Touchless attitude correction for satellite with constant magnetic moment

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ScienceDirect Advances in Space Research xxx (2017) xxx–xxx www.elsevier.com/locate/asr

Touchless attitude correction for satellite with constant magnetic moment Hou-jun Ao ⇑, Le-ping Yang, Yan-wei Zhu, Yuan-wen Zhang, Huan Huang National University of Defense Technology, Changsha, Hunan 410073, PR China Received 10 February 2017; received in revised form 19 May 2017; accepted 25 May 2017

Abstract Rescue of satellite with attitude fault is of great value. Satellite with improper injection attitude may lose contact with ground as the antenna points to the wrong direction, or encounter energy problems as solar arrays are not facing the sun. Improper uploaded command may set the attitude out of control, exemplified by Japanese Hitomi spacecraft. In engineering practice, traditional physical contact approaches have been applied, yet with a potential risk of collision and a lack of versatility since the mechanical systems are missionspecific. This paper puts forward a touchless attitude correction approach, in which three satellites are considered, one having constant dipole and two having magnetic coils to control attitude of the first. Particular correction configurations are designed and analyzed to maintain the target’s orbit during the attitude correction process. A reference coordinate system is introduced to simplify the control process and avoid the singular value problem of Euler angles. Based on the spherical triangle basic relations, the accurate varying geomagnetic field is considered in the attitude dynamic mode. Sliding mode control method is utilized to design the correction law. Finally, numerical simulation is conducted to verify the theoretical derivation. It can be safely concluded that the no-contact attitude correction approach for the satellite with uniaxial constant magnetic moment is feasible and potentially applicable to on-orbit operations. Ó 2017 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: Touchless attitude correction; Configuration design; Electromagnetic torque; Sliding mode control

1. Introduction In recent years, on-orbit service of faulty spacecraft has gained great concern. Traditional contact operation method has potential risk of collision, and lacks versatility. To overcome these shortcomings, a great deal of research has been done on touchless force, mainly including electromagnetic force, Coulomb force and flux-pinned force. Among them, the operating distance of flux-pinned force is too small and Coulomb force cannot provide torque directly. Trevor Bennett surveyed the prospects and ⇑ Corresponding author.

E-mail addresses: [email protected] (H.-j. Ao), [email protected] (L.-p. Yang), [email protected] (Y.-w. Zhu), [email protected] (Y.-w. Zhang), [email protected] (H. Huang).

challenges of detumbling large debris objects near Geostationary Earth Orbit for active debris remediation with Coulomb force, using an electron gun to charge the servicer and debris (Bennett et al., 2015). While mission success depends on the debris’ geometry, inertia properties and material properties. Electromagnetic force is capable of 6 degree of freedom (6-DOF) control naturally, which would be the best choice for touchless attitude correction. With the development of high temperature superconductor technology, electromagnetic coils could produce much stronger dipole moment. Thus stronger control ability and further operating distance becomes available (Kwon and Sedwick, 2011). Many satellites use flywheels as main attitude control system actuator and magnetic torquer as the second (Huang, 1997). However, if any fault occurs on the target satellite, making its flywheels shut off and the

http://dx.doi.org/10.1016/j.asr.2017.05.040 0273-1177/Ó 2017 COSPAR. Published by Elsevier Ltd. All rights reserved.

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magnetic torquer out of control (can only offer constant magnetic moment), then the external control torque is needed to correct the target’s attitude to rescue it. In another case of Clarke station (an artificial gravity space station at the earth-moon L1 point), the most prominent external torque is from solar pressure, which is on the order of 10
2 N m (Ashmore et al., 2001). For counteracting the disturbance, it is inadvisable to utilize reaction wheels since mechanical failure may cause great loss of life. Magnetorquers are also unsuitable because of strong magnetic fields. In the literature (Ashmore et al., 2001), hot gas thrusters have been chosen for counteracting the disturbance. While 56 thrusters and 300 kg propellant are needed, leading to increased complexity and limited operating life. In contrast, touchless attitude control with electromagnetic torque is ideal for the proposed scheme as the station spins stably and only minor attitude corrections are necessary. We proposed a method of touchless attitude correction, using small on-orbit service satellites to correct the attitude of high-value target satellite through inter-satellite magnetic torque. As magnetic torque is a function of the cross-product of magnetic moment and magnetic field (Forbes and Damaren, 2011), the magnetic field of active spacecraft is designed perpendicular to the magnetic moment of the chaser satellite to produce larger strength magnetic torque. And then, two correction configurations (i.e. ‘‘One-to-One” mode and ‘‘Two-to-One” mode) are introduced to avoid magnetic force producing at the target satellite. A number of literatures have studied the attitude stabilization and control method using magnetic torque based on its interaction with geomagnetic field. The attitude control system of nano-satellite ‘‘Tian Tuo 1” was designed with a pitch bias wheel and three magnetic coils as actuators (Ran et al., 2014). The test approval and on-orbit flying showed that the satellite is featured with three-axis stabilization control capability and it is suitable, robust and feasible. Ovchinnikov proposed semi-analytic method utilizing Floquet theory to optimize the parameters of PD-controller for three-axis magnetic control (Ovchinnikov et al., 2015a, 2015b). By selecting a suitable spin rate and orbital maneuver duration, Ousaloo demonstrated the process of increasing the perigee altitude by the use of high-magnitude thrust for orbital maneuver and the use of low-torque magnetic actuators for attitude maneuver. The attitude disturbance caused by the orbital maneuvering procedure is eliminated via four magnetic torquers, and the spacecraft easily executes its successive orbital maneuvers at the apogee point (Ousaloo, 2014). In addition, some researches focus on the hybrid application of magnetic actuators and flywheels. Forbes designed the attitude controller for a spacecraft equipped with both reaction wheels and magnetic torque rods, where the control torques are distributed between wheels and magnetic torque rods in a natural way guided by the physical constraints imposed by magnetic actuation (Forbes and Damaren, 2011). Sliding mode control has also been intro-

