Equilibrium attitude and nonlinear attitude stability of a spacecraft on a stationary orbit around an asteroid

Accepted Manuscript Equilibrium attitude and nonlinear attitude stability of a spacecraft on a station‐ ary orbit around an asteroid Yue Wang, Shijie Xu PII: DOI: Reference:

S0273-1177(13)00464-X http://dx.doi.org/10.1016/j.asr.2013.07.035 JASR 11449

To appear in:

Advances in Space Research

Received Date: Revised Date: Accepted Date:

8 May 2013 22 July 2013 24 July 2013

Please cite this article as: Wang, Y., Xu, S., Equilibrium attitude and nonlinear attitude stability of a spacecraft on a stationary orbit around an asteroid, Advances in Space Research (2013), doi: http://dx.doi.org/10.1016/j.asr. 2013.07.035

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Equilibrium attitude and nonlinear attitude stability of a spacecraft on a stationary orbit around an asteroid Yue Wang , Shijie Xu Room B1024, New Main Building, Department of Guidance, Navigation and Control, School of Astronautics, Beijing University of Aeronautics and Astronautics, 100191, Beijing, China

Abstract The classical problem of attitude stability in a central gravity field is generalized to that on a stationary orbit around a uniformly-rotating asteroid. This generalized problem is studied in the framework of geometric mechanics. Based on the natural symplectic structure, the non-canonical Hamiltonian structure of the problem is derived. The Poisson tensor, Casimir functions and equations of motion, which govern the phase flow and phase space structures of the system, are obtained in a differential geometric method. The equilibrium of the equations of motion, i.e. the equilibrium attitude of the spacecraft, which corresponds to a stationary point of the Hamiltonian constrained by Casimir functions, is determined from a global point of view. Nonlinear stability conditions of the equilibrium attitude are obtained in a modified energy-Casimir method. The nonlinear attitude stability is then investigated versus three parameters of the asteroid, including the ratio of the mean radius to the orbital radius, the harmonic coefficients C20 and C22. We find that when the spacecraft is located on the intermediate-moment principal axis of the asteroid, the nonlinear stability domain can be totally different from the classical Lagrange region on a

*Corresponding author. Tel. +86 10 8233 9751. E-mail addresses: [email protected] (Y. Wang), [email protected] (S. Xu). 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

circular orbit in a central gravity field. With the Lie group framework uncovered by geometric mechanics, besides determination of the equilibrium from a global point of view and the energy-Casimir method for nonlinear stability, several other powerful techniques can be performed, such as the variational integrators for greater accuracy in the numerical simulation and the geometric control theory for control problems. Keywords: Asteroids, Equilibrium attitude, Nonlinear stability, Geometric mechanics, Non-canonical Hamiltonian structure, Energy-Casimir method

1 Introduction Attitude stability of spacecraft subjected to the gravity gradient torque in a central gravity field has been one of the most fundamental problems in space engineering. The problem of attitude stability on a circular orbit in a central gravity field has been studied by Beletskii (1958), DeBra and Delp (1961), Hughes (1986) and many other authors. Brucker and Gurfil (2007) showed that the classical attitude stability domain can be modified in the restricted three-body problem by the extra primary body. Over the last two decades, the growing interest in the scientific exploration of asteroids and the near-Earth object (NEO) hazard mitigation has translated into an increasing number of asteroids missions. All the major space agencies are involved on missions to NEOs and several missions are under development (Barucci et al. 2011). A thorough understanding of the dynamical behavior of spacecraft near asteroids is necessary prior to the mission design. Due to the significantly non-spherical mass distribution and the fast rotation of the asteroid, the orbital and attitude dynamics of the spacecraft are much more complex than that around the Earth. This point has been 2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

shown by many works on the orbital dynamics around asteroids, such as Hirabayashi et al. (2010), Hu (2002), Hu and Scheeres (2004), San-Juan et al. (2002), Scheeres (1994, 2012), Scheeres and Hu (2001), Scheeres et al. (1996, 1998, 2000), as well as by several works on the attitude dynamics around asteroids, such as Kumar (2008), Misra and Panchenko (2006), Riverin and Misra (2002), Wang and Xu (2012a, 2013b, 2013c). Therefore, detailed investigations on the orbital and attitude dynamics near asteroids are of great interest and value. GoĨdziewski and Maciejewski (1998), and Maciejewski (1997) have studied the attitude dynamics of a rigid body around a spheroid planet, whose gravity field is truncated on the zonal harmonic J2= ±C20. The effect of the harmonic coefficients J2 of the gravity field on the attitude motion and stability was investigated in details. However, their works are not applicable to the case around asteroids, because the J2 gravity field is not precise enough for an asteroid, the ellipticity coefficient C22 of which is as significant as the oblateness coefficient C20. Therefore, both the harmonic coefficients C20 and C22 of the gravity field of the asteroid need to be considered in the attitude dynamics of the spacecraft, just as in previous works Kumar (2008), Misra and Panchenko (2006), Riverin and Misra (2002), Wang and Xu (2012a, 2013b, 2013c). The attitude dynamics of a spacecraft around asteroids can also be considered as a restricted model of the dynamics of two rigid bodies orbiting each other interacting through the mutual gravitational potential, i.e. the Full Two Body Problem (F2BP). A spherically-simplified model of the F2BP has been studied broadly, in which one body

3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

is assumed to be a homogeneous sphere and the gravity field of the other body is truncated on the second-order terms (Balsas et al. 2008, 2009; Barkin 1979; Kinoshita 1970, 1972a, 1972b), or the other body is assumed to be a general rigid body (Beletskii and Ponomareva 1990; Koon et al. 2004; Scheeres 2006) or an ellipsoid (Bellerose and Scheeres 2008a, 2008b; Scheeres 2004). There are also several works on the more general models of the F2BP, in which both the bodies are non-spherical, such as Boué and Laskar (2009), Maciejewski (1995), McMahon and Scheeres (2013), Mondéjar and Vigueras (1999), Koon et al. (2004) and Scheeres (2009). In our problem, we assume that the motion of the primary body, i.e. the asteroid, is not affected by the secondary body, i.e. the spacecraft. As shown by Kumar (2008), Misra and Panchenko (2006), Riverin and Misra (2002), the non-central gravity field and rotational state of the asteroid disturbed the attitude motion strongly, and the attitude resonance could exist. A full fourth-order model of the gravity gradient torque was derived by Wang and Xu (2013a) by taking into account of higher-order inertia integrals of the spacecraft. The equilibrium attitude and linear stability on a stationary orbit around an asteroid were studied by Wang and Xu (2013b, 2013c) based on the linearized equations of motion. It was found that the linear stability domain was modified significantly in comparison with, even totally different from, the classical linear stability domain on a circular orbit in a central gravity field. The full nonlinear attitude dynamics on a stationary orbit around an asteroid was analyzed via the canonical Hamiltonian formalism and dynamical systems theory by Wang and Xu (2012a).

