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1
ATTITUDE STABILIZATION OF A CHARGED SPACECRAFT
SUBJECT TO LORENTZ FORCE
Yehia A. Abdel-Aziz,* and Muhammad Shoaib†
In this paper, the possibility of the use of Lorentz force, which acts on charged
spacecraft, is investigated as a means of attitude control. We assume that the
spacecraft is moving in the Earth's magnetic field in an elliptical orbit under the
effects of the gravitational and Lorentz torques. We derived the equation of the
attitude motion of a charged spacecraft in pitch direction. The effect of the
orbital elements on the attitude motion is investigated with respect to the
magnitude of the Lorentz torque. The oscillation of angular velocity in pitch
direction due to Lorentz force is given for various values of charge to mass ratio.
The stability of the attitude orientation is analyzed; and regions of stability are
provided for various values of charge to mass ratio. Finally, an analytical
method is introduced to study the behavior of all the equilibrium positions. The
numerical results confirm that the charge to mass ratio can be used as a semi-
passive control for Lorentz-Augmented spacecraft.
INTRODUCTION
The problem of charged spacecraft subject to Lorentz force has recently received renewed atten-
tion in the Literature (see, References 1-2). Most recent results concerns the use of Lorentz force
for orbital perturbation and controlling the relative motion (see, References 3-8). However, for
the problem of attitude stabilization of a charged spacecraft a few important results have also
been derived (see, References 9-12).
The present work considers the attitude orientation of an electrostatically charged spacecraft. We
have taken into account the effects of gravitational and Lorentz Torques to study the attitude
stablization of spacecraft moving in low Earth orbit (LEO). We are using the same model for
Lorentz torque as in reference 12 to analyize the effects of orbital elements on the magnitude of
Lorentz torque, and invistigate the oscillation in angular velocity in pitch direction for various
values of charge to mass ratio. Both numerical and analytical techniques are used to identify
stable and unstable regions for equlibrium positions for various values of charge to mass ratios.
* Associate Professor, National Research Institute of Astronomy and Geophysics (NRIAG), Elmarsed Street 11721,
Helwan, Cairo Egypt. Email: yehia@nriag.sci.eg
† Assistant Professor, Department of Mathematics, University of Ha'il, PO BOX 2440, Ha'il, Saudi Arabia, email:
safridi@gmail.com
IAA-AAS-DyCoSS2-04-11
2
SPACECRAFT MODEL AND TORQUE DUE TO LORENTZ FORCE
A rigid spacecraft is considered whose center of mass moves in the Newtonian central
gravitational field of the earth in an elliptic orbit. We suppose that the spacecraft is equipped with
an electrostatically charged protective shield, having an intrinsic magnetic moment. The
rotational motion of the spacecraft about its center of mass is analyzed, considering the influence
of gravity gradient torque
G
T
and the torque
L
T
due to Lorentz forces. The torque
L
T
results
from the interaction of the geomagnetic field with the charged screen of the electrostatic shield.
The rotational motion of the satellite relative to its center of mass is investigated in the orbital
coordinate system
0
z
o
y
o
x
C
with
o
x
C
tangent to the orbit in the direction of motion,
o
y
C
lies
along the normal to the orbital plane, and
o
z
C
lies along the radius vector
r
of the point
E
O
relative to the center of the Earth. The investigation is carried out assuming the rotation of the
orbital coordinate system relative to the inertial system with the angular velocity
Ω
. As an
inertial coordinate system, the system
XYZ
O
is taken, whose axis
)(kOZ
is directed along the
axis of the Earth’s rotation, the axis
)(iOX
is directed toward the ascending node of the orbit,
and the plane coincides with the equatorial plane. Also, we assume that the satellite’s principal
axes of inertia
b
z
b
y
b
x
C
are rigidly fixed to a satellite
),,(
bbb
kji
. The satellite’s attitude may be
described in several ways, in this paper the attitude will be described by the angle of yaw
ψ
the
angle of pitch
θ
, and the angle of roll
ϕ
, between the axes
b
z
b
y
b
x
C
and
XYZ
O
. The three angles
are obtained by rotating satellite axes from an attitude coinciding with the reference axes to
describe attitude in the following way:
- The angle of precession
ψ
is taken in plane orthogonal to
Z
-axis.
-
θ
is the notation angle between the axes
Z
and
.