duced to magnetic control systems in some research. Sofyali proposed a modified sliding mode control algorithm for purely magnetic attitude control of small satellites (Sofyali and Jafarov, 2012). Janardhanan put forward a second order sliding mode control with a nonlinear sliding surface for the three-axis attitude control of magnetic actuated rigid spacecraft and demonstrated that super-twisting based control with proposed nonlinear sliding surface is able to generate sufficient control torque, which results in the three-axis controllable spacecraft (Janardhanan et al., 2012). All of these researches are based on the premise that the magnetic actuators are controllable. Magnetic force has also been studied in satellite formation control and orbit correction (Ahsun, 2007; Cai et al., 2013; Huang et al., 2014; Zhang et al., 2012, 2014). Robert proposed an approach to use alternating fields and currents to induce finely-controlled forces on the satellites, and gave a general drive satellite configuration which can provide full position and orientation control of a number of satellites (Robert et al., 2013). However, up to now, few work have been reported that investigate inter-satellite magnetic torque used in attitude correction, especially for the target satellite with failed flywheels and uncontrollable magnetic coils, which will be the focus of the work presented here. This paper is organized as follows. The next section designs the correction configuration and establishes the rotational dynamics model including the disturbance torque. In Section 3, the control law based on the sliding mode control method is proposed. Section 4 gives the method of solving active magnetic moment and flywheel speed, and in Section 5, the simulation results are provided and discussed. Section 6 completes the paper with conclusion remarks. 2. Problem formulation In this paper, we consider the target satellite that can only provide uniaxial constant magnetic moment and its attitude control system fails. Active satellites, which are equipped with three orthogonal electromagnetic coils and reaction wheels, are used to correct the attitude of target satellite in a circular orbit. In fact, uniaxial constant magnetic moment means several cases including one, two or even more uncontrollable constant magnetic torques on the target satellite, because different constant magnetic moment can be composited into only one. Of the main environment torques in the low orbit, which include magnetic torque, gravity gradient torque and aerodynamic torque, the first one has a greater impact than others on the satellite with electromagnetic coils (discussed in Section 2.3). Therefore, in following theoretical derivation, the geomagnetic torque is described accurately in the mathematical model and the other environment torques are treated as stochastic disturbance. The non-contact attitude correction approach is depicted as Fig. 1. Firstly, correction configuration is

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pivot on the direction perpendicular to lt . Simultaneously, by comparing the two far field equations we can find that when lt , la and rta are perpendicular to each other, the interaction force is zero and at the same time the torque exists. Based on this conclusion, the first operational configuration is designed as Fig. 2. In this configuration, two conditions need to be satisfied to keep the electromagnetic force zero. Firstly, la and lt must remain perpendicular to rta . Secondly, la must rotate around rta with the same rate as the target satellite to keep orthogonal. However, in the actual process, these conditions cannot be always satisfied due to the uncertainties such as disturbance torque and time delay of the control system. To solve this problem, we introduce a ‘‘two-toone” mode, namely using two identical active satellites (represented with a1 and a2) to correct the attitude of one target satellite. The electromagnetic force and torque on the target satellite due to active satellites can be expressed as:

Correction Configuration Design Producing larger strength magnetic torque without magnetic force

Two-to-One

Mode

Introducing Reference Frame & Considering Accurate Varying Geomagnetic Field

Attitude kinematics and dynamics Sliding Mode Control

Correction Control Law Additional Constraint

Solution of magnetic moment and flywheel speed Fig. 1. Diagram of the correction approach.

designed as ‘‘One-to-One” mode. To produce larger strength magnetic torque without magnetic force acts on the target satellite, ‘‘Two-to-One” mode is proposed to overcome the disadvantage of ‘‘One-to-One” mode. Then a reference frame based on the initial and desired attitude of target satellite is introduced to simplify the correction process. And attitude kinematics and dynamics mode are established after that. A control law based on the sliding mode control method is proposed based on the dynamics mode. In the end, the magnetic moment and flywheel speed are solved under additional constraint. 2.1. Attitude correction strategy In order to reduce the impact on the orbit of the target during attitude correction, the electromagnetic force acting should be avoided. Using the far-field model, the interaction force/torque can be calculated by Ahsun et al. (2010): 8 h i lt rta la rta ðlt rta Þðla rta Þ > 0 lt la < F ta ¼
3l r þ l þ l
5 r ta ta 5 5 5 7 a t 4p rta r rta rta  ta  l0 lt 3rta ðla rta Þ la > : sta ¼ 4p 
r3 r5 ta

3

ta

ð1Þ Eq. (1) gives the force and torque on the target satellite due to the active satellite, where lt , la are the dipole strength of the target and the active respectively (lt is constant, la is the vector sum of three orthogonal coils on the active satellite, we can control the current in each coil to orient this dipole vector arbitrarily), rta the relative distance vector between two satellites, l0 the permeability of free space, ‘‘” the dot-product and ‘‘” the cross-product operator. Since lt is constant, it can be seen from the far field model and the nature of cross-product that no electromagnetic torque is generated along the direction of lt . Hence the touchless attitude correction can only control the target