4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

The linear attitude stability of a spacecraft on a stationary orbit around an asteroid has been studied thoroughly by Wang and Xu (2013c), however, since the system is conservative and only the necessary conditions of stability can be obtained via the linearized equations of motion, the linear stability domain obtained there are only infinitesimally stable, but the stability can not be guaranteed for the finite motions. Therefore, the more practical nonlinear attitude stability, which can be guaranteed for the finite motions, needs to be investigated. In this paper, the equilibrium attitude and nonlinear stability on a stationary orbit around an asteroid are studied in the framework of geometric mechanics. As in previous works mentioned above, both the harmonic coefficients C20 and C22 of the gravity field of the asteroid are considered in this paper. According to Wang and Xu (2013a), the 2nd degree and order-gravity field is precise enough for a fourth-order gravity gradient torque model. The higher order harmonic coefficients of the gravity field of the asteroid affect the attitude motion through the fifth and higher order terms. The asteroid is assumed to be rotating uniformly about its maximum- moment principal axis, which is the rotational state of most asteroids in the Solar System. The tools of geometric mechanics have had enormous successes in many areas of mechanics (Marsden 1992). The geometric mechanics has also been used in widelyranged problems in the celestial mechanics and space engineering, such as Beck and Hall (1998), Guirao and Vera (2010a, b), Hall (2001), Maciejewski (1995), Mondéjar and Vigueras (1999), Mondéjar et al. (2001), Vera (2008, 2009, 2010), Vera and Vigueras (2004, 2006), Wang et al. (1991, 1995). Within the framework of geometric

5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

mechanics, several powerful techniques can be performed, such as the reduction of the symmetric system, the determination of the relative equilibria from a global point of view, the energy-Casimir method for determining the stability, the variational integrators for greater accuracy in the numerical simulation and the geometric control theory for control problems (Koon et al. 2004; Wang and Xu 2012b). Starting from the basic settings of the problem, we uncover the Lie group framework of the problem through the derivation of the Poisson tensor and Casimir functions. Based on this Lie group framework, two of the powerful techniques mentioned above, the determination of the equilibria and the energy-Casimir method for determining the nonlinear stability, are performed in this paper. It is worth mentioning that a modified energy-Casimir method was adopted by Beck and Hall (1998), and Hall (2001) in the studies of attitude stability. This method was also discussed in the Appendix C of Wang et al. (1991), where it was called Lagrange multiplier approach. This modified method, in which the stability problem is considered as a constrained variational problem, is more convenient for applications than the original energy-Casimir method because there is no requirement to search for a particular Casimir function (Wang et al. 1991). Using this modified method, we obtain the conditions of nonlinear stability, which are more practical than the linear results by Wang and Xu (2013b, 2013c). Then the nonlinear attitude stability is investigated in details in a similar manner to Wang and Xu (2013c) versus three important parameters of the asteroid, including the ratio of the mean radius to the orbital radius, the harmonic coefficients C20 and C22.

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

2 Statement of the Problem

Fig. 1. The spacecraft on a stationary orbit around the asteroid.

As described by Fig. 1, we consider a rigid spacecraft B moving on a stationary orbit around the asteroid P. The body-fixed reference frames of the asteroid and the spacecraft are defined as SP={u, v, w} and SB={x, y, z} with O and C as their origins respectively. The origin of the frame SP is at the mass center of the asteroid, and the coordinate axes are chosen to be aligned along the principal moments of inertia of the asteroid. The principal moments of inertia of the asteroid are assumed to satisfy the following inequations

I P,ww ! I P,vv , I P,ww ! I P,uu .

(1)

Then the 2nd degree and order-gravity field of the asteroid can be represented by the harmonic coefficients C20 and C22 with other harmonic coefficients vanished, since the origin of the frame SP is fixed at the mass center of the asteroid, and the coordinate axes are chosen to be aligned along the principal moments of inertia of the asteroid (Hu 2002). The harmonic coefficients C20 and C22 are defined by C20



1  2I P,ww  I P,uu  I P,vv  0 , C22 2Mae2

1  I P,vv  I P,uu , 4Mae2

(2)

where M and ae are the mass and the mean radius of the asteroid respectively. The 7

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

practical ranges of the harmonic coefficients C20 and C22 considered in this paper are

0.5  C20  0 , 0.25  C22  0.25 ,

(3)

which should cover most asteroids in Solar System. The frame SB is attached to the mass center of the spacecraft and coincides with the principal axes reference frame. We assume that the mass center of the asteroid is stationary in the inertial space, and the asteroid is in a uniform rotation around its maximum-moment principal axis, i.e. the w-axis. The spacecraft is on a stationary orbit, and the orbital motion is not affected by the attitude motion. According to the orbital theory in Hu (2002), a stationary orbit in the inertial space corresponds to an equilibrium in the body-fixed frame of the asteroid. There are two kinds of stationary orbits: those that lie on the intermediate-moment principal axis of the asteroid, and those that lie on the minimum-moment principal axis of the asteroid. In this paper, we assume that the spacecraft is located on the v-axis of the asteroid. Thus, a negative C22 corresponds to a stationary orbit lying on the minimummoment principal axis, and a positive C22 corresponds to a stationary orbit lying on the intermediate-moment principal axis. According to Hu (2002), the radius of the stationary orbit RS satisfies the following equation RS5 

where P

P § 2 3 · R  W  9W 2 ¸ 0 , ZT2 ¨© S 2 0 ¹

GM , G is the Gravitational Constant, W 0

ae2C20 , W 2

(4)

ae2C22 and ZT is

the angular velocity of the uniform rotation of the asteroid. As described by Fig. 1, the orbital reference frame is defined by So={xo, yo, zo} with its origin coinciding with C, the mass center of the spacecraft. zo points towards 8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

the mass center of the asteroid, yo is in the opposite direction of the orbital angular momentum, and xo completes the orthogonal triad.

3 Symplectic Structure and Non-canonical Hamiltonian Structure The attitude of the spacecraft is described with respect to the orbital frame So by A, A [i , j, k ] [Į, ȕ, Ȗ ]T  SO(3) ,

(5)

where the vectors i, j and k are components of the unit axial vectors x, y and z of the frame SB in the frame So respectively, Į , ȕ and Ȗ are coordinates of the unit vectors xo, yo and zo in the frame SB, and SO(3) is the 3-dimensional special orthogonal group. The matrix A is the coordinate transformation matrix from the body-fixed frame SB to the orbital frame So. Therefore, the configuration space of the problem is the Lie group Q

SO(3) .

(6)

The velocity phase space of the system is the tangent bundle TQ with elements ( A; A) , where A TA SO(3) .

