? = +j? , the complex representation. In
addition, for convenience of derivation of an approximate
characteristic root by the perturbation method, the state variable of
the system is given as
( ) T
B x=??? j? h J dt?? (13)
The infinitesimal term of equation (12) is ( )2 B h J ; therefore,
by 1storder perturbation of this term, the approximate characteristic
root of the satellite nutation S
is given as
{ }
{ }
4 2
2 2 2 2 2
( ) (2 )
( ) (2 )
S
B
B B P i C B P
B B B P i C B P
j h
J
h J J K K h j h h J K
J J J K K h h h J K
=
+ ? ? +
+
+ + +
@@
(14)
The approximate root locus of the satellite nutation is shown in
Figure 3; compared to the root locus in Figure 2(b), approximate
characteristic root S
can completely describe the satellite
nutation.
Because equation (12) is the complex representation, an
imaginary number of the characteristic root based on equation (14)
and Figure 3 is a positive number. This shows that the direction of
the satellite motion and that of the MBW rotor angular momentum
are the same.
Moreover, equation (14) shows that both the crossfeedback
control C K of the MBW rotor gimbal angle and the integrator i
in the ordinary magnetic bearing controller are the instability factors
responsible for the satellite nutation.
For the stabilization of satellite nutation, the
crossfeedback control of the satellite angular rate is
assumed to be effective. With this crossfeedback control,
the magnetic bearing control torque is given as
S =? j?K (15)
where S K denotes the crossfeedback gain of the satellite angular
rate and x y = + j? , the ordinary magnetic bearing control
torque. The characteristic equation corresponding to equation (12) is
given as
( )2 0
det ( ) 0
0 1
B B
C
S Pi
S B P
B
j h J h J
K
hhK j hhh K J K jKh
J h
?? ? ?
? ?
?? ? + ??
?? + ? + ??=
? ?
? ?
?? ? ??
c(16)
Fig. 3 Approximate root locus of satellite nutation with an
ordinary magnetic bearing control
By 1storder perturbation of the ( )2 B h J term of equation (16),
an approximate characteristic root S
with crossfeedback control
of the satellite angular rate is given as
{ }
{ }
3
2 2 2 2
( ) ( ) [(2 ) ]
( ) [ (2 ) ]
S
B
S B B P i C S B P
B B B P i C S B P
j h
J
h h K J J K K h j h h h K J K
J J J K K h h h h K J K
=
? + ? ? ? +
+
+ + ? +
c(17)
The crossfeedback gain S K is larger than the MBW rotor angular
momentum h ; hence, the satellite nutation can be stabilized.
However, in general, the magnetic bearing controller cannot
observe the satellite angular rate. Therefore, in practice, the observer
estimates the satellite angular rate, and the crossfeedback controller
of the estimated angular rate is applied.
The state equations of the approximated model corresponding to
equation (12) are expressed as
[ ]
0 1
0 0 1
1 0
B
d j h
dt J
? ? ? ? ? ? ? ? ?
?? ??=?? ?? ?? ??+??? ??
? ?
= ? ?
? ?
(18)
Based on equation (18), a minimalorder observer of the satellite
angular rate is given asP
[10]
P
2 1
?
B
dz Lz L j L
dt J h
z L
? ?
= + ?? + ?
? ?
= +
(19)
where z denotes the state variable of the observer; L , the
observer gain; and ? , the estimated satellite angular rate.
Because the natural angular frequency of the satellite is almost
equal to B h J in equation (14), the observer gain can be given as
( 1)
B
L h
J
=? @@> (20)
By substituting equation (20) into equation (19), the estimated
satellite angular rate is given as
? 1
B B
hs j
Js h Js h
? ? + ?
= +
+ +
(21)
where the capital letters denote Laplace transforms. However, the
0 rpm
6000 rpm
~10P
5
P
6
6000 rpm
0 rpm
0 rpm
~10P
4
P
0 rpm
6000 rpm
6000 rpm
orders of the rightside terms in equation (21) are estimated as
hs?10?1??(?1+j?)?102 (22)
Therefore, the first term on the righthand side can be neglected, and
the satellite angular rate can be estimated by magnetic bearing
torques, as follows.
? 1
B
j
Js h
? + ?
?
+
(23)
Equation (23) shows that the satellite angular rate can be estimated
by lowfrequency components of the magnetic bearing torques
transformed in the complex plane.