8   la rta1 ðlt rta1 Þðla rta1 Þ > lt rta1 3l0 lt la1 > 1 1 > F ¼
r þ l þ l
5 r ta1 ta1 < ta1 a1 t 4p r7ta r5ta r5ta r5ta 1 1 1 1   > lt la la rta2 ðlt rta2 Þðla rta2 Þ lt rta 3l > 2 > r : F ta2 ¼
4p0 r5 2 rta2 þ r5 2 la2 þ r25 lt
5 ta 7 2 r ta2

ta2

ta2

ta2

ð2Þ

8   3rta1 ðla rta1 Þ la > l0 lt > 1 1 >
r3 < sta1 ¼ 4p  r5ta ta1 1   > 3rta2 ðla rta2 Þ la l0 lt > 2 2 >
r3 : sta2 ¼ 4p  r5 ta2

ð3Þ

ta2

It can be found from Eqs. (2) and (3) that F ta2 ¼
F ta1 ; sta2 ¼ k  sta1 if rta2 ¼
k  rta1 ; la2 ¼ k 4  la1 , where k is determined by rta1 and rta2 . Thus, the ‘‘two-toone” configuration is designed as Fig. 3. In this configuration, the constraints of ‘‘one-to-one” mode are nonexistent. The resultant electromagnetic force on the target is zero and resultant torque is 1 þ k times of the torque due to satellite-a1. Furthermore, the resultant electromagnetic torques on satellite-a2 and satellite-a3 have the following relation: sa2 t ¼ k  sa1 t ; sa2 a1 ¼ sa1 a2 . In this paper, we set k ¼ 1, which means two active satellites are located at the same distance from the target satellite. Thus the resultant electromagnetic torques on the target satellite is twice of the torque due to one active satellite. The torques on satellite-a1 and satellite-a2 are identical.

Target Satellite

Active Satellite

A

T

rTA

AT

TA

Fig. 2. ‘‘One-to-One” mode configuration.

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H.-j. Ao et al. / Advances in Space Research xxx (2017) xxx–xxx YR t initial

Satellite-a1

a1 y

Satellite-t

Satellite-a2

XR

a2 y

t-desired

a1 x

a2 x

ZR

a1 z

a2 z

Fig. 3. ‘‘Two-to-One” mode configuration.

2.1.1. Reference frame For the convenience of configuration design and control, a reference frame R (shown in Fig. 3) is introduced ^t
desired (the desired as follows: the XR axis is aligned with l direction of the magnetic dipole on the target), YR axis is perpendicular to XR axis and in the plane that is deter^t
desired and l ^t
initial (the initial direction of the mined by l magnetic dipole on the target), the ZR axis completes the right hand system. Thus, the attitude correction can be considered as the rotation of lt in XR-YR plane. Actually during the control process, vector lt may get off XR-YR plane due to external disturbances, but it would not deviate far under the effect of control. Two active satellites are placed at the points ½ 0 0 rta1  and ½0 0
rta1  to ensure k ¼ 1. Assuming that the translational movement of the active satellites is controlled well by the thrusters assembled on them, thereby the change of the relative position is ignored in this paper, our research mainly concentrates on the attitude motion. Three orthogonal reaction wheels equipped on the active satellites are used to counter the magnetic torque to maintain their attitude relative to the frame R, and the wheels’ gyroscopic effect will also be included in the dynamic equations. Under the constraint la2 ¼ k 4  la1 , the target attitude is corrected through synchronous control of la1 and la2 .

2.2. Dynamic model exploiting inter-craft electromagnetic and geomagnetic fields 2.2.1. Coordinate frames and transformation relation To build the dynamic model, several coordinate systems in common use are introduced as shown in Figs. 4 and 5, where the Earth Centered Inertial (ECI) frame is noted as I, Earth-Centered Earth-Fixed (ECEF) frame E, North-East-Down (NED) frame N, Local-Vertical LocalHorizontal (LVLH) frame L and Body frame B. Frame Bi’s origin is attached to the i-th (i ¼ t; a1; a2) spacecraft center of mass Obi , and the XBi, YBi, ZBi axes are fixed with the principal inertia axes of the spacecraft. The transformation matrices of the coordinate systems are defined as follows: 8 I > M E ¼ M 3 ð
/E Þ > >   > E > > M N ¼ M 2 p2 M 1 ð
uS ÞM 2 p2
hS > > > E M L ¼ M 3 ð
/SE ÞM 1 ð
iS ÞM 3 ð
wS Þ > > > > R > M L ¼ M 3 ðjÞM 2 ðrÞM 1 ðmÞ > > > : Bi R M ¼ M 1 ðci ÞM 2 ðbi ÞM 3 ðai Þ where /E is the angle between OEXE and OEXI, which can be acquired through astronomical calendar table, uS , hS , /S , iS and wS are the longitude, colatitudes, right ascension

ZE

Z I (Z E )

ZL YL

XN

XL iS φE

OE

φS

YN

YE

θS

ψS

φSE

θS YI

OE

ZN

ϕS

YE

XI XE

XE Fig. 4. Relative coordinate systems.

Fig. 5. NED coordinate system.