The image of vector v 

3

by standard isomorphism between Lie Algebras

3

with cross product and so(3) is denoted by vÖ , where so(3) is the Lie Algebras of Lie group SO(3). That is to say,

vÖ

ª 0 « 3 «v « v 2 ¬

v 3 0 v1

v2 º » v1 » . 0 »¼

(7)

A1 A  so(3) ,

(8)

The left translation of A to so(3) gives

Ö ȍ r

TA LA1 A

where ȍr is the relative angular velocity of the spacecraft with respect to the orbital

9

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

frame So expressed in the body-fixed frame SB. We identify the tangent bundle TSO(3) with SO(3) u by left trivialization and the standard isomorphism ^ :

3

3

with elements ( A; ȍr ) o so(3) . Therefore, the

elements of TQ can be written as ( A; ȍr ) . The angular velocity of the spacecraft with respect to the inertial space expressed in the body-fixed frame of the spacecraft SB, ȍ can be calculated by ȍ

ȍr  AT ȍOrbit

ȍr  AT >0 ZT

T

0@

ȍr  ZT ȕ ,

(9)

where ȍOrbit is the angular velocity of the orbital frame So expressed in itself. The rotational kinetic energy of the spacecraft is a function T : TQ o T

1 T ȍ Iȍ 2

1 T 1 ȍr Iȍr  ZT ȍrT Iȕ  ZT2 ȕ T Iȕ , 2 2

given by (10)

where the inertia tensor I is given by I

diag ^I xx , I yy , I zz ` ,

(11)

with the principal moments of inertia of the spacecraft I xx , I yy and I zz . The gravitational potential of the spacecraft is the function V : Q o V

V (Į, ȕ, Ȗ ) .

(12)

According to Wang and Xu (2013b), due to the significantly non-spherical shape and the rapid rotation of the asteroid, the effects of the harmonic coefficients C20 and C22 are as significant as that of the central component of the gravity field of the asteroid, while effects of the third and fourth-order inertia integrals of the spacecraft could be neglected. Therefore, we only consider the moments of inertia I xx , I yy and I zz in the gravitational potential, with the third and fourth-order inertia integrals of

the spacecraft neglected.

10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Based on the results in Wang and Xu (2013a), through some rearrangements, the explicit formulation of the attitude-dependent part of the gravitational potential V (Į, ȕ, Ȗ ) is given by

V (Į , ȕ , Ȗ )

3PW 0 § T 3P T 5 · 3PW 2 17Ȗ T IȖ  2ĮT IĮ . (13) Ȗ IȖ  ȕ Iȕ  Ȗ T IȖ ¸  3 5 ¨ 5  2 RS 2 RS © 2 ¹ 2 RS

Then, the Lagrangian of the system L : TQ o

is given as follows:

L T V W ,

(14)

where W :TQ o Q is the canonical projection. The (momentum) phase space is the cotangent bundle T Q , which can be written as ( A; Į A ) with Į A  TA SO(3) . By left trivialization, the elements of T Q can be written as ( A; Į) with Į Te LAĮ A  so(3) , where so(3) is the dual space to Lie Algebra so(3) . We identify so(3) with

3

using the standard isomorphism ^ ,

and the pairing between so(3) and so(3) is defined as the dot product on a, b

1 tr (aÖ T bÖ) . 2

a ˜b

3

(15)

This pairing is extended to T SO(3) and TSO(3) by left translation as follows: 1 tr (aTAbA ) 2

a A , bA

1 tr (aÖ T bÖ) a ˜ b , 2

(16)

where aÖ Te LAa A  so(3) and bÖ TA LA1 bA  so(3) . By the means of Legendre transformation, we can obtain the conjugate momentum as follows: Ȇ

wL wȍr

Iȍr  ZT Iȕ ,

(17)

where wf wv represents the gradient of the function f with respect to the vector

v . Ȇ is the angular momentum of the spacecraft with respect to the inertial frame

11

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

expressed in the body-fixed frame SB. belongs to so(3) and Ȇ

Notice that Ȇ

is not the natural conjugate

momentum, which should belong to TA SO(3) . Since ȍr  so(3) and Ȇ  so(3) , the pairing between them can be written as Ȇ ˜ ȍr

1 tr ( ȆÖ T ȍÖ r ) 2

1 tr ( ȆÖ T A1 A) 2

1 tr 2

 AȆÖ A . T

(18)

Since A  TA SO(3) , using Eqs. (16) and (18) we can obtain

AȆÖ

TA LA1 ȆÖ  TA SO(3) ,

(19)

which is actually the natural conjugate momentum. Therefore, elements of the phase space T Q can be written as

Ȅ

( A; AȆÖ ) .

(20)

The phase space T Q carries a natural symplectic structure Z , defined as

Z Z SO (3) .

(21)

The canonical bracket associated to the symplectic structure Z can be written in the coordinates Ȅ as follows: { f , g}T Q ( Ȅ )

DA f , DAȆÖ g  DA g , DAȆÖ f

(22)

for any f , g  C f (T Q) , where ˜, ˜ is the pairing between T SO(3) and TSO(3) , and DB f is a matrix whose elements are the partial derivates of f with respect to the elements of matrix B respectively. By the means of Legendre transformation, the Hamiltonian of the system H : T Q o

is obtained as follows: H

1 T 1 Ȇ I Ȇ  Ȇ T  ZT ȕ  V W T Q , 2

(23)

where W T Q : T Q o Q is the canonical projection. We can see that the first term in

12

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Eq. (23) is the Hamiltonian of the free-spin motion, and the attitude dynamics of the spacecraft is perturbed both by the second and third terms of the Hamiltonian in Eq. (23). The third term V represents the perturbation due to the gravity gradient torque; the second term  Ȇ T  ZT ȕ represents the perturbation due to the precession of the orbital frame, which is consistent with the results by Gurfil et al. (2007), Wang and Xu (2012a). Although the coordinates Ȅ in Eq. (20), the symplectic structure Z in Eq. (21) and the canonical bracket Eq. (22) are natural and intrinsic, they are not convenient for applications, since the variables for the attitude motion are given in the matrix form and the calculations of the pairing between T SO(3) and TSO(3) in Eq. (22) are tedious. The non-canonical Hamiltonian structure with variables in the vector form is more convenient for applications. We can choose a set of coordinates of the phase space T Q instead of Ȅ as

z

T

ª¬ Ȇ T , ĮT , ȕ T , Ȗ T º¼ 

12

.



(24)



There exists a Poisson diffeomorphism < : T Q, {˜, ˜}T Q ( Ȅ ) o 

12

, {˜, ˜} 12 ( z) ,

defined as follows:

<( A; AȆÖ )

T

ª¬ Ȇ T , ĮT , ȕ T , Ȗ T º¼ ,

(25)

where {˜, ˜} 12 ( z ) is the Poisson bracket in coordinates z . These two brackets satisfy

{ f , g} 12 ( z) < { f <, g <}T Q ( Ȅ ) for any f , g  C f (

12

(26)

) . We write Poisson bracket {˜, ˜} 12 ( z ) in the following form { f , g} 12 ( z )

T

’z f

B( z )  ’ z g ,

(27)

with the Poisson tensor B( z ) (see Appendix for the derivation of B( z ) ) given by

13

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

B( z )

ª ȆÖ « « ĮÖ « ȕÖ « «¬ ȖÖ

ĮÖ 0 0 0

ȕÖ ȖÖ º » 0 0» . 0 0 »» 0 0 »¼

(28)

The 12×12 antisymmetric and degenerated Poisson tensor B( z ) has six geometric integrals as independent Casimir functions C1 ( z )

1 T Į Į , C2 ( z ) 2

1 T ȕ ȕ , C3 ( z ) 2

1 T Ȗ Ȗ , C4 ( z ) ĮT ȕ , C5 ( z ) ĮT Ȗ , C6 ( z ) 2

ȕT Ȗ .