Based on the above discussion, a new magnetic bearing controller
(motion of radial rotation) for the stabilization of the satellite nutation
is proposed, as shown in Figure 4. In this controller, a broken line
block is added to the ordinary magnetic bearing controllerP
[5]
P; based
on Figure 4, the magnetic torques are expressed as
_ _ ( ) ( )( ) x xmag Y X y mag =FɃ ?G K K (241)
_ _ ( )( ) ( ) y X Y xmag ymag =G K K +FɃ (242)
where
[ ]
[ ]2 2
( ) ( )
( )
( )
B B S
B S S
J h J K h
F
J K h K
+ ? ?
=
? ? +
(251)
[ ]2 2
( )
( )
( )
S B
B S S
G K J h
J K h K
+
=
? ? +
(252)
In this case, equations (6),(71)`(73), and (24) are the equations of
motion of the MBWsatellite system.
Fig. 4 Block diagram of the proposed magnetic bearing
control
With regard to equations (6), (71)`(73), and (24), the
numerical analysis of the root loci of the MBW precession and the
satellite nutation are shown in Figure 5.
Here,
3
500
( )2
S
B Bxx Byy
K Nms rad
J J J
? =
? = ??
? = +
(26)
Figure 5 shows that both the MBW motion and the satellite motion
become stable by the proposed magnetic bearing controller.
(a) MBW precession
(b) Satellite nutation
Fig. 5 Root loci of the MBWsatellite system with the
proposed magnetic bearing control
3E2 With The Disturbance Feedback Controller The
stability of the system in which the disturbance feedback controller
is added to the proposed magnetic bearing controlleriFigure 4j is
considered.
In this case, the magnetic bearing torques are given as equation
(27); equations (6),(81)`(83), and (27) are the equations of
motion of the MBWsatellite system.
( ) ( )
( ) ( )
_ _
1 2 1 2
1 2 1 2
( ) ( )( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
x xmag Y X y mag Y D P
W u u xw u u yw
PW zw u u yw u u xw
F G K K K K K
J F C G C G C F C
K
J F C G C G C F C
Ƀ
= ? + ~
?? ??? + + + ?? ??
? ?
??? ??? ? + + + ????
???? ????
?? ?? u
@@@@
(271)
( ) ( )
( ) ( )
_ _
1 2 1 2
1 2 1 2
( )( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
y X Y x mag y mag X D P
W u u yw u u xw
PW zw u u xw u u yw
G K K F KK K
J F C G C G C F C
K
J F C G C G C F C
Ƀ
= + + ~
?? ??? + ? + ?? ??
? ?
??? ??? + + + ????
???? ????
?? ?? u
@@@@
(272)

0,1, 2,4, 5,
j m z
em
m
e ?
= ? ?
??
XYP cmd K c
c s
+
2
1
P z Js ?j?Js
*
XYM cmd K
_
0,1, 2,4, 5, _
z
z
j m
biasm
j m
m biasm
e
e
?
+
? ?
= ? ? ?
? ?
???+ ???
??
D K
cmd i
P
K s
s
+
d
d
s
s
+
 
r z ?j?K
a z j?K
?
cmd
1
B
j
Js h
? + ?
S + ?j?K
?
7
6000 rpm
0 rpm
0 rpm
~10P
4
P
0 rpm
6000 rpm
6000 rpm
0 rpm
6000 rpm
6000 rpm
Based on these equations of motion, the numerical analysis of the
root loci of the MBW precession, satellite nutation, and }1N
disturbance feedback controller are shown in Figure 6 with the same
parameters in Figure 5.
A comparison between Figure 5 and Figure 6 shows that the
motion properties of the MBW rotor precession and the satellite
nutation are not affected by the disturbance feedback controller, and
they are found to be stable. It is assumed that the third term on the
righthand side of equation (27)Cwhich is caused by the disturbance
feedback controller, affects only the uN componenti}1N
component in Figure 6j.
On the other hand, the disturbance feedback controller, which is
added to the proposed magnetic bearing controller, remains stable;
therefore, the MBWsatellite system is stable with the disturbance
feedback controller.
(a) MBW precession
(b) Satellite nutation
(c) }1N disturbance feedback control
Fig. 6 Root loci of the MBWsatellite system with the
proposed magnetic bearing control and
disturbance feedback control
4. CONCLUSION
The dynamic interactions between the satellite and the
MBW with inclined magnetic poles are considered. Based
on the stability analysis of the MBWsatellite system, it is
shown that both the crossfeedback control of the rotor
gimgal angle and integrator in an ordinary magnetic bearing
controller make the satellite nutation unstable.