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R

M L ¼ M 3 ðjÞM 2 ðrÞM 1 ðmÞ

ð5Þ

For analysis convenience, we make the following assumptions: (a) The magnetic moment of the target satellite lt is on the direction of XBt, i.e. Bt lt ¼ ½ lt 0 0 T , where the presuperscript denotes the coordinate frame. (b) The electromagnetic coils and reaction flywheels of the active satellites are installed along their principal inertia axes. (c) The X, Y, Z axes of frame Ba1 and Ba2 are kept in the same direction with those of frame R, since the active satellites’ attitude relative to the frame R is maintained by the reaction flywheels. Thus we have:

Bi

M R ¼ E;

i ¼ a1; a2

ð6Þ

Without subscript, B defaults to the body reference frame of the target satellite. 2.2.2. Effect of the Earth’s magnetic field There are various ways to simulate the earth magnetic field and among these ways the most accurate one is the harmonic coefficients and the earth magnetic field mathematical model (Navabi and Nasiri, 2011). In this study, the International Geomagnetic Reference Field (IGRF12) magnetic field model (apply to 2015–2020) is adopted. The geomagnetic field is a potential field and can be written as: nþ1 1 X n  X RE V ¼ RE ðgmn cos mu þ hmn r n¼1 m¼0  sin muÞPmn ðcos hÞ

ð7Þ

where r is the geocentric distance, RE the equatorial radius of earth, gmn and hmn spherical harmonics, n the degrees, m the order, Pmn ðxÞ the Schmidt normalized associated Legendre polynomials. Generally, a satisfactory approximate solution can be obtained with the first three polynomials as: V ¼

R3E 0 ½g cos h þ ðg11 cos k þ h11 sin kÞ sin h r2 1

8 N > Dx ¼ 1r > > < N

> > > :N

@V @h

R3

¼
r3E ½g01 sin h
ðg11 cos u þ h11 sin uÞ cos h

1 Dy ¼
r sin h

@V @u

Dz ¼
@V ¼ @r

2R3E r3

¼

R3E r3

ðg11 sin u
h11 cos uÞ

½g01 cos h þ ðg11 cos u þ h11 sin uÞ sin h ð9Þ

where D is the magnetic flux density. The approximate geomagnetic field distribution at the altitude of 500 km can be calculated by Eqs. (8) and (9) as shown in Fig. 6. As a result, the geomagnetic strength at 500 km altitude is about 2  5  10
5 T. Assume the magnetic moment on the satellite is 100 A m2 (off-the-shelf electromagnetic torquers exist up to a strength of 400 A m2 (Voirin et al., 2012), then it can be seen from electromagnetic torque formula se ¼ l  De that the geomagnetic torque reaches maximum when the magnetic moment is perpendicular to the local geomagnetic field, and the value is on the order of 10
3 N m. 2.2.3. Spacecraft attitude kinematics and dynamics The spacecraft attitude kinematics and dynamics are developed in frame B. For a rigid body rotating about its center of mass, its attitude dynamics can be described by the Euler equation in frame I (Hughes, 1986) as: _ ¼s H

ð10Þ

which can be converted into frame B by the Transport Theorem as (Ahsun, 2007): _ þ B xB=I  H ¼ s H

ð11Þ T

where B xB=I ¼ ½ x1 x2 x3  is the angular velocity of the satellite with respect to the inertial frame, s is the external torque acting on the satellite (including electromagnetic control torque sm , geomagnetic torque se and uncertain

-5

x 10 5

Geomagnetic strength (T)

of ascending node, inclination and argument of the target satellite, /SE ¼ /S
/E . In this paper, the attitude of the satellites is described in the orbital coordinate system. As frame R is determined by the initial and desired direction of the magnetic dipole on the target satellite, the transformation matrix from L to ^t
initial and l ^t
desired are decided. R would be constant once l Assume frame R could be obtained by rotating L as 1-2-3 Euler angles (m, r, j):

5

4.5 4 3.5 3 2.5 2 200

200 100

150

ð8Þ

The gradients of V along northward, eastward and downward are the components in three axes of frame N:

0

100 -100

50

Latitude (deg)

0

-200

Longitude (deg)

Fig. 6. Profile of the geomagnetic strength at 500 km altitude.

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disturbance torque sd ), H is the total angular momentum of the satellite due to its rotation about its center of mass: 8 > < H ¼ H B þ H RW H B ¼ I B  B xB=I ð12Þ > : H RW ¼ I RW X where I B and I RW represent the inertial matrixes of the T spacecraft and reaction wheels, X ¼ ½ X1 X2 X3  is the spin rate vector of the reaction wheels. Substituting (12) into (11) yields the attitude dynamics as: 8 I 1 x_ 1 þ ðI 3
I 2 Þx2 x3 þ x2 I RW 3 X3
x3 I RW 2 X2 > > > > ¼ s þ s þ s
I X_ > mx ex dx RW 1 1 > > > < I x_ þ ðI
I Þx x þ x I X
x I X 2 2 1 3 1 3 3 RW 1 1 1 RW 3 3 ð13Þ _ > ¼ s my þ sey þ sdy
I RW 2 X2 > > > > > > > I 3 x_ 3 þ ðI 2
I 1 Þx1 x2 þ x1 I RW 2 X2
x2 I RW 1 X1 : ¼ smz þ sex þ sdz
I RW 3 X_ 3 Then, we express the geomagnetic torque se and angular velocity B xB=I in frame B. T