The six-dimensional invariant manifold or symplectic leaf of the system can be defined in 6

12

by Casimir functions

­ ®z  ¯

12

| C1 ( z ) C2 ( z ) C3 ( z )

1 ½ , C4 ( z) C5 ( z) C6 ( z) 0 ¾ . 2 ¿

(29)

The symplectic structure on this symplectic leaf is defined by restriction of the Poisson bracket {˜, ˜} 12 ( z ) to 6 . The vectors generating the nullspace of B( z ) are actually the gradients of the Casimir functions with respect to z. Then, the six-dimensional nullspace of B( z ) can be got from Casimir functions as follows:

N > B( z )@

­§ 0 · § 0 · § 0 · § 0 · § 0 · § 0 · °¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ 0 0 ȕ Ȗ 0 ° Į span ®¨ ¸ , ¨ ¸ , ¨ ¸ , ¨ ¸ , ¨ ¸ , ¨ ¸ °¨ 0 ¸ ¨ ȕ ¸ ¨ 0 ¸ ¨ Į ¸ ¨ 0 ¸ ¨ Ȗ ¸ °¯¨© 0 ¸¹ ¨© 0 ¸¹ ¨© Ȗ ¸¹ ¨© 0 ¸¹ ¨© Į ¸¹ ¨© ȕ ¸¹

½ ° ° ¾. ° °¿

(30)

The Hamiltonian of the system in coordinates z is given by Eq. (23). The equations of motion can be written as

z

B( z )’ z H ( z ) .

(31)

The explicit equations of the attitude motion can be obtained from Eqs. (23) and (31) as follows:

14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

ªȆ º « » «Į» «ȕ» « » ¬Ȗ ¼

ª ȆÖ I 1 Ȇ  ĮÖ  wV wĮ  ȕÖ  wV wȕ  ȖÖ  wV wȖ º « » ĮÖ  I 1 Ȇ  ZT ȕ « » « » .(32) ȕÖ  I 1 Ȇ « » « » ȖÖ  I 1 Ȇ  ZT ȕ «¬ »¼

ª I 1 Ȇ  ZT ȕ º « » wV wĮ » B( z ) « «wV wȕ  ZT Ȇ » « » wV wȖ ¬ ¼

The term ĮÖ  wV wĮ  ȕÖ  wV wȕ  ȖÖ  wV wȖ in Eq. (32) is actually the gravity gradient torque of the spacecraft TB expressed in the body-fixed frame SB, the explicit formulation of which is given as follows: TB

3PW Ö 3PW 3P 5 Ö  5 0 §¨ ȕIȕ Ö ·¸  5 2 17ȖIȖ Ö  2ĮIĮ Ö . ȖIȖ  ȖIȖ 3 2 RS RS © ¹ RS

(33)

4 Equilibrium Attitude and Conditions of Nonlinear Stability 4.1 Equilibrium attitude The equilibrium attitude of the spacecraft corresponds to a stationary point of the Hamiltonian constrained by the Casimir functions. The stationary point can be determined by the first variation conditions of the variational Lagrangian ’F  ze 0 , where the variational Lagrangian F  z is given by F z

6

H  z  ¦ Pi Ci  z ,

(34)

i 1

where the subscript e is used to denote the value at the equilibria. By using the formulations of the Hamiltonian and the Casimir functions, the equilibrium conditions are obtained as follows: I 1 Ȇ e  ZT ȕe

0,

(35a)

6PW 2 IĮe  P1Įe  P4 ȕe  P5Ȗ e RS5

ZT Ȇ e 

0,

3PW 0 Iȕe  P2 ȕe  P4 Įe  P6 Ȗ e RS5

(35b) 0,

§ 3P 15PW 0 51PW 2 ·  ¨ 3 ¸ IȖ e  P3Ȗ e  P5Įe  P6 ȕe 2 RS5 RS5 ¹ © RS 15

(35c) 0.

(35d)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Eq. (35a) implies that Ȇe

ZT Iȕe ,

(36)

and the spacecraft is stationary with respect to the orbital frame, i.e. this solution is an equilibrium attitude of the spacecraft. Taking the dot product of ȕe with Eq. (35b)

 6PW

yields P4

P4

3PW

0

2

RS5 ȕeT IĮe , while the dot product of Įe with Eq. (35c) yields

RS5  ZT2 ĮeT Iȕe . Here we consider a general case when

3PW 0 6PW  ZT2 z 5 2 , 5 RS RS

(37)

by Eq. (4) which is equivalent to 2

§ Rs · 9 ¨ ¸ z C20  3C22 . 2 © ae ¹

In this case we have P4 method, we will have P5

(38)

0 due to the symmetry of the inertia tensor. In the same 0 and P6

0 in the general cases when

2

2

§ Rs · § Rs · 5 ¨ ¸ z C20  19C22 , ¨ ¸ z 3C20  15C22 2 © ae ¹ © ae ¹

(39)

respectively. Then the equilibrium conditions (35b)-(35d) imply that Įe , ȕe and Ȗ e must be principal axes of the inertial tensor, i.e. the orbital frame is parallel to the body-fixed frame. This contains 24 equilibrium attitudes, only one of which is found by Wang and Xu (2013b) in the linear method, since geometric mechanics adopted here allows the determination of the equilibrium attitude from a global point of view. 4.2 Conditions of Nonlinear Stability Without of loss of generality, we choose one of the equilibrium attitudes as follows for stability conditions

16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

T

ª¬0,  ZT I yy , 0º¼ , Įe

Ȇe

6PW 2 I xx , P2 RS5

P1

T

>1, 0, 0@

T

>0,1, 0@

, ȕe

§ 3PW 0 2· ¨ 5  ZT ¸ I yy , P3 © RS ¹

, Ȗe

T

>0, 0,1@

,

§ 3P 15PW 0 51PW 2 ·  ¨ 3 ¸ I zz . RS5 ¹ 2 RS5 © RS

(40a) (40b)

Following the modified energy-Casimir method adopted by Beck and Hall (1998), and Hall (2001), we will obtain the conditions of nonlinear stability. The Hessian of the variational Lagrangian is calculated as follows:

’2 F  z ª I 1 « « 0 « « «Z I « T 3u3 « « « 0 ¬

0

ZT I 3u3

6 PW 2 I  P1I 3u3 RS5

 P4 I 3u3

 P4 I 3u3

3PW 0 I  P 2 I 3u3 RS5

 P5I 3u3

 P6 I 3u3

º » »  P5I 3u3 » » » , (41)  P6 I 3u3 » » § 3P 15PW 0 51PW 2 · »  ¨ 3 ¸ I  P3I 3u3 » 5 5 RS ¹ 2 RS © RS ¼ 0

where I 3u3 is the is the 3™3 identity matrix. At the equilibrium attitude, we have