0
z
-
φ
is angle of self -rotation around the
Z
-axis
According to13, we can write the relationship between the reference frames
b
z
b
y
b
x
C
and
0
z
o
y
o
x
C
as given by the matrix A which is the matrix of direction cosines
iii
γβα
,,
,
1,2,3).=(i
,=
321
321
321
γγγ
βββ
ααα
A
(1)
where
3
,cos=
,cossin=
,sinsin=
),cossin=
,coscoscossinsin=
,sincoscoscossin=
,sinsin=
,cossincossincos=
,cossinsincoscos=
3
2
1
3
2
1
3
2
1
θγ
φθγ
φθγ
ψθβ
φψθφψβ
ϕψθϕψβ
ψθα
φψθφψα
θφψφψα
−
+−
+
−−
−
(2)
and
,=,=,= 321321321 bbbbbbbbb kjikjikji
γγγγββββαααα
++++++
(3)
As stated above, it is assumed that the spacecraft is equipped with elctrostatic cahrge, therefore
we can write the torque due to Lorentz force as follows12.
),(=),,(=
0orel
T
LzLyLxL
BVAqTTTT
××
ρ
(4)
or
()
,
,,=),,(=
0T
LLL
T
LzLy
LxL
NT
RAT
TTT ×
ρ
(5)
.==
1
0000
dSqkzjyix
S
bbb
ρσρ
∫
−
++
(6)
0
ρ
is the radius vector of the charged center of the spacecraft relative to its center of mass and
T
A
is the transpose of the matrix
A
of the direction cosines
α
,
β
,
γ
,
( )
, ,
LL L
RTN
are the
components of Lorentz force into the radial, transverse, and normal components respectively
yields,
( ) ( )
2
3
22
0
2
= [1 ] / cos 1 cos ,
sin sin
Le
B
q
R i f p ief
mr
ω ωµ
− +− +
(7)
( )
( ) ( )( )
2
33
0
2
2
/ cos 1 cos 2 /
=,
sin cos 1 cos
sin
e
L
rp ief p
B
qr
Tmpi f f ef
µ ωµ
µωω
++ ×
+ ++
(8)
4
()()
( ) ()
( ) ( ) ()
( ) ()
2
3
22
2
33
0
22
32
24
322
2 [1 ] / cos 1 cos
sin sin
cos
= / cos 1 cos / .
1sin sin
sin
sin cos
1 cos 2 1 cos
1sin cos
e
L
i f p ief
B
q ri
N p ief p
mr
pif
if f
ef ef
p if
ω ωµ
µµ
µω
ωω
µ
ω
− +− + ×
++ ×
−+
++
+− +
−+
(9)
where,
0
B
is the strength of the magnetic field in Wb.m., / is the charge-to-mass ration of the
spacecraft,
µ
is the Earth’s gravitational parameter
a
,e,
i
,
ω
and
f
a
are semi-major axis,
eccentricity, inclination of the orbit on the equator, argument of the perigee, and the true anomaly
of the spacecraft orbit respectively.
SPACECRAFT ATTITUDE MOTION EQUATIONS
The attitude motion of the spacecraft is expressed by Euler’s equations13. Therfore, the
equation of the attitude dynamics of a rigid spacecraft due to gravity gradient and Lorentz torques
is expressed as
=,
GL
I IT T
ω ωω
+× +
(10)
where
ITG
γ
γ
×Ω2
3=
is the well known formula of the gravity gradient torque,
=I
diag
),,( CBA
is the inertia matrix of the spacecraft,
Ω
is the orbital angular velocity,
ω
is the
angular velocity vector of the spacecraft. The angular velocity of the spacecraft in the inertial
reference frame is
),,(= rqp
ω
, where
,
.cos=
sincossin=
cossinsin=
φθψ
φθφθψ
φθφθψ
+
−
+
r
q
p
(11)
The system of Eq. (10) admits Jacobi´s integral14
0
1- =h
2IV
ωω
⋅
, (12)
Where
0
V
is the potential of the problem and takes the following expression15
( ) ( )
T
20 LL L 123
20 L1 0L 2 0 L3
3( I) R ,T , N , ,
2
3( I) R T z N ,
2
0
V
xy
= Ω γ⋅γ +ρ ⋅ β β β
= Ω γ⋅γ + β + β + β
(13)
5
where, the first part is potential due to the gravitational potential and the rest of the part due to
Lorentz force.
ATTITUDE MOTION IN THE PITCH DIRECTION
Assume the attitude motion of the charged spacecraft in the pitch direction, i.e.
0. 0,== ≠
θφψ
Applying this condition in Euler equation of the attitude motion of the
spacecraft in Eq (10), we derive the second order differential equation of the motion.