(1) B se ¼ ½ se1 se2 se3  As the distance between the satellites (generally in meter-scale) is far smaller than that from the satellite to the center of the earth, it can be assumed that three satellites are in the same electromagnetic environment Be . To calculate the geomagnetic torque, we need to express the magnetic flux density in the body frame, which can be obtained by transforming N B using the transformation matrices as: B

B ¼ BM N  N B

ð14Þ

where B

M N ¼ BM R  RM L  LM I  I M E  EM N ¼ M 1 ðcÞM 2 ðbÞM 3 ðaÞM 3 ðjÞM 2 ðrÞM 1 ðmÞM 3 ðwS Þ T

M 1 ðiS ÞM 3 ð/S ÞM 3 ð/E Þ . . . p p  T M2 M 1 ðuS Þ M 2
hS 2 2

ð15Þ

/E ; wS ; uS and hS can be calculated by Huang (1997): 8 /E ¼ /E0 þ nE t > > > u ¼ / þ ¼ /
/ þ arctanðtan w cos i Þ S > S SE S E S > : hS ¼ p=2
¼ p=2
arcsinðsin wS sin iS Þ where /E0 and wS0 denote the initial values, nE is the rotapffiffiffiffiffiffiffiffiffiffiffiffi tional angle velocity of the earth, nS ¼ lE =r3S is the mean motion of the satellites, lE is the gravitational coefficient of the earth, rS is the radius of the orbit (see Fig. 7). Substituting Eqs. (15) and (16) into Eq. (14), we can obtain the magnetic flux density in the body frame B B. Then the geomagnetic torque se can be calculated as: B

se ¼ B l  B B e

ð17Þ

(2) B xB=I ¼ ½ x1

x2

T

x3 

The rotational angular velocity vector of frame B with respect to frame I can be expressed as a composition of three relative angular velocities: B

xB=I ¼ B xB=R þ B xR=L þ B xL=I

ð18Þ

where B xR=L ¼ 0 as the transformation matrix of frame R and frame L is fixed, B xL=I can be obtained based on T L xL=I ¼ ½ 0 0 nS  and the defined rotating matrix R M L as: B

xL=I ¼ B M R  R M L  L xL=I

ð19Þ

To calculate B xB=R , we need to discuss the active satellites and target satellite separately. (i) Active satellite According to the previous discussion, two active satellites share geomagnetic environment and electromagnetic torque. Hence we just study one of them, and take satellite-a1 for instance. As the active satellites’ attitude relative to frame R is maintained by the reaction flywheels, we can get that Ba1 xBa1 =R ¼ 0. Meanwhile, Ba1 M R ¼ E and R M L is constant, so the rotational angular velocity vector Ba1 xBa1 =I ¼ R M L  L xL=I is also constant, which means that Ba1 x_ Ba1 =I ¼ 0. Thus the dynamic model of the active satellite can be expressed as: 8 ðI 3
I 2 Þx2 x3 þ x2 I RW 3 X3
x3 I RW 2 X2 > > > > ¼ s þ s þ s
I X_ > > mx ex dx RW 1 1 > > > > ðI
I Þx x þ x I > 1 3 1 3 3 RW 1 X1
x1 I RW 3 X3 > > > < ¼ smy þ sey þ sdy
I RW 2 X_ 2 ð20Þ ðI 2
I 1 Þx1 x2 þ x1 I RW 2 X2
x2 I RW 1 X1 > > > > > > ¼ smz þ sez þ sdz
I RW 3 X_ 3 > > > T T > > ½ x x2 x3  ¼ R M L  ½ 0 0 nS  > > 1 >  T : T ½ sex sey sez  ¼ lx ly lz  Ba1 D (ii) Target satellite Based on the definitions of Euler angles (ai ; bi ; ci ) and rotating matrix Bi M R , the rotational angular velocity vector of frame B with respect to frame R expressed in B can be written as: 2 3 c_
a_ sin b 6 7 B xB=R ¼ 4 b_ cos c þ a_ sin c cos b 5 ð21Þ
b_ sin c þ a_ cos c cos b Thus the rotational angular velocity vector of frame B with respect to frame I is: 2 3 2 3 c_
a_ sin b b1 6 _ 7 B R6 7 ð22Þ x ¼ 4 b cos c þ a_ sin c cos b 5 þ M 4 b2 5
b_ sin c þ a_ cos c cos b b3

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2.3. Disturbance torque

Z I (Z E )

For the satellites in the near-earth orbit, there exists two main interference torques, namely gravity gradient and aerodynamic torque. Gravity gradient torque can be calculated by the following formula: 2 3 ðI 3
I 2 Þr2 r3 3l 6 7 s g ¼ 5 4 ðI 1
I 3 Þr1 r3 5 ð25Þ r ðI 2
I 1 Þr1 r2