’ 2 F  ze ª I 1 « « 0 « « « «Z I « T 3u3 « « « « 0 « «¬

0

ZT I 3u3

6PW 2  I  I xx I3u3 RS5

0

0

3PW 0  I  I yy I3u3 RS5 ZT2 I yy I 3u3

0

º » » 0 » » » » 0 » . (42) » » § 3P 15PW 0 51PW 2 · »  ¨ 3 ¸ RS5 ¹ » 2 RS5 © RS » »¼ u  I  I zz I 3u3 0

0

The Hamiltonian system is non-canonical, and the phase flow of the system is constrained on the six-dimensional invariant manifold or symplectic leaf 6 . Therefore, rather than considering general perturbations in the phase space, we need to restrict consideration to perturbations on T6 z , the tangent space to the invariant e

17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

manifold 6 at the equilibrium, i.e. the range space the Poisson tensor B( z ) at the equilibrium, denoted by R  B( ze ) . According to Beck and Hall (1998), the projected Hessian matrix is given by

P  ze ’2 F  ze P  ze , where the projection operator is 1

P  ze I12u12  K ( ze )  K ( ze )T K ( ze ) K ( ze )T , K ( ze )

ª¬’ z C1  z e ª0 0 0 «Į « e 0 0 « 0 ȕe 0 « ¬ 0 0 Ȗe

’ z Ci  z e 0 ȕe Įe 0

0 Ȗe 0 Įe

(43)

’ z C6  z e º¼

0º 0 »» . Ȗe » » ȕe ¼

(44)

The 12™12 projected Hessian matrix will have six zero eigenvalues associated with the six-dimensional nullspace N > B( ze )@ , i.e. the complement space of the tangent space to the invariant manifold at the equilibrium. The remaining six eigenvalues are associated with the six-dimensional tangent space to the invariant manifold T6 z , e

and if they are all positive, then ze is a constrained minimum on the invariant manifold 6 and the equilibrium attitude is nonlinearly stable. Through some calculations, we get the projected Hessian matrix as follows:

18

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

P  ze ’ 2 F  ze P  ze ª 1 « I « xx « « 0 « « « 0 « « 0 « « 1 Z « 2 T « « 0 « 1 « ZT « 2 « 0 « « 0 « « 0 « « « 0 « ¬ 0

0

0

1 0  ZT 2

0

1 ZT 2

0

0

0

0

1 I yy

0

0

0

0

0

0

0

0

0

0

1 I zz

0

0

0

0

0

1 ZT 2

0

1  ZT 2

0

0

0

0

0

0

0

0

0

0

0

0

0

M1

0

M1

0

0

0

0

0

0

0

0

M2

0

0

0

M 2

0

0

0

0

M1

0

M1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

M3

0

M 3

0

0

M 2

0

0

0

M2

0

0

0

0

0

0 M 3

0

M3

0

0

0

0

0

0

0

1 ZT 2 0 1  ZT 2 0

0 0 0 0

0

º 0» » » 0» » » 0» » 0» » 0» » » 0 » , (45) » 0» » 0» » 0» » 0» » » 0» » 0¼

where 1 6PW 2 1 3PW 0 1 I  I xx  I  I xx  ZT2 I yy , 5  yy 5  yy 4 RS 4 RS 4

(46)

1 6PW 2 1 § 3P 15PW 0 51PW 2 · I  I xx  ¨ 3   ¸  I zz  I xx , 5  zz RS5 ¹ 4 RS 4 © RS 2 RS5

(47)

1 3PW 0 1 § 3P 15PW 0 51PW 2 · 1 2 I  I yy  ¨ 3   ¸  I zz  I yy  ZT I yy . 5  zz 5 5 RS ¹ 4 RS 4 © RS 2 RS 4

(48)

M1

M2

M3

The eigenvalues of the projected Hessian matrix are calculated as follows:

^0, 0, 0, 0, 0, 0,1 I

yy

, 2 M 2 , V1, V 2 , V 3 , V 4 ` ,

(49)

where

V1

1 1 ­ ½ ®1  2M1I xx  ª1  2M1I xx 2  2ZT2 I xx2 º 2 ¾ , 2 I xx ¯ ¬ ¼ ¿

19

(50)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

V2

1 1 ­ ½ ®1  2M1I xx  ª1  2M1I xx 2  2ZT2 I xx2 º 2 ¾ , 2 I xx ¯ ¬ ¼ ¿

(51)

V3

1 1 ­ ½ ®1  2M 3 I zz  ª1  2M 3 I zz 2  2ZT2 I zz2 º 2 ¾ , 2 I zz ¯ ¬ ¼ ¿

(52)

V4

1 1 ­ ½ ®1  2M 3 I zz  ª1  2M 3 I zz 2  2ZT2 I zz2 º 2 ¾ . 2 I zz ¯ ¬ ¼ ¿

(53)

The six zero eigenvalues are associated with the six-dimensional complement space of the tangent space to the invariant manifold at the equilibrium, and the remaining six eigenvalues are associated with the six-dimensional tangent space to the invariant manifold T6 z . Therefore, the conditions of nonlinear stability are that all the e

remaining six eigenvalues are positive. Notice that 1 I yy is always positive, V 2 ! 0 implies V 1 ! 0 , and V 4 ! 0 implies V 3 ! 0 , we will have the conditions of nonlinear stability as follows: M 2 ! 0, V 2 ! 0, V 4 ! 0 .

(54)

According to Eqs. (51) and (53), the conditions of nonlinear stability Eq. (54) can be written as

M 2 ! 0, 4M1 ! ZT2 I xx , 4M 3 ! ZT2 I zz .

(55)

According to Eqs. (46)-(48), we can write the conditions of nonlinear stability Eq. (55) further as

§ 3P 15PW 0 57 PW 2 ·  ¨ 3 ¸  I xx  I zz ! 0 , RS5 ¹ 2 RS5 © RS

(56a)

§ 6PW 2 3PW 0 2· ¨ 5  5  ZT ¸  I yy  I xx ! 0 , RS © RS ¹

(56b)

§ 3P 21PW 0 51PW 2 ·   ZT2 ¸  I yy  I zz ! 0 . ¨ 3 5 5 RS 2 RS © RS ¹

(56c)

20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

According to Eq. (4), we have

ZT2

P 3 s

R



P §3

· ¨ W 0  9W 2 ¸ . R ©2 ¹ 5 s

(57)

Then, Eqs. (56b) and (56c) can be written as follows:

§ P 9PW 0 3PW 2 ·  5 ¸  I yy  I xx ! 0 , ¨ 3 5 RS ¹ © Rs 2 RS

(58a)

§ P 3PW 0 15PW 2 · ¨ 3 5  ¸  I yy  I zz ! 0 . RS RS5 ¹ © RS

(58b)

Keeping in mind that W 0

ae2C20 and W 2

ae2C22 , we can write the stability

conditions Eqs. (56a), (58a) and (58b) as follows:

ª § a ·2 § 5 ·º «1  ¨ e ¸ ¨ C20  19C22 ¸ »  I xx  I zz ! 0 , ¹ »¼ «¬ © RS ¹ © 2

(59a)

ª § a ·2 § 9 ·º «1  ¨ e ¸ ¨ C20  3C22 ¸ »  I yy  I xx ! 0 , ¹ »¼ «¬ © RS ¹ © 2

(59b)