.cos
)
(sin) (cossin
1))(3
(
=
000
0
2
2
2
θ
θθθ
θ
LLLL
T
zN
yTyNzB
C
dt
d
A−
+−+−Ω
−
(14)
Let
, = 00 zky
(15)
where is arbitrary number. Then equation (12) takes the following form.
.
cos)
(
sin)
(cos
sin1)
)(3(
=
00
2
2
2
θ
θθ
θ
θ
L
L
LL
T
kNz
kT
N
zB
C
dt
d
A−+
−+
−Ω
−
(16)
In right hand side of this equation, the first term represents the gravity gradient torque, the second
and third terms represent the Lorentz torque.
Using equation (12, 13), we can write the Jacobi´s integral as follows
222 20
13
( ) ( sin cos ) ( cos sin )
22
LL
d
h B C z kN T
dt
θθ θ θθ
= −Ω + − −
(17)
The quantity h corresponds to the energy in the rotating refrence frame, which we can use to
calculate the minimum and maximum energy required for stable equilibrium positions, and the
energy needed to move from unstable position to stable position.
EFFECT OF ORBITAL ELEMENTS ON TORQUE
In this section, we study the effects of orbital elements on the magnitude of Torque due to Lo-
rentz force.
Using Equation (5), and
0
1, 1.1, 0.001, 15 , 60 .z k e i and f= = = = =
We can calculate the
components of the Lorentz torque, and the magnitude of the torque as function of semimajor axis
and charge to mass ratio =/ .
(,)=.×.
, (18)
(,)=(.×.×.
)
, (19)
6
(,) = (.×.×.
)
. (20)
Using the values
0
1, 1.1, 6900 , 15 , 60z k a km i and f= = = = =
, we get the components of
Lorentz torque as a function of eccentricity and charge to mass ration () asthe following:
(,) = ( .)(.×.×.×.())
()
, (21)
(,)=
()(1 + 0.98)(3.8148 ×10(1 + 0.98)
+(1.06 + 1.22 ×10(1 + 0.98)
)), (22)
(,)=
()1.31 2.561.256.06 ×101+1.36
3.61 ×1012.56 + 2.41 ×101+5.12 2.39 ×
101+1.2 5.96 ×101.
(23)
Using Equations (18-20), we can calculate the magnitude of the Lorentz torque as a function
of semi-major axis and charge to mass ratio. Figure (1-left) shows the magnitude of Lorentz
torque for semi-major (a) between 6500 km to 12000 km with three different values of = 0.1,
= 0.2, and = 0.02. The figure shows that the magnitude of Lorentz torque is decreasing
with the altitude of the satellite increases and the magnitude of the torque is affected by charge to
mass ratio. Similarly, from equations (21-23) we can calculate the magnitude of the torque as a
function of eccentricity and charge to mass ratio; shown in Figure (1-rigth). This figure shows the
magnitude is nearly constant up to = 0.1and when > 0.1, the magnitude of torque increases
with the increasing value of eccentricity.
Figure 1. Magnitude of the Lorentz torque as function of (Left) semi-major axis and (right) eccen-
tricity
Using the values
0
1, 1.1, 0.001, 6900 , 60
z k e a km and f= = = = =
, we get the
components of Lorentz torque as a function of inclination and charge to mass ratio () asthe
following:
7
(,)=sin (1.55 ×10+ 1.37 ×10sin+..×
(.)
(1.71 ×102.27 sin )10.9 sin)
(24)
(,)
=((8.3 × 10+ 2.39 cos )sin 7.50 ×10sin+ (1.37 1.24 sin)10.9 sin)
10.9 sin,
(25)
(,)=171.37(10.9 sin)0.01 +.×
.+sin 1.55 ×10
1.37 ×10sin.×
(.)2.27 sin 10.9 sin. (26)
Similarly using the values
0
1, 1.1, 6900 , 15 6900z k a km i and a km= = = = =
, we get the
components of Lorentz torque as a function of eccentricity and charge to mass ratio () asthe
following:
(,)=(0.53 2.63 ×10cos ), (27)
(,)=(1.88 + 0.03 cos + 1.2 × 10cos), (28)
(,)=(1.3 0.02 cos 1.2 × 10cos). (29)
Figure 2. Magnitude of the Lorentz torque as function of (Left) inclination and (right) true anomaly
Using Equations (24-26), we can calculate the magnitude of the Lorentz torque as a function
of inclination and charge to mass ratio. Figure (2-left) shows the magnitude of Lorentz torque
when = 0.1,0.5,0.8. The figure shows that the magnitude of Lorentz torque is increasing with
inclination increasing till 50° and after that the torque is decreasing with increasing value of in-
clination. Similarly, from equations (27-29) we can calculate the magnitude of the torque as a
function of true anomaly and charge to mass ratio. Figure (2-rigth) shows the magnitude of torque
is decreasing with the increasing value of true anomaly up to 180° and when f > 180°, the
magnitude of the torque starts increasing.