ZL YL

XL

S

iS φE

OE

YE

θS

ψS

φSE

φS

YI

B

A

XI XE

Fig. 7. Geometry relationship of /E ; wS ; uS and hS . T

where ½ b1 b2 b3  ¼ R M L  L xL=I is constant. Transforming Eq. (22), we can get the expression of _ c_ Þ as: _ b; ða; 2 3 2 a_ 0 6_7 6 4b5 ¼ 40 c_

sin c sec b cos c

1 sin c tan b 2
cos a tan b 6 þ4 sin a
cos a sec b

cos c sec b

7

32

x1

3

76 7
sin c 54 x2 5 x3 cos c tan b
sin a tan b
cos a
sin a sec b


1

32

b1

3

76 7 0 54 b2 5 0 b3 ð23Þ

It can be seen from Eq. (23) that there would be singular value if b ¼ p=2. However, due to the design of the correction configuration, the target satellite mainly revolves around ZR axis (corresponds to a) in actual correction process. The initial value of b is 0 and its change can be suppressed by the control magnetic torque. So the Euler angle b floats near 0, avoiding the occurrence of singular value. Since there is no operative flywheel assembled on the target satellite, its dynamic model can be expressed as: 8 I 1 x_ 1 þ ðI 3
I 2 Þx2 x3 ¼ smx þ sex þ sdx > > > > > I 2 x_ 2 þ ðI 1
I 3 Þx1 x3 ¼ smy þ sey þ sdy > > > > > I 3 x_ 3 þ ðI 2
I 1 Þx1 x2 ¼ smz þ sez þ sdz > > > > > > < a_ ¼ x2 sin c sec b þ x3 cos c sec b
b1 cos a tan b
b2 sin a tan b
b3 > > _ > b ¼ x2 cos c
x3 sin c þ b1 sin a
b2 cos a > > > > > c_ ¼ x1 þ x2 sin c tan b þ x3 cos c tan b > > > > >
b1 cos a sec b
b2 sin a sec b > > >  T : T ½ sex sey sez  ¼ lx ly lz  B D

ð24Þ

In the orbit at an altitude of 500 km, assuming the difference between the rotational inertias is 10 kg m2, the gravity gradient torque would be 10
5 N m scale. And previous studies have shown that the aerodynamic torque is on the same order with gravity gradient (Huang, 1997), which is much smaller than the geomagnetic torque (10
3 N m scale). Therefore, these two torques will be treated as 10
5 N m scale stochastic disturbance in the simulation process. 3. Control algorithm design Sliding mode variable structure control is a special kind of nonlinear control in essence. Sliding mode can be designed and is unrelated to the object parameter and disturbance. It has the advantages of quick response, insensitiveness to parameter change and disturbance, no need of on-line identification of parameters, and simple physical implementation (Huang, 1997; Zeng and Hu, 2012; Hu and Zeng, 2011). Thus it is widely used in practical engineering, especially in the fields of motor and power system control, robot control, aircraft control and satellite attitude control, etc. In this paper, the sliding mode control is applied to the attitude correction control law design. Note that the magnetic moment of the target satellite ltarget is on the direction of XB, hence no electromagnetic torque generated along XB. We just need to design the algorithm on the direction of YB and ZB. Take the sliding mode control hyperplane as (Liu, 2012):

sy ¼ cy ey þ e_ y ð26Þ sz ¼ cz ez þ e_ z where ey ¼ b
bt arg et ; ez ¼ a
at arg et are the system error (the determination of at arg et and bt arg et could be achieved by the 3-D camera equipped on the active satellites). The sliding mode control law is designed as T B smt ¼ ½ 0 smy smz  :

smy ¼ ðI 1
I 3 Þx1 x3
sdy
ey sy
gy signðsy Þ ð27Þ smz ¼ ðI 2
I 1 Þx1 x2
sdz
ez sz
gz signðsz Þ To eliminate high frequency chattering effect, we replace the sign function signðsÞ with a saturation function satðs=0:01Þ, written as

y if jyj 6 1 satðyÞ ¼ ð28Þ sgnðyÞ otherwise

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H.-j. Ao et al. / Advances in Space Research xxx (2017) xxx–xxx

4. Solution of magnetic moment and flywheel speed In the previous section, the desired electromagnetic control torque acting on the target satellite has been designed. Then we need to solve the required magnetic moment and flywheel speed of the active satellites. As discussed in Section 2.1, the electromagnetic torques acting on the target by the two active satellites are equal, i.e. smt ¼ sta1 þ sta2 ¼ 2sta1 . The magnetic moment of satellitea1 can be calculated in frame B, which is B la1 , and then transformed to frame Ba1. Using the far-field model, the electromagnetic torque acting on satellite-1 can be written as: B

smt ¼ 2sta

2 3 0 l0 lt 6 7 B B 2 2 B ¼ 4
½3rx ry  la1 x þ 3ry rz  la1 y þ ð3rz
rta1 Þ  la1 z  5 2pr5ta1 3rx ry  B la1 x þ ð3r2y
r2ta1 Þ  B la1 y þ 3rz ry  B la1 z ð29Þ T