ª § a ·2 º «1  ¨ e ¸  3C20  15C22 »  I yy  I zz ! 0 . «¬ © RS ¹ »¼

(59c)

The conditions of nonlinear stability Eqs. (59a)-(59c) can be rearranged further as follows:

AastV x ! 0 , BastV y ! 0 , CastV z ! 0 ,

(60)

where V x , V y and V z are defined as

Vx

§ I yy  I zz ¨ © I xx

· ¸, ¹

(61a)

Vy

§ I xx  I zz ¨¨ © I yy

· ¸¸ , ¹

(61b)

Vz

§ I yy  I xx · ¨ ¸. © I zz ¹

(61c)

and the parameters Aast , Bast and Cast are defined by 21

2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Aast

§a · 1  3 ¨ e ¸  C20  5C22 , © Rs ¹

(62a)

2

Bast

§ a · §5 · 1  ¨ e ¸ ¨ C20  19C22 ¸ , ¹ © Rs ¹ © 2

Cast

§ a · §9 · 1  ¨ e ¸ ¨ C20  3C22 ¸ . ¹ © Rs ¹ © 2

(62b)

2

(62c)

The parameters Aast , Bast and Cast are defined same as in Wang and Xu (2013c). Obviously, the ranges of V x , V y and V z are all from -1 to 1. Taking into account that V y ! 0 is equivalent to V x ! V z , we can write the conditions of nonlinear stability Eq. (60) as follows:

AastV x ! 0 , Bast V x  V z ! 0 , CastV z ! 0 .

(63)

When the problem is reduced to the attitude motion on a circular orbit in a central gravity field, we have Aast

Bast

Cast

1. Then, Eq. (63) is reduced to

V x ! 0, V x ! V z , V z ! 0 .

(64)

Hence, we have obtained the nonlinear stability domain in the V x -V z plane, i.e. the classical Lagrange region, which has already been obtained by Hughes (1986), Beck and Hall (1998) in the studies on the nonlinear attitude stability on a circular orbit in a central gravity field. The differences between our results Eq. (63) and the classical results Eq. (64) are due to the parameters Aast , Bast and Cast , i.e. the non-central gravity field of the asteroid. 4.3 Some Discussions on the Parameters According to Eq. (63), the signs of functions Aast , Bast and Cast have important qualitative effects on the conditions of nonlinear stability. According to Eqs. 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

(62a)-(62c), the functions Aast , Bast and Cast are determined by three basic parameters of the asteroid, including the ratio of the mean radius to the orbital radius

ae Rs , the harmonic coefficients C20 and C22. Precisely speaking, the ratio ae Rs depends on the harmonic coefficients C20 and C22, the average density and the rotational period of the asteroid. However, from an approximate point of view, the parameter ae Rs can be roughly determined by the average density and the rotation period of the asteroid, with the effects of the harmonic coefficients C20 and C22 neglected. Wang and Xu (2013c) have made a rough estimate of the range of the ratio ae Rs , and have shown that the range from 0.2 to 0.8 should cover most asteroids in our Solar System. The ratio ae Rs will be treated as the third parameter of the asteroid in the conditions of nonlinear stability described by Eq. (63), besides the harmonic coefficients C20 and C22. Therefore, the practical ranges of the three parameters in Eq. (63) are as follows:

­ 0.2  ae Rs  0.8, ° ® 0.5  C20  0, °0.25  C  0.25. ¯ 22

(65)

In a similar manner to Wang and Xu (2013c), we can divide the range of the parameter ae Rs into three parts as follows: (a).

0.2 

ae 2 , d Rs 19

(66)

in the case of which the functions Aast , Bast and Cast are all positive in the domain of the harmonic coefficients given by 0.5  C20  0 , 0.25  C22  0.25 ; (b).

a 2 2  e d , 19 Rs 15

23

(67)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

in the case of which the functions Aast and Cast are both positive, but Bast can be positive or negative in the domain 0.5  C20  0 , 0.25  C22  0.25 ; a 2  e  0.8, 15 Rs

(c).

(68)

in the case of which the functions Cast is positive, but Aast and Bast can be positive or negative in the domain 0.5  C20  0 , 0.25  C22  0.25 .

5 Nonlinear Stability Domain in the V x -V z Plane The domain of the harmonic coefficients 0.5  C20  0 , 0.25  C22  0.25 can be divided into four parts according to the signs of the functions Aast , Bast and Dast , where Dast is defined as 2

Dast

§a · §3 · 1  ¨ e ¸ ¨ C20  9C22 ¸ . ¹ © Rs ¹ © 2

(69)

According to Eq. (57), we have

ZT2

P Rs3

Dast .

(70)

Therefore, Dast should be positive, and the case of Dast  0 does not exist in the real physical situation. Fig. 2 has shown the regions I, II, III and IV in the case of ae Rs straight lines Aast

0 , Bast

0 , Cast

0 , Dast

0.8 . The

0 on the C20  C22 plane are the

same as in Wang and Xu (2013c), therefore the regions I, II, III and IV on the

C20  C22 plane are same as in Wang and Xu (2013c). According to Wang and Xu (2013c), the ratio ae Rs has important effects on the conditions of the nonlinear stability. In the case of 0.2  ae Rs d 2 of

the

harmonic

coefficients;

the

19 , only the region I can exist in the domain regions 24

I

and

II

can

exist

when

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

2

19  ae Rs d 2

15 ; in the case of 2

15  ae Rs d 2 3 , the regions I, II and III

can exist; in the case of 2 3  ae Rs d 0.8 , all the four regions can exist.

Fig. 2. The domain of the harmonic coefficients is divided into region I, region II, region III and region IV according to the sign of Aast , Bast and Dast in the case of ae Rs

0.8 (right

upper side is the negative side for Aast , Bast and Dast ).

In the region I of the domain of the harmonic coefficients in Fig 2, the functions

Aast , Bast and Cast given by Eqs. (62a)-(62c) are all positive. Then, the conditions of nonlinear stability Eq. (63) can be written as Eq. (64), the classical nonlinear stability conditions on a circular orbit in a central gravity field. Therefore, in this case the nonlinear stability domain in the V x -V z plane is the classical Lagrange region, same as the classical results on a circular orbit in a central gravity field, as shown by Fig. 3.

25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Fig. 3. The nonlinear stability domain in the

V x -V z plane in the region I of Fig. 2.

In the region II of the domain of the harmonic coefficients in Fig 2, the functions

Aast and Cast are positive, and Bast is negative. Then, the conditions of nonlinear stability Eq. (63) can be written as

V x ! 0 , V x V z  0 , V z ! 0 .

(71)

Therefore, the nonlinear stability domain in the V x -V z plane is an isosceles right triangle region above the straight line V x  V z

0 in the first quadrant, as shown by

Fig. 4. Obviously, due to the non-spherical mass distribution of the asteroid, the stability domain is totally different from the classical results in Hughes (1986), Beck and Hall (1998) on a circular orbit in a central gravity field. This result is very important for the design of attitude control system of the asteroid missions.

26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Fig. 4. The nonlinear stability domain in the

V x -V z plane in the region II of Fig. 2.