8
Figure 3. Oscillation in
d
dt
θ
due to Lorentz torque when (left) = ,=.,=
°, =.,=.,= °, =, B < C and (right) =,<.
Figures (3-4) shows the oscillation in angular velocity due to Lorentz torque. It is clear from Fig-
ure (3-left) that the oscillation in angular velocity is between -0.004 and 0.004 when =1,
and B > C but when = 1(Figure 3,right) and B > C the oscillation is between -0.002 and
0.002. Similarly (Figure 4-left) the oscillation in angular velocity is between -0.3 and 0.3 when
= 1and B> C. In the case of figure (4-right) where =1, and B> C, the oscillation is
between -0.3 and 0.3. It is obvious that the difference in behavior in the oscillation in Figure (3,
left) is due to charge to mass ratio. In addition, the effects of the components of moment of inertia
of the charged spacecraft are clear when we compare figure (3) and figure (4).
Figure 4. Oscillation in
d
dt
θ
due to Lorentz torque when =6900 ,=.001,=15°, =
.,=.,= °, (left) =, B > C and (right) =, >.
9
Figure 5. (Left) Oscillation in angular velocity of LAGEOS with the orbital elements =
.,=,=.,=.°, =.,=,= °. (Right)
Oscillation in angular velocity of LARES with the orbital elements =.,=,=
.,=.°, =.,=.,= °.
Figure 5, shows the comparison between the oscillation in angular velocity for an artificially
charged LAGEOS satellite (5-left) and oscillation in angular velocity for an artificially charged
LARES satellite (5-right) due to Lorentz torque. It is clear from the figure that the oscillation in
LARES is two time the oscillation in LAGEOS for =..
EXISTENCE AND STABILITY OF EQUILIBRIUM SOLUTIONS
In this section, we discuss the existence and stability of equilibrium positions of a general
shape spacecraft under the influence of gravitational torque and Lorentz torque. We will identify
regions in the phase space in which a continuous family of stable equilibrium positions exists.
Existence of equilibrium positions
To find the equilibrium positions, take the right hand side of equation (14) equal to zero which
reduces to the following equation when = 1, =6878137,=12 °, =51°and cb = C B.
(,,,)=(1.74 + 0.93)cos +(0.93 1.74)sin 0.5cb sin 2. (30)
If we solve (,,,)= 0, we will obtain all the equilibrium positions for all values
of ,,,and .For = 1, after some algebraic manipulation (,,,)= 0 is reduced to
the following equation.
cb(sin )1.62cb(sin )cb1.32(sin )+ 1.62cbsin 0.66= 0.(31)
The above equation is quartic in sin and can theoretically be solved but is very complicated and
beyond the scope of this study. Therefore, we will study it numerically for fixed values of
and . As is apparent from equation (31), the values of and will have a significant ef-
fect on the existence equilibrium solutions. For example, when (0.07,0.07),there are four
equilibrium positions for all values of except when (0.065,0.065),and when =
0.1and||> 0.07, the number of equilibrium positions reduces to two. Two typical examples
are given in figure (5). In figure (5, left), when= 0.05,there are four equilibrium positions for
10
all values of except when (0.065,0.065). For > 0.065, the equilibriums occur
around = 1.77 rad, 3.44 rad, 4.4rad, and 6.1rad. For <0.065,the first equilibrium, oc-
curs around = 0.32 rad and converges to zero as the value of decreases to -1. The other
three equilibriums occur around = 1.3 rad, 3rad, and 5.8rad. In figure (5, right), when=
0.1, there are four equilibrium positions for very small values of and two when || > 0.1.
These equilibrium positions occur at = 2.4 and = 5.4 when the spacecraft is negatively
charged and at = 2.2 and = 5.57 for a positively charged spacecraft
Figure 6. Family of equilibrium positions when (left) =. and (right) =..
Stability analysis of equilibrium solutions
To discuss the stability of the equilibrium position identified we convert equation (14) to a
system of two first order equations.
= y, and y=(,,,).
After linearization, we can write the Jacobian of the above system as below.