where ½ rx ry rz  ¼ B M R  ½ 0 0 rta1 . Combining Eqs. (27) and (29), there are two independent equations, but three unknowns B la1 x ; B la1 y ; B la1 z , constituting an overdetermined equation system. To get the sole solution to the equations, an additional constraint is required. In fact, it can be concluded from the design of ‘‘Two-toOne” configuration that, the direction of lt turns mainly in the XR-YR plane during the correction (small angle deviates from the plane due to the disturbance torque). Therefore, XB keeps approximately perpendicular to rta1. From the far-field model (Eq. (1)), we can find that the interaction torque generated by the active magnetic moment on the direction of XB is zero if B la1 x is perpendicular to rta1. Hence the effects of B la1 x are negligible (i.e. the electromagnetic torque generated by B la1 x would be very small even if B la1 x is very large). Thus we can make B la1 x ¼ 0 to ensure a sole solution to the equations. Since the reaction wheels equipped on the active satellites are used to counter the magnetic torque to maintain their attitude relative to the frame R, with Eq. (20) the output torque and speed of the flywheels can be calculated through iterative method as: 8 > X_ 1 ðk þ 1Þ ¼ ½smx ðkÞ þ sdmx ðkÞ þ sdux ðkÞ
I 1 x_ 1 > > > > >
ðI 3
I 2 Þx2 x3
x2 I RW 3 X3 ðkÞ > > > > > þx3 I RW 2 X2 ðkÞ=I RW 1 > > > > _ > X2 ðk þ 1Þ ¼ ½smy ðkÞ þ sdmy ðkÞ þ sduy ðkÞ
I 2 x_ 2 > > > <
ðI 1
I 3 Þx1 x3
x3 I RW 1 X1 ðkÞ ð30Þ > þx1 I RW 3 X3 ðkÞ=I RW 2 > > > > > X_ 3 ðk þ 1Þ ¼ ½smz ðkÞ þ sdmz ðkÞ þ sduz ðkÞ
I 3 x_ 3 > > > > >
ðI 2
I 1 Þx1 x2
x1 I RW 2 X2 ðkÞ > > > > > > þx2 I RW 1 X1 ðkÞ=I RW 3 > > : Xi ðk þ 1Þ ¼ Xi ðkÞ þ X_ i ðk þ 1Þ  Dt; i ¼ 1; 2; 3

To avoid saturation of the reaction wheels speed, electric propulsion (EP) thrusters are required for wheel offloading. Once the cumulative speed exceeds the limit, a constant torque by EP of 70 mN m (which is within the capability of a force-free arrangement of the RIT-10 thrusters (Voirin et al., 2012) would be provided to absorb the angular momentum until the speed is reduced to a low level. 5. Numerical simulations and results analysis In this section, we provide a numerical example to demonstrate the correction method. The satellite parameters are: ( I 1 ¼ diagð½ 50 51 52 Þkg m2 ð31Þ I 2 ¼ diagð½ 20 21 22 Þkg m2 The satellites are in a circular orbit with an altitude of 500 km, inclination of 30 deg, right ascension of ascending node of 60 deg, initial argument of latitude of 30 deg, initial angle between OEXE and OEXI of 25 deg. The distance between the target satellite and the active satellites is 5 m; magnetic moment of the target satellite is 200 A m2; rotational inertia of the flywheel is 0.05 kg m2 and its maximum output torque is 0.1 N m. A constant torque by EP of 70 mN m would be provided if the flywheel speed exceeds 1500 rpm until it is reduced to 100 rpm. Assume direction of the target magnetic moment is along the local geomagnetic field direction, and the purpose is to control it to earth direction. Then the Euler angles of frame L and Frame R can be determined as (not unique): m ¼ 72:06 deg; r ¼ 0;

j¼0

ð32Þ

And the initial and terminal conditions can also be calculated as: Initial :

a ¼ 76:6 deg;

Terminal :

b ¼
10 deg;

x1 ¼ 0; x2 ¼ 1 red=s; a ¼ 0; b ¼ 0

c ¼ 10 deg;

x3 ¼
1 red=s ð33Þ

The control parameters are selected as: ey ¼ 2;

gy ¼ 5:1  10
4 ; cy ¼ 0:025

ez ¼ 1:5; gz ¼ 5:2  10
4 ;

cz ¼ 0:033

ð34Þ

The simulation results are shown as Figs. 8–13. It can be seen from Fig. 8 that a and b converge to 0 in 5 min with no singular value problems. The Euler angle c (the rotation around XR) floats in a small angle since the direction of the target satellite magnetic moment stabilizes along with XR and there is no control torque generated in this direction. Thus the rotation around XR is uncontrolled, which can also be reflected from Fig. 9 where the target’s angular velocity of the X direction is fluctuating due to the effect of disturbance torque. Its angular velocity

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H.-j. Ao et al. / Advances in Space Research xxx (2017) xxx–xxx

0 -100

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of the Y direction converges to approximately 0.06 rad/s, which is because in the terminal state, the target rotates with frame L and keeps the same angel velocity. From the simulation results, we can also find that during the

correction process, the magnetic moment on the active satellite is less than 12,000 A m2 (which can be achieved with superconductor magnet as literature (Kaneda et al.,

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H.-j. Ao et al. / Advances in Space Research xxx (2017) xxx–xxx

2004) has demonstrated the feasibility of the superconductor magnets of 60,000 A m2); the maximum output torque and cumulative speed of the reaction flywheel is smaller than 0.08 N m and 1500 rpm, respectively, conforming to the actual situations. The numerical simulations show that the proposed correction method is feasible with a high controlling accuracy and short control convergence time. 6. Conclusion and prospects Based on the theoretical study and numerical simulation analysis, the following conclusions could be drawn. (1) The proposed ‘‘Two-To-One” operational configuration is able to counteract the impact of electromagnetic force on the target satellite during the attitude correction process. (2) The introduction of reference frame can effectively avoid the singular value problem and simplify the control process. (3) The sliding mode variable structure control is proved to have fast convergence speed and high accuracy, thus feasible for the attitude correction. In the next step of work, we will take orbital control into account to study the 6-DOF problem involved in noncontact attitude correction method. As electromagnetic coils enable touchless orbit control and attitude control, we can install them as standardized part on future satellites. It will bring great convenience to on-orbit operations, including orbit correction, attitude correction, failed satellites detumbling and de-orbit. It is of important engineering value for spacecraft rescue and space debris removal. Acknowledgments The first author would like to acknowledge his colleague Ms. PENG Wangqiong for her revising suggestions in the paper’s English writing. This work is supported in part by a grant from the National Natural Science Foundation of China (11502287) and the Scientific Research Program of National University of Defense Technology (JC15-01-05). References Ahsun, U., 2007. Dynamics and Control of Electromagnetic Satellite Formations (Ph.D. Dissertation). Massachusetts Institute of Technology, USA. Ahsun, U., Miller, D.W., Ramirez, J.L., 2010. Control of electromagnetic satellite formations in near-earth orbits. J. Guid., Control, Dyn. 33 (6), 1883–1891. Ashmore, Matthew, Barkmeyer, Daniel, Daddino, Laurie, et al., 2001. Clarke Station: An Artificial Gravity Space Station at the Earth-Moon L1 Point. Available: .