In the region III of the domain of the harmonic coefficients in Fig 2, the function

Cast is positive, and Aast and Bast are negative. Then, the conditions of nonlinear stability Eq. (63) can be written as

V x  0 , V x V z  0 , V z ! 0 .

(72)

Therefore, the nonlinear stability domain in the V x -V z plane is the second quadrant, as shown by Fig. 5. Due to the non-spherical mass distribution of the asteroid, this stability domain is also totally different from the classical results in Hughes (1986), Beck and Hall (1998) on a circular orbit in a central gravity field. In the region IV of the domain of the harmonic coefficients in Fig 2, Dast is negative. As shown above, this case does not exist in the real physical situation.

27

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Fig. 5. The nonlinear stability domain in the

V x -V z plane in the region III of Fig. 2.

In Table 1, we give a summary of the stability domains in the V x -V z plane in different regions of the domain of the harmonic coefficients. Notice that the nonlinear attitude stability is more practical than the linear attitude stability in Wang and Xu (2013c). Since the system is conservative and only the necessary conditions of stability can be obtained via the linearized equations of motion, the linear stability domain obtained there are only infinitesimally stable, but the stability can not be guaranteed for the finite motions; whereas the nonlinear attitude stability obtained in this paper can be guaranteed for the finite motions. We can find that in regions I and II, the nonlinear stability domain in the V x -V z plane is only a part of the linear stability domain obtained in Wang and Xu (2013c), i.e. the part located in the first quadrant of the V x -V z plane. Therefore, in the linear 28

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

stability regions located in the third quadrant of the V x -V z plane in Wang and Xu (2013c), the stability can only be guaranteed for the infinitesimal motions. This result is consistent with the conclusion in the classical result in the central gravity filed in Hughes (1986). Whereas in region III, the nonlinear stability domain in the V x -V z plane is the same as the linear stability domain obtained in Wang and Xu (2013c). Table 1 Stability domains in the V x -V z plane in regions of the C20  C22 domain Regions of the

C20  C22

Stability domains in the V x -V z plane

Descriptions of the stable attitude

domain Maximum-moment principal axis perpendicular to orbital plane; Minimum-moment principal axis points toward the asteroid;

I

Same as the classical nonlinear stability domain in a central gravity field. Maximum-moment principal axis perpendicular to orbital plane; Minimum-moment principal axis parallel to orbital velocity;

II

Totally different from the classical nonlinear stability 29

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

domain in a central gravity field. Maximum-moment principal axis points toward the asteroid; Minimum-moment principal axis parallel to orbital velocity;

III

IV

±±±

Totally different from the classical nonlinear stability domain in a central gravity field. ±±±

6 Conclusions The equilibrium attitude and the nonlinear stability of a rigid spacecraft on a stationary orbit around a uniformly-rotating asteroid have been studied in the framework of the geometric mechanics. In the studied problem, the harmonic coefficients C20 and C22 of the gravity field of the asteroid were considered. The tools of the geometric mechanics adopted in the paper provided a method for determining the equilibrium attitude from a global point of view and the energy-Casimir method for the conditions of the nonlinear stability. Starting from the natural symplectic structure, we have derived the non-canonical Hamiltonian structure of the problem. The Poisson tensor, Casimir functions and equations of motion were obtained in a differential geometric method. 24 equilibrium attitudes of the spacecraft, which corresponds to stationary points of the Hamiltonian constrained by Casimir functions, were determined from a global point of view.

30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

The conditions of the nonlinear stability of the equilibrium attitude were obtained in a modified energy-Casimir method. Nonlinear stability of the equilibrium attitude was then investigated versus three basic parameters of the asteroid, including the ratio of the mean radius to the orbital radius, the harmonic coefficients C20 and C22. We have found that due to the significantly non-spherical shape and the rapid rotation of the asteroid, the nonlinear attitude stability domain in the V x -V z plane is modified significantly in comparison with the classical nonlinear stability domain, i.e. the Lagrange region. In the different regions of the domain of the harmonic coefficients, the nonlinear stability properties of the equilibrium attitude can be totally different. Especially, when the spacecraft is located on the intermediate-moment principal axis of the asteroid, i.e. C22 ! 0 , the nonlinear stability domain in the

V x -V z plane can be an isosceles right triangle region above the straight line

V x V z

0 in the first quadrant or the second quadrant, totally different from the

classical Lagrange region on a circular orbit in a central gravity field. With the Lie group framework of the problem uncovered by the geometric mechanics, besides the determination of equilibrium from a global point of view and the energy-Casimir method for the nonlinear stability, several other powerful techniques can be performed, such as the variational integrators for greater accuracy in the numerical simulation and the geometric control theory for control problems. The nonlinear attitude stability in this paper is more practical than the linear attitude stability in the previous result. The linear stability domain obtained in the previous result is only infinitesimally stable, but the stability can not be guaranteed

31

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

for the finite motions; whereas the nonlinear attitude stability obtained in this paper can be guaranteed for the finite motions. Notice that a 2nd degree and order-gravity field may not be precise enough for many irregular asteroids, and the higher order harmonic coefficients of the gravity field can be significant especially when the spacecraft is very close to the asteroid. The effects of the higher order harmonic coefficients will make some modifications on the stability regions obtained in this paper through the fifth and higher order terms of the gravity gradient torque. The investigation on the effects of the higher order harmonic coefficients on the attitude stability is beyond the scope of this paper and worthy of detailed studies in the future. Our results are very useful for the design of attitude control system in the future asteroid missions.

Appendix: Calculation of Poisson Tensor B(z) The Poisson tensor B( z ) can be obtained in a differential geometric method following the idea in the Appendix B of Wang et al. (1991) and the Appendixes of Wang and Xu (2012b). The natural canonical bracket is given by Eq. (22) on the manifold T Q( Ȅ ) . We define f , g  C f T Q( Ȅ ) as follows:

f where f , g  C f 

12

f <, g

g <,

(A.1)

( z ) . According to Eqs. (22) and (26), we have

{ f , g} 12 ( Ȇ, Į, ȕ, Ȗ )

DA f , DAȆÖ g  DA g , DAȆÖ f .

(A.2)

In order to calculate the differential of an arbitrary smooth function f , we consider W  TȄ (T Q) defined as follows:

32

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

 AvÖ , A(vÖ ȆÖ  vÖ ) T

W

1

1

( A, AȆÖ )

2

(T Q) .

(A.3)

W generates the curve in T Q

 Ae

tvÖ1



, AetvÖ1 ( ȆÖ  tvÖ2 )  T Q .

(A.4)

Thus, the differential is given by df ˜W

d |t dt

0

f

 Ae

tvÖ1

, AetvÖ1 ( ȆÖ  tvÖ2 )

d |t dt







f AetvÖ1 , Ȇ  tv2 .