(,)=0 1
0
Let be an eigenvalue of (,).The characteristic equation (|J(y, ) I| = 0) is given
by= 0. It is well known that if at least one of the eigenvalues is positive then the corre-
sponding equilibrium will be unstable. If both the eigenvalues are pure imaginary then corre-
sponding equilibrium will be a center. In our case > 0 will imply real eigenvalues with one of
them positive and hence instability and < 0 will imply a stable center. Therefore, to study the
stability of all the equilibrium positions we need to write (,,,).
(,,,)=(0.15 0.63)cos +(0.63 0.15)sin cb (cos )+
cb(sin ).
11
Before we give a complete picture of the stability and unstable regions, we give specific examples
of the artificial satellites LARES and LAGEOS 1. We assume that these two satellites are artifi-
cially charged. The orbital elements of LARES are
=7820,= 0.0005,=69.5°,=0.1, = 1.1, =60 °, =299°.
Let it has a charge to mass ratio of = 0.3. There are two equilibrium positions for LARES at
= 2.67 and 5.33. It is a straightforward exercise to show that |.is positive and hence
= 2.67 is an unstable equilibrium. The value of |.is negative and hence this equilibrium
is a stable center. To support this conclusion we also give a phase diagram in in
figure (7, left) which confirms both the conclusions. The orbital elements for LAGEOS 1 satellite
are =12300,= 0.0044709,=109.804°,=0.1, = 2, =182°,=345°.
Let it has a charge to mass ratio of =0.3.There are two equilibrium positions for LARES at
= 2.7 and 6.1. The equilibrium position at = 2.7 is stable and = 6.1 is unstable as the ei-
genvalues are imaginary and positive respectively.
To completely analyze the stability and its dependence on and , we provide the stability
diagrams for various values of . In figures (8 to 10), the blue lines correspond to the family of
equilibrium positions and the shaded regions are the regions where the equilibrium positions will
be stable. Figure (8, left) and figure (8, right) are given for =0.001 and = 0.001. The
locations of the stable regions are nearly unaffected by the changing values of the charge to mass
ratios but the locations and hence stability of equilibrium positions are significantly affected. For
example in the upper half of both parts of figure 8, there are four equilibriums each and two of
them stable but for =0.001, the first equilibrium is at 0 which is stable and for = 0.001
there is no such equilibrium. In fact there is an at 0 but for negative values of . Similar behav-
ior is shown by the equilibrium = 2. There is no change in the behavior of the equilibrium
close to = 3. The only change in this case is when cb is very close to 0.
Figure 7. Phase diagram in of (left) LARES satellite with =. and (right)
LAGEOS satellite with =.. The red dots in both figures correspond to the unstable orbit and
the black dot corresponds to the stable orbit.
12
Figure 8. Stability regions when ==,=.,=°, =.,=° and (Left)
=. and (right) =.
Figure (9, left) and figure (9, right) are given for = 1 and =1. It is readily noticed that,
both the locations of the stable regions and locations of equilibrium positions are revered by the
changing values of the charge to mass ratios. When = 1, there are four equilibriums when
< 0 and there equilibriums when > 0 with two and one of them lying in the stable regions
respectively. Similarly, when =1, there are four equilibriums when > 0 and three equi-
libriums when < 0 with two and one of them lying in the stable regions respectively. Similar
conclusions can be drawn from figures (10) which is given for = ±5.
In summary, we can conclude from figures 8,9 and 10 that and significantly affect both
the stability and locations of the equilibrium solutions and hence can be used as semi-passive
control.
Figure 9. Stability regions when ==,=.,=°, =.,=° and (Left) =
and (right) =
13
Figure 10. Stability regions when ==,=.,=°, =.,=° and (Left)
= and (right) =
CONCLUSIONS
In this work, we analyzed the problem of attitude dynamics of an electrostatically charged
spacecraft under the effects of gravity gradient and Lorentz torque in pitch direction. The effects
of orbital elements on the magnitude of Lorentz torque is analyzed for various value of charge to
mass ratio. The oscillation in angular velocity in pitch direction for various values of charge to
mass ratio is considered for different operational satellite (LARES and LAGEOS 1).The stable
and unstable regions concerning the effects of Lorentz toreque for different satellite orbits are
identified. Both stable and unstable equilibrium poitisions are identified for LARES and
LAGEOS 1 satellite. It is shown that for an artificialy charged LARES and LAGEOS 1 satellite
there are two possible equilibrium positions with one each stable and one unstable. As from the
general analysis it is possible to chnge the location of the equilibrium positions by varying the
values of charge to mass ratio or the moment of inertia. This is demonstrated by various
numerical studies performed for various values of orbital elements. Depending on the values of
and moment of inertia the number of equilibrium positions vary between two and four.
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