Bennett, Trevor, Stevenson, Daan, Hogan, Erik, et al., 2015. Prospects and challenges of touchless electrostatic detumbling of small bodies. Adv. Space Res. 56, 557–568. Cai, W.W., Yang, L.P., Zhu, Y.W., et al., 2013. Optimal satellite formation reconfiguration actuated by inter-satellite electromagnetic forces. Acta Astronaut. 89, 154–165. Forbes, J.R., Damaren, C.J., 2011. Linear time-varying passivity-based attitude control employing magnetic and mechanical actuation. J. Guid., Control, Dyn. 34 (5), 1363–1372. Hu, M., Zeng, G.Q., 2011. Finite-time control for electromagnetic formation of fractionated spacecraft based on terminal sliding mode technique. Aerosp. Control 29 (6), 22–33. Huang, Z.G., 1997. Spacecraft Attitude Dynamics. National University of Defense Technology Press, China. Huang, H., Yang, L.P., Zhu, Y.W., et al., 2014. Collective trajectory planning for satellite swarm using inter-satellite electromagnetic force. Acta Astronaut. 104, 220–230. Hughes, Peter C., 1986. Spacecraft Attitude Dynamics. John Wiley and Sons, New York. Janardhanan, S., Nabi, M., Tiwari, P.M., 2012. Attitude control of magnetic actuated spacecraft using super-twisting algorithm with nonlinear sliding surface. In: 12th IEEE Workshop on Variable Structure System, January 12–14, 2012, Mumbai. Kaneda, Ryosuke, Yazaki, Fumito, Sakai, Shin-ichiro, et al., 2004. The relative position control in formation flying satellites using superconducting magnets. In: Proceedings of 2nd International Symposium on Formation Flying Missions & Technologies, Washington D.C., U. S.. Kwon, Daniel W., Sedwick, Raymond J., 2011. Electromagnetic formation flight testbed using superconducting coils. J. Spacecraft Rockets 48, 124–134. Liu, Jinkun, 2012. Sliding Mode Control Design and Matlab Simulation. Tsinghua University Press. Navabi, M., Nasiri, N., 2011. Simulating the earth magnetic field according to the 10th generation of IGRF coefficients for spacecraft attitude control applications. In: 5th International Conference on Recent Advances in Space Technologies, 2011 June 9–11. Ousaloo, H.S., 2014. Magnetic attitude control of dynamically unbalanced spinning spacecraft during orbit raising. J. Aerosp. Eng. 27, 262–278. Ovchinnikov, M.Yu., Roldugin, D.S., Ivanov, D.S., 2015a. Choosing control parameters for three axis magnetic stabilization in orbital frame. Acta Astronaut. 116, 74–77. Ovchinnikov, M.Yu., Roldugin, D.S., Penkov, V.I., 2015b. Three-axis active magnetic attitude control asymptotical study. Acta Astronaut. 110, 279–286. Ran, Dechao, Sheng, Tao, Cao, Lu, Chen, Xiaoqian, 2014. Attitude control system design and on-orbit performance analysis of nanosatellite—‘‘Tian Tuo 1”. Chin. J. Aeronaut. 27 (3), 593–601. Robert, C., Mark, A., Stanley, O., 2013. Alternating magnetic field forces for satellite formation flying. Acta Astronaut. 84, 197–205. Sofyali, A., Jafarov, E.M., 2012. Purely magnetic spacecraft attitude control by using classical and modified sliding mode algorithms. In: 12th IEEE Workshop on Variable Structure System; 2012 January 12– 14, Mumbai. Voirin, Thomas, Kowaltschek, Steeve, Dubois-Matra, Olivier, 2012. NoMAD: a contactless technology for active large debris removal. In: 60th International Astronautical Congress, October 2012, Naples, Italy. Zeng, G.Q., Hu, M., 2012. Finite-time control for electromagnetic satellite formations. Acta Astronaut. 74, 120–130. Zhang, Y.W., Yang, L.P., Zhu, Y.W., et al., 2012. Nonlinear 6-DOF control of spacecraft docking with inter-satellite electromagnetic force. Acta Astronaut. 77, 97–108. Zhang, Y.W., Yang, L.P., Zhu, Y.W., Huang, H., 2014. Dynamics and solution for multispacecraft electromagnetic orbit correction. J. Guid., Control, Dyn. 37 (5), 1604–1610.

Please cite this article in press as: Ao, H.-j., et al. Touchless attitude correction for satellite with constant magnetic moment. Adv. Space Res. (2017), http://dx.doi.org/10.1016/j.asr.2017.05.040