In order to solve this problem, a new magnetic bearing controller
is proposed. In this controller, the crossfeedback control of the
satellite angular rate estimated by the minimalorder observer from
the magnetic bearing control torques is added to the ordinary
magnetic bearing controller. The proposed magnetic bearing
controller can stabilize both the MBW and the satellite motion,
including the satellite nutation; further, even if the disturbance
feedback controller is added to the proposed magnetic bearing
controller, the MBWsatellite system remains stable.
REFERENCES
[1] Saito, M. et al., Development of Magnetic Bearing
Wheel (MBW) with Inclined Magnetic Poles (1st
Report, Magnetic Bearing Control and Compensation
of Surface Distortion of Sensors and Magnetic Poles),
Transaction of the Japan Society of Mechanical
Engineers, Series C, Vol. 71, No. 705 (2005), pp.
1429?1437.
[2] Saito, M. et al., Development of Magnetic Bearing
Wheel (MBW) with Inclined Magnetic Poles (2nd
Report, Analysis of Disturbance Factors and Motion of
MBW), Transactions of the Japan Society of
Mechanical Engineers, Series C, Vol. 72, No. 716
(2006), pp. 1345?1353.
[3] Saito, M. et al., Development of Magnetic Bearing
Wheel (MBW) with Inclined Magnetic Poles (3rd
Report, Low Disturbance Control Based on Distur
Bance Feedback), Transactions of the Japan Society of
Mechanical Engineers, Series C, Vol. 72, No. 715
(2006), pp.698?705.
[4] Saito, M. et al., Development of Magnetic Bearing
Wheel (MBW) with Inclined Magnetic Poles (4th
Report, Fine Tuning of Magnetic Bearing Controller),
Transactions of the Japan Society of Mechanical
Engineers, Series C, Vol. 73, No. 730 (2007),
pp.1691?1698.
[5] Saito, M. et al., Equations of Motion of a Magnetic
Bearing Wheel (MBW) with Inclined Magnetic Poles,
Transactions of the Japan Society of Mechanical
Engineers, Series C, under contribution.
[6] Inoue, M. and Ninomiya, K., Stability and Control of
Attitude Motion of a Satellite Equipped with a
Magnetic Wheel, Transactions of the Society of
Instrument and Control Engineers, Vol. 25, No. 10
(1989), pp.1104?1110.
[7] Kane, T. R. and Levinson, D. A., Dynamics: Theory
and Applications, (1985), pp.158?189, McGrawHill.
[8] Tazima, H., Fundamentals of Multibody Dynamics,
(2006), pp., 202?212, Tokyo Denki University Press.
[9] Inoue, M., Highspeed Test and the Gyroscopic
Stability of the Magnetically Suspended Wheel,
Transactions of the Society of Instrument and Control
Engineers, Vol. 23, No. 3 (1987), pp.294?300.
[10] Yoshikawa, T. and Imura, J., Modern Control
Theory (in Japanese), (1994), pp.119?130, Shokodo.
8
APPENDIX INTERNAL STRUCTURE OF THE FREQUENCY TRANSFER FUNCTION
A. Magnetic Bearing Controller Assuming that the polar
form of the variable in the stationary state is a?e t , the internal
structures of the frequency transfer functions of the magnetic
bearing controller are given as
( ) ( ) c c c W = +
( ) ( ) ( ) c d d D =W ? +
( ) ( ) ( ) c i K =W ?+
where
c
Fbreak frequency of the 1storder system affecting
the electromagnetic force
d
Fbreak frequency of the defective differentiation
i
Fbreak frequency of the integrator
B. uN Disturbance Feedback Controller The internal
structures of frequency transfer functions of the uN disturbance
feedback controller are given as
( )
( )( )
j u
IR dis lpf
Ru
zw dis zw lpf
K e
C
j u j u
?
=
? ? + ? ? +
( )
( )( )
j u
IZ dis lpf
Zu
zw dis zw lpf
K e
C
j u j u
?
=
? ? + ? ? +
( )
( )( )
j u
I dis lpf
u
zw dis zw lpf
K e
C
j u j u
ƃ
?
?
=
? ? + ? ? +
1
( ) ( )
2
Ru Ru
Ru
C C C
+ ? ?
= C 2
( ) ( )
2
Ru Ru
Ru
C C C
j
? ? ?
=
1
( ) ( )
2
u u
u
C C C
? ?
+
= ? C 2
( ) ( )
2
u u
u
C C C
j
? ?
?
= ?
where
subscript on the upperrightside of the equations? Fconjugate
complex
lpf Fbreak frequency of the lowpass filter
dis Fbreak frequency of the defective integrator
, , IR IZ I K K K Fgain of the defective integrator
, , u u u Fphase lead of the translation/rotation command