0

(A.5)

Eq. (A.5) can be written further as follows: T

df ˜W

§ wf · ¨ ¸ © wĮ ¹

T

§ wf ·  AvÖ1 1, :  ¨ ¸ © wȕ ¹ T

T

T

T § wf · § wf · T ¨ ¸  Į vÖ1  ¨ ¸ © wĮ ¹ © wȕ ¹

where

 AvÖ1 j , :

T

§ wf ·  AvÖ1 2, :  ¨ ¸ © wȖ ¹ T

T

wf  AvÖ1 3, :  §¨ ·¸ v2 © wȆ ¹ T

T

T

 ȕ vÖ T

1

T

T § wf · § wf ·  ¨ ¸  Ȗ T vÖ1  ¨ ¸ v2 , (A.6) © wȆ ¹ © wȖ ¹

is the jth row of the matrix AvÖ1 . Rearranging Eq. (A.6) yields T

§ wf · E Tf v1  ¨ ¸ v2 , © wȆ ¹

df ˜W

(A.7)

where ^

^

Ef

^

§ wf · § wf · § wf · ¨ ¸ Į¨ ¸ ȕ ¨ ¸ Ȗ , © wĮ ¹ © wȕ ¹ © wȖ ¹

(A.8)

^

where  ˜ is the isomorphism defined by Eq. (7). The element df  T( A, AȆÖ ) (T Q) can be denoted by

df

Ö Ö  aÖ , AbÖ .  A  bȆ

(A.9)

And then we get

df ˜W



Ö Ö  aÖ AvÖ1 , A bȆ



 AbÖ, A(vÖ1 ȆÖ  vÖ2 )

1 T Ö Ö  aÖ  1 tr tr  AvÖ1 A bȆ 2 2 T T a v1  b v2 .







According to Eqs. (A.7) and (A.10), we have

33

 AbÖ A(vÖ ȆÖ  vÖ ) T

1

2

(A.10)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

a

wf . wȆ

Ef , b

(A.11)

According to Eq. (A.9), the derivatives of f can be obtained as follows: DA f



Ö Ö  aÖ A bȆ



§ § wf ·^ A¨ ¨ ȆÖ   E f ¨ © wȆ ¸¹ ©



^

· ¸¸ , DAȆÖ f ¹

^

AbÖ

§ wf · A¨ ¸ . (A.12) © wȆ ¹

DA g and DAȆÖ g can be obtained in the same way as Eq. (A.12). Then Eq. (A.2) can

be written as follows: { f , g} 12 ( z )

§ § wf · ^ A¨¨ ȆÖ   E f ¨ © wȆ ¸¹ ©



^

· § wg · ^ ¸¸ , A ¨ ¸ ¹ © wȆ ¹

^ § § wg · ^ ^· § wf · Ö  A¨¨ Ȇ   Eg ¸ , A ¨ ¨ © wȆ ¸¹ ¸ © wȆ ¸¹ © ¹ ^

wg wf § wf · wg Ȇ ¨  E Tf  E gT . ¸ wȆ wȆ © wȆ ¹ wȆ T

(A.13)

We rewrite Eq. (A.13) in the following form

{ f , g} 12 ( Ȇ, Į, ȕ, Ȗ )

T

’z f

ª ȆÖ « « ĮÖ « ȕÖ « «¬ ȖÖ

ĮÖ 0 0 0

ȕÖ ȖÖ º » 0 0» ’z g . 0 0 »» 0 0 »¼

(A.14)

Now we have obtained the Poisson tensor B( z ) .

Acknowledgements This work is supported by the Innovation Foundation of BUAA for PhD Graduates and the Fundamental Research Funds for the Central Universities.

References Balsas, M.C., Jiménez, E.S., Vera, J.A. The motion of a gyrostat in a central gravitational field: phase portraits of an integrable case. J. Nonlinear Math. Phy. 15(s3), 53±64, 2008. Balsas, M.C., Jiménez, E.S., Vera, J.A. et al. Qualitative analysis of the phase flow of an integrable approximation of a generalized roto-translatory problem. Cent. Eur. J. Phys. 7(1), 34

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

67±78, 2009. Barkin, Y.V. Poincaré periodic solutions of the third kind in the problem of the translational-rotational motion of a rigid body in the gravitational field of a sphere. Astronom. Zh. 56, 632±640, 1979. Barucci, M.A., Dotto, E., Levasseur-Regourd, A.C. Space missions to small bodies: asteroids and cometary nuclei. Astron. Astrophys. Rev. 19(1):48, 2011. Beck, J.A., Hall, C.D. Relative equilibria of a rigid satellite in a circular Keplerian orbit. J. Astronaut. Sci. 40(3), 215±247, 1998. Beletskii, V.V. Motion of an artificial satellite about its center of mass. Artificial Satellite of Earth 1, 13±31, 1958. Beletskii, V.V., Ponomareva, O.N. A parametric analysis of relative equilibrium stability in a gravitational field. Kosm. Issled. 28(5), 664±675, 1990. Bellerose, J., Scheeres, D.J. Energy and stability in the full two body problem. Celest. Mech. Dyn. Astron. 100, 63±91, 2008a. Bellerose, J., Scheeres, D.J. General dynamics in the restricted full three body problem. Acta Astronaut. 62, 563±576, 2008b. Boué, G., Laskar, J. Spin axis evolution of two interacting bodies. Icarus 201, 750±767, 2009. Brucker, E., Gurfil, P. Analysis of gravity-gradient-perturbed rotational dynamics at the collinear Lagrange points. J. Astronaut. Sci. 55(3), 271±291, 2007. DeBra, D.B., Delp, R.H. Rigid body attitude stability and natural frequencies in a circular orbit. J. Astronaut. Sci. 8(1), 14±17, 1961. GoĨdziewski, K., Maciejewski, A.J. Semi-analytical model of librations of a rigid moon orbiting

35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

an oblate planet. Astron. Astrophys. 339, 615±622, 1998. Guirao, J.L.G., Vera, J.A. Dynamics of a gyrostat on cylindrical and inclined Eulerian equilibria in the three-body problem. Acta Astronaut. 66, 595±604, 2010a. Guirao, J.L.G., Vera, J.A.: Lagrangian relative equilibria for a gyrostat in the three-body problem: bifurcations and stability. J. Phys. A: Math. Theor. 43, 195203, 2010b. Gurfil, P., Elipe, A., Tangren, W. et al. The Serret-Andoyer formalism in rigid-body dynamics: I. symmetries and perturbations. Regul. Chaotic Dyn. 12(4), 389±425, 2007. Hall, C.D. Attitude dynamics of orbiting gyrostats. In: PrĊWND-Ziomek, H., Wnuk, E., Seidelmann, P.K. et al. (eds.) Dynamics of Natural and Artificial Celestial Bodies, pp. 177±186. Kluwer Academic Publishers, Dordrecht, 2001. Hirabayashi, M., Morimoto, M.Y., Yano, H. et al. Linear stability of collinear equilibrium points around an asteroid as a two-connected-mass: Application to fast rotating Asteroid 2000EB14. Icarus 206, 780±782, 2010. Hu, W. Orbital Motion in Uniformly Rotating Second Degree and Order Gravity Fields, Ph.D. dissertation. Department of Aerospace Engineering, The University of Michigan, Michigan, 2002. Hu, W., Scheeres, D.J. Numerical determination of stability regions for orbital motion in uniformly rotating second degree and order gravity fields. Planet. Space Sