Attitude dynamics of a pendulum-shaped charged satellite

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Acta Astronautica

Acta Astronautica 70 (2012) 77–84

0094-57

doi:10.1

$ Thi� Cor

E-m

hachiya

bando.m

journal homepage: www.elsevier.com/locate/actaastro

Attitude dynamics of a pendulum-shaped charged satellite$

Hiroshi Yamakawa a,b, Shinji Hachiyama a, Mai Bando b,�

a Research Institute for Sustainable Humanosphere, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japanb Unit of Synergetic Studies for Space, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan

a r t i c l e i n f o

Article history:

Received 3 February 2011

Received in revised form

16 July 2011

Accepted 19 July 2011Available online 17 August 2011

Keywords:

Attitude control

Lorentz force

Charged satellite

65/$ - see front matter & 2011 Elsevier Ltd. A

016/j.actaastro.2011.07.019

s paper was presented during the 61st IAC in

responding author.

ail addresses: [email protected] (

[email protected] (S. Hachiyama),

[email protected], [email protected] (

a b s t r a c t

In this paper, we investigate the possibility of the use of the Lorentz force, which acts on

charged satellite when it is moving through the magnetic field, as a means of satellite

attitude control. We first derive the equations of attitude motion of charged satellite

and then investigate the stability of the motion. Finally we propose an attitude control

method using the Lorentz force. Our method requires moderate charge level for future

Lorentz-augmented satellite.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Orbiting artificial satellites naturally tend to accumulateelectrostatic charge which may in some circumstances causemalfunctions in measuring instruments and other devices onthe satellite body and ultimately lead to difficulties insatellite operation. Various research efforts have led to thedevelopment of technology for active mitigation of thesatellite charge through control of the charge. The questiontherefore arises as to whether such charge control technologymight be applied, rather, to satellite operation. Variousinteractions occur between a charged satellite and its envir-onment. Coulomb forces arise between charged satellites,and Lorentz forces arise between a charged satellite and theEarth’s magnetic field. It may therefore be possible to applycharge control techniques to the utilization of these Coulomband Lorentz forces for satellite orbital control, and research isin progress [1–8]. Most of the references require high specificcharge levels around 0.1–10 C/kg [3–8]. We assume specificcharges 9q=m9� 1:0� 10�8

21:0� 10�6 C=kg, which will

ll rights reserved.

Prague.

H. Yamakawa),

M. Bando).

likely be more feasible applications of Lorentz-augmentedspacecraft in the near future.

In this paper, we consider the possibility of a differentapplication, involving the use of the Lorentz force forsatellite attitude control. Gravity gradient torque hasalready been used as a means of satellite attitude control,but the scope of control by this means is rather limited. Ifthe torque due to the Lorentz force (‘‘Lorentz torque’’) canbe utilized, it may open the way to a more generallyapplicable control method. As with the application ofgravity, centrifugal force, or other natural forces toattitude control, the use of the Lorentz force wouldgreatly reduce concerns about fuel supply which areinherent to the conventional thrusters like ion engines.In this light, we investigate the possibility of simulta-neous use of the Lorentz torque and gravity gradienttorque for satellite attitude control, through simulationand analysis using a small satellite model with a dumb-bell pendulum configuration conceived for this purpose.

In Section 2, we introduce the Lorentz force on chargedsatellites. The general equations of satellite attitudemotion are presented in Section 3, and in Section 4 wederive the equations of external torque acting on chargedsatellites in a circular orbit. In Section 5, we investigatethe stability of the linearized equations of motion andnumerical examples are shown based on equations ofmotion of charged satellites in a circular orbit. Then we

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dx.doi.org/10.1016/j.actaastro.2011.07.019

r B0

x

z y

qvrel B0

r3FL =

q (> 0)

vrel

Fig. 2. Lorentz force acting on the charged satellite.

H. Yamakawa et al. / Acta Astronautica 70 (2012) 77–8478

discuss their application to attitude control. The conclu-sions of this study are summarized in Section 6.

2. Satellite charge and the Lorentz force

2.1. Satellite charging and charge control

To utilize the Lorentz force for control of the satelliteattitude, it requires free and active control of amount ofcharge, and thus the capability to hold it constant, raise orlower it to a given level, or eliminate it (by charge mitiga-tion) and thus obtain a non-charged state as appropriate. Inthis study, we assume the use of an ion or electron emitterfor this purpose. Such emitters draw ions or electrons from aplasma chamber inside the satellite and emit them to space,and can therefore be used to charge control. The pendulumsatellite assumed in the present study is small, with a totalweight of about 50 kg, and the emitter devices describedabove are also lightweight in comparison with the conven-tional thrusters, and therefore appropriate for control of thesatellite attitude. Such emitters have actually been used insatellites for control charge and mitigation (neutralization)[3], and are capable of charging and discharging in thepresent study.

2.2. The Lorentz force acting on a charged satellite in the

Earth’s magnetic field

In this study, we consider the attitude motion of asatellite in Earth orbit. A charged satellite will be subjectedto the Lorentz force in the Earth’s magnetic field. Forsimplicity, we approximate the Earth’s magnetic field as adipole magnetic field, where the magnetic north pole andthe true north pole are perfectly aligned. Fig. 1 shows theLorentz force exerted on a charge q by a dipole magneticfield, in a coordinate system having its origin at the centerof the Earth. The components of the magnetic field at pointr are expressed as

BrðrÞ ¼ 2B0

r3cos y

ByðrÞ ¼B0

r3sin y

9>>=>>; ð1Þ

q

r

x

z

y

θN

B

Br

FL = qv×B

B� v

Earth

Fig. 1. Earth magnetic field.

where B0 is the magnetic dipole moment which is taken asbeing negative when north is taken to be in the þz

direction. If a satellite carrying a charge q moves in thisdipole magnetic field with a velocity v, then the Lorentzforce on the charged satellite is given by

FL ¼ qv � B ð2Þ

However, because the Earth’s magnetic field rotatestogether with the Earth at a rotational speed oE, theLorentz force on the charged satellite moving in the Earth’smagnetic field as approximated by a dipole magnetic fieldbecomes

FL ¼ qðv�oEn̂ � rÞ � B ð3Þ

where n̂ is a unit vector in the þz direction. The satellitein the present study is assumed to orbit in the equatorialplane, and therefore as shown in Fig. 2, with vrel as thespeed of the satellite relative to the magnetic field. For acircular equatorial orbit, vrel ¼ r0n, where r0 is circularorbit of radius and n¼ ðm=r3Þ

1=2 is the orbit rate of thereference orbit. The Lorentz force on the satellite

FL ¼qvrelB0

r3ð4Þ

is obtained. In the following, we investigate the influence ofthis external force on the attitude motion of the satellite.

3. Satellite attitude motion equations

Satellite attitude may be defined in terms of therelation between the coordinate system of the satelliteorbit (the ‘‘reference frame’’) and that of the satellite itself(the ‘‘body-fixed frame’’).

In this section, we first derive the equations of three-dimensional attitude motion for satellites, and then considerthe one-dimensional equations of motion for the pendulumsatellite, focusing on rotation about its pitch axis.

3.1. Attitude motion equations

The attitude motion of the body is expressed byEuler’s equation [9]. If we align the three axes of theorthogonal coordinate system of the rigid body withthe principal axes of inertia, then Euler’s equation in thebody-fixed frame FSðxS,yS,zSÞ rotating at angular velocity

H. Yamakawa et al. / Acta Astronautica 70 (2012) 77–84 79

x is given by

Ix _oxþðIz�IyÞoyoz ¼Mx

Iy _oyþðIx�IzÞozox ¼My

Iz _ozþðIy�IxÞoxoy ¼Mz

9>=>; ð5Þ

where Ix, Iy, Iz are the moments of inertia around the roll,pitch, and yaw axes, respectively. Eq. (5) thus shows theequations of motion around the roll (xS), pitch (yS), andyaw (zS) axes, respectively.

3.2. Pendulum satellite model

The satellite model used in this study has the form of adumbbell pendulum. For simplicity, we consider a circularequatorial Earth orbit (zero orbital inclination). The pen-dulum satellite model is shown in Fig. 3. The chargedportions are the two mass points A and B (each of massm), which are connected by a rigid rod of negligible massand length 2a. Its midpoint lies on the circular Earthorbit of radius r0. We consider the attitude motionof the satellite in the orbital plane, and thus the attitudemotion about its pitch axis, and assume the effect of theattitude motion on the orbital motion (constant angularvelocity O) to be negligible. With the roll angle f and theyaw angle c taken as zero, the angular velocity andangular acceleration of the satellite become

x¼

ox

oy

oz

0B@

1CA¼

0_y�O

0

0B@

1CA,

dxdt¼

_ox

_oy

_oz

0B@

1CA¼

0€y0

0B@

1CA ð6Þ

Substituting this into Eq. (5), we obtain the equation ofattitude motion (Ix ¼ Iy ¼ 2ma2, Iz¼0) about the pitch axis(relating to pitch angle y)

Iyd2ydt2¼My ð7Þ

where My is the external torque. As the inertial moment Iy

is known, if we obtain the external torque My, then we canderive the equation for the pitch angle y.

In the following, we consider the pitch motion only.For the linearized motion around the equilibrium point,the pitch motion is independent of the roll and yaw

X

Z

Y

zs

r0

xs

ys

a

a

roll

pitch

yaw

Earth

�

�

Fig. 3. Pendulum-shaped satellite.

motion and is good approximation. However our analysiscannot be applied to yaw and roll motion because thesemotion are coupled.

4. External torques

We next examine the external torques acting on thependulum satellite. As the external torques, we considerthe gravity gradient torque (torque due to gravity andcentrifugal forces) and the Lorentz torque (the torque dueto the Lorentz force). In the present study, we focus on theforces in the orbital plane, and thus the torques actingabout the satellite’s pitch axis.

4.1. Gravity gradient torque

The gravitational and centrifugal forces acting on thependulum satellite are illustrated in Fig. 4. The forcesacting on point masses A and B may be, respectively,expressed as

A : FAG ¼mm=r2A

B : FBG ¼mm=r2B ð8Þ

where m is the Earth’s gravitational constant and rA and rB

are the distances from the Earth’s center to point massesA and B, respectively. Using the cosine theorem,

r2A ¼ r2

0 1þ2a

r0cos yþ

a2

r20

!

r2B ¼ r2

0 1�2a

r0cos yþ

a2

r20

!9>>>>>=>>>>>;

ð9Þ

and r0ba,we obtain the following components perpendi-cular to the rod:

FAG? ¼mm sin y

r20

1�3a

r0cos y

� �

FBG? ¼mm sin y

r20

1þ3a

r0cosy

� �9>>>>=>>>>;

ð10Þ

The second term in each equation arises from the differ-ent distances of mass points A and B from the center ofthe Earth. This difference in gravitational force thus actsas the gravitational torque which is given by

MG ¼ ðFAG?�FBG?Þa¼�3m2r3

0

Iy sin 2y ð11Þ

N

Earth

gravitational forcecentrifugal forcerB

rA

B

A

a

a

m

m

r0

��B

�A

Fig. 4. Gravitational force and centrifugal force.


The term Iy ¼ 2ma2 represents the inertial moment aboutthe pitch axis. The term 3m=2r3

0 is constant in Eq. (11), andthe equilibrium points (the points at which MG ¼ 0) arey¼ np=2. The gravitational force on the mass points hasno component perpendicular to the rod at y¼ np from Eq.(10), and the distance from the Earth’s center to the twomass points is equal at y¼ p=2þnp; the gravitationaltorque is therefore zero at those points.

The centrifugal forces on mass points A and B are,respectively,

A : FAC ¼mmrA=r30

B : FBC ¼mmrB=r30 ð12Þ

and, as with the gravitational force, their componentsperpendicular to the rod are, respectively, given by

FAC? ¼ FBC? ¼mm sin y

r20

ð13Þ

As these perpendicular components are the same inmagnitude and direction, they effectively exert no nettorque on the satellite.

4.2. Lorentz torque

We next consider the torque due to the interactionbetween the charged satellite and the Earth’s magneticfield. The charges on mass points A and B, which are thecharged components of the satellite, are equal in quantityand may be either positive or negative in sign. We there-fore consider the cases in which their signs are the same(‘‘same-sign’’), and those in which the signs are opposite(‘‘opposite-sign’’). As noted in Section 2, the Earth’smagnetic field is approximated by a dipole magnetic field.The Lorentz forces acting on each of the mass points areshown in Fig. 5.

4.2.1. The charges on the mass points have opposite sign

We next consider the case in which the signs of thecharges on mass points A and B are positive and negative,respectively (qA ¼ þq, qB ¼�q, q40). As in the casewhere the charges are of the same sign, the magnitudesof the Lorentz forces may be expressed as

A : FALd ¼ qvrelB0=r3A

B : FBLd ¼ qvrelB0=r3B ð14Þ

N

Earth

same-signopposite-signrB

rA

B

A

a

a

m

m

r0

qA

qB

�

�B

�A

Fig. 5. Lorentz force acting on charged satellite.

In the same way as described above, the componentsperpendicular to the rod are

FALd? ¼qvrelB0 sin y

r30

1�3a

r0cos y

� �

FBLd? ¼qvrelB0 sin y

r30

1þ3a

r0cos y

� �9>>>>=>>>>;

ð15Þ

and the resulting torque on the satellite is

MLd ¼ ðFALd?þFBLd?Þ � a¼2qvrelB0a

r30

sin y ð16Þ

The equilibrium points (MLd ¼ 0) for this torque are there-fore at y¼ np (i.e., where the Lorentz forces have nocomponent perpendicular to the rod), and are thus dif-ferent from those of the gravity gradient torque. Again,the magnitude can be changed by varying the chargequantity.

If the signs of the charges on mass points A and B arereversed from the above case (qA ¼�q, qB ¼ þq, q40),the torque on the satellite has the same magnitude asabove but is opposite in direction.

We have derived the equations for the torque on themoving pendulum satellite due to the Lorentz force, withcharges of equal quantity and the opposite sign on its twomass points. The satellite itself is biased in sign and itsorbital motion is therefore affected, in contrast to themutually cancelling effects of the charges in opposite-signconfigurations. For this reason, we will consider only theopposite-sign configurations in this paper.

5. Analysis of satellite attitude motion

5.1. Differential equation for pitch angle

From the equations of attitude motion about the pitchaxis derived in Section 3 and the external torques (gravitygradient torque and Lorentz torque) derived in Section 4,we can obtain the differential equation for the pitch angley of the pendulum satellite. As we consider only the casesin which the charges on the two mass points are oppositein sign, it follows that in Eq. (7) My ¼MGþMLd, and fromEqs. (11) and (16), we thus find that

Iyd2ydt2¼�

3m2r3

0

Iy sin 2y72qvrelB0a

r30

sin y ð17Þ

By rearrangement, this yields

d2ydt2þ

3

2

mr3

0

sin 2y7q

m

vrelB0

ar30

sin y¼ 0 ð18Þ

The signs in this equation are dependent on the chargesigns. The second term represents the gravity gradienttorque, and the third term represents the Lorentz torque.Let a be the ratio between the coefficients of these twoterms which represents the ratio between the magnitudesof the gravity gradient torque and the Lorentz torque:

a¼ q

m

2vrelB0

3mað19Þ

Fig. 6. Trajectory in phase plane (a¼3). (For interpretation of the

references to color in this figure legend, the reader is referred to the

web version of this article.)


Then Eq. (19) becomes

d2ydt2þ

3m2r3

0

ðsin 2yþa sin yÞ ¼ 0 ð20Þ

We shall apply Eq. (20) to analysis of the behavior of thepitch angle y.

5.2. Stability analysis

In this section, the stability analysis is carried outbased on the linearized equation of Eq. (20). It followsfrom Eq. (20) that the angle y at the equilibrium pointsatisfies the condition

sin 2yþa sin y¼ 0 ð21Þ

and thus

sin yð2 cos yþaÞ ¼ 0 ð22Þ

The equilibrium points may therefore be expressed byðy, _yÞ ¼ ðnp,0Þ ðn 2 ZÞ, ðya,0Þ where ya satisfies cos ya ¼�a=2. In the vicinity of equilibrium points, the linearizedequations of motion are given by

d2ydt2þ

3m2r3

0

ð2þaÞy¼ 0 at y¼ 2np ð23Þ

d2ydt2þ

3m2r3

0

ð2�aÞy¼ 0 at y¼ ð2nþ1Þp ð24Þ

d2ydt2þ

3m2r3

0

ðsin 2yaþa sin yaÞy¼ 0 at ya ð25Þ

According to the value of a, we classify it into sevencategories (denoted by case (i)–(vii)) and Table 1 sum-marizes the stability properties of the linearized equa-tions for case (i)–(vii). In the next section, the results ofstability analysis are discussed in detail.

5.3. Numerical simulations

In this section, we show the simulation results of theattitude motion of the pendulum satellite based onEq. (20). The parameters are given as r0 ¼ 6900 km, vrel

¼ 7:112� 103 m=s, a¼3 m, B0¼�8.0�1015 Wb m. Thecharge value for 50 kg satellite which corresponds toa¼ 1 is q¼ 3:15n10ð�5Þ C=kg. It is regarded as the moder-ate value for future Lorentz-augmented satellite. Thevalidity of the simulations was verified by showing the

Table 1The stability properties of the linearized equations.

a y¼ 2np y¼ ð2nþ1Þp y¼ ya

Case (i): 2oa Stable Unstable –

Case (ii): a¼ 2 Stable Unstable –

Case (iii): 0oao2 Stable Stable Unstable

Case (iv): a¼ 0 Stable Stable ya ¼ np: stable

ya ¼ npþp=2:

unstable

Case (v): �2oao0 Stable Stable Unstable

Case (vi): a¼�2 Unstable Stable –

Case (vii): ao�2 Unstable Stable –

integral of Eq. (19)

J¼1

2

dydt

� �2

þ3

4

mr3

0

ð1�cos 2yÞ7q

m

vrelB0

ar30

ð1�cos yÞ ð26Þ

The quantity J corresponds to the energy in the rotatingreference frame. Figs. 6–12 show the trajectory corre-sponding to cases (i)–(vii), respectively, in phase planewhere y=p as the horizontal axis and dy=dt as the verticalaxis. In each figure, the closed curves represent oscillatorymotion, curves with _y40 (y monotonically increasing) or_yo0 (y monotonically decreasing) represent rotationalmotion, and the integral J is constant on each curve. Theclosed black dots and open red dots on the straight line inFigs. 6–12 indicate the equilibrium points.

Case (i): 2oa. The result for a¼ 3 is shown in Fig. 6.The stable equilibrium points are at ðy, _yÞ ¼ ð2np,0Þ (theclosed black dot). On the other hand, the equilibriumpoints ðy, _yÞ ¼ ðð2nþ1Þp,0Þ (the open red dot) are unstable.On the straight line _y ¼ 0, in fact, the value of the energy J










Fig. 10. Trajectory in phase plane (a¼�1). (For interpretation of the

references to color in this figure legend, the reader is referred to the web

version of this article.)








has a maximum at unstable equilibrium points and aminimum at stable equilibrium points. If the energy at anunstable equilibrium point is even slightly smaller than J

then it will result in oscillatory motion, and if it is slightlylarger it will result in rotational motion.

Case (ii): a¼ 2. The result for a¼ 2 is shown in Fig. 7.The stable equilibrium points are at ð2np,0Þ and unstableat ðð2nþ1Þp,0Þ.

Case (iii): 0oao2. The result for a¼ 1 is shown inFig. 8. The stable equilibrium points are at ðy, _yÞ ¼ ðnp,0Þ.In contrast, an unstable equilibrium occurs at ðya,0Þ.Depending on the value of a, ya exist in the rangeð2nþ 1

2Þpoyo ð2nþ1Þp and ð2nþ1Þpoyo ð2nþ 32Þp. For

example, when a¼ 1, ya ¼ 23pþ2np, 4

3pþ2np. In thevicinity of y¼ ya, the gravity gradient torque and theLorentz torque act in opposite directions, and theirmagnitudes balance at y¼ ya. In the phase plane, pertur-bation Dy from the equilibrium point will invariablyinvolve a bias in the magnitude of one torque, resulting


in oscillatory motion, and perturbation D _y (and thushigher energy) will result in rotational motion.

Case (iv): a¼ 0 (gravity gradient torque alone). Theresult for a¼ 0 is shown in Fig. 9. The stable equilibriumpoints are at ðy, _yÞ ¼ ðnp,0Þ.

Case (v): �2oao0. The result for a¼�1 is shown inFig. 10. The stable equilibrium points are at ðy, _yÞ ¼ ðnp,0Þ.The unstable equilibrium points are at ðya,0Þ. Dependingon the value of a, one ya exists in the range 2npoyoð2nþ 1

2Þp and ð2nþ 32Þpoyo2np. For example, when

a¼�1, ya ¼ 13pþ2np, 5

3pþ2np.Case (vi): a¼�2. The result for a¼�2 is shown in

Fig. 11. The stable equilibrium points are at ðy, _yÞ ¼ðð2nþ1Þp,0Þ and the unstable equilibrium points are atð2np,0Þ.

Case (vii): ao�2. The result for a¼�3 is shown inFig. 12. The stable equilibrium points are at ðð2nþ1Þp,0Þand unstable equilibrium points are at ð2np,0Þ.

5.4. Application to attitude control

We have to this point elucidated the attitude dynamicsof the pendulum satellite under the influence of a gravitygradient torque and a Lorentz torque. Attitude controlusing the gravity gradient torque is mainly used for thepurpose of achieving satellite stability in the direction ofthe orbit radius. As we have seen, however, it is alsopossible to obtain a torque on a charged satellite as aneffect of the Lorentz force. In this section, we investigatethe possibility to utilize Lorentz torque for attitude con-trol by varying the amount of charge on the satellite,using a specific example.

Here we consider the attitude control using Lorentztorque to desired angle. At first, the attitude motion areassumed to be rotational motion with a¼ 3 and desiredstate is set to ðy, _yÞ ¼ ð53p,0Þ. Fig. 13 shows two character-istic curves with a¼ 3 and �1 in the phase plane. Weconsider two trajectories in phase plane shown by ‘‘curve1’’ and ‘‘curve 2’’ in Fig. 13 which are characteristic curvesfor a¼ 3 and �1, respectively. Since the trajectories crossat ‘‘point 1’’, shifting the state from curve 1 to curve 2 bychanging the charge will lead to the convergence to the

Fig. 13. Lorentz force acting on charged satellite.

equilibrium point ðy, _yÞ ¼ ð53p,0Þ (‘‘point 2’’). From Eq. (26),with R¼ 3

2m=r30 , the energies at a¼ 3 and �1, which are

expressed by J3 and J�1, respectively, are given by

J3 ¼1

2

dydt

� �2

þR

2ð1�cos 2yÞþ3Rð1�cos yÞ ð27Þ

J�1 ¼1

2

dydt

� �2

þR

2ð1�cos 2yÞ�Rð1�cos yÞ ð28Þ

Curve 1 passes through ðp,0:0015Þ, and curve 2 passesthrough ð53p,0Þ, and it is therefore possible to obtainenergies J3 and J�1. Let point 1 be ðy1, _y1Þ, then we find

J3�3Rð1�cos y1Þ ¼ J�1þRð1�cos y1Þ ð29Þ

and the solution for y1 is given by

y1 ¼ arccos 1�J3þ J�1

4R

� �ð30Þ

Thus it is shown that the desired attitude control can beperformed by varying the amount of charge so as tochange the a from 3 to �1. Numerically, this correspondsto R¼1.82�10�6 s�2, J3¼1.20�10�5 s�2, J�1¼4.55�10�7 s�2, y1C5p=4. This example shows that it is indeedpossible to perform attitude control by varying theamount of charge.

6. Conclusions

In this paper, we propose utilization of a Lorentztorque together with the conventional gravity gradienttorque, for charged satellite attitude motion control, andinvestigate the influence of these torques on the attitudemotion of a ‘‘pendulum satellite’’ having the shape of adumbbell pendulum, consisting of two charge-carryingmass points of equal mass connected by a rigid rod ofnegligible mass, with the center of mass at the midpointof the rod and thus at the center of the satellite. Thesatellite is assumed to be in a circular orbit in theequatorial plane, with the center of the satellite lying onthe orbit. The attitude motion of the satellite is derived,focusing on its pitch motion within the equatorial plane.We investigate the stability of the equilibrium points andpropose an attitude control method using the character-istics induced by changing the charge amount.

Acknowledgments

This research is supported by the Grant-in-Aid forChallenging Exploratory Research (No. 21656222) of theJapan Society for Promotion of Science.

References

[1] L.B. King, G.G. Parker, S. Deshmukh, J.H. Chong, Study of interspace-craft coulomb forces and implications for formation flying, Journal ofPropulsion and Power 19 (3) (2003) 497–505.

[2] A. Natarajan, H. Schaub, Linear dynamics and stability analysis of acoulomb tether formation, Journal of Guidance Control, andDynamics 29 (4) (2006) 831–839.

[3] B. Streetman, M.A. Peck, New synchronous orbits using the geomag-netic Lorentz force, Journal of Guidance Control, and Dynamics 30(6) (2007) 1677–1690.


[4] B. Streetman, M.A. Peck, Gravity-assist maneuvers augmented by theLorentz force, Journal of Guidance Control, and Dynamics 32 (5)(2009) 1639–1647.

[5] J.A. Atchison, M.A. Peck, Lorentz-augmented Jovian orbit insertion,Journal of Guidance Control, and Dynamics 32 (2) (2009) 418–423.

[6] J.W. Gangestad, G.E. Pollock, J.M. Longuski, Propellantless station-keeping at enceladus via the electromagnetic Lorentz force, Journalof Guidance Control, and Dynamics 32 (5) (2009) 1466–1475.

[7] Y. Utako, Y. Hiroshi, Two-craft coulomb-force formation dynamicsand stability analysis with debye length characteristics, in: AIAA/AASAstrodynamics Specialist Conference, AIAA 2008-7361, 2008.

[8] Y. Hiroshi, Y. Katsuyuki, B. Mai, Spacecraft formation dynamics underthe influence of geomagnetic Lorentz force, in: Twenty-seventhInternational Symposium on Space Technology and Science, 2009.

[9] B. Wie, Space vehicle dynamics and control, AIAA, Reston, Virginia,1998.

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Attitude dynamics of a pendulum-shaped charged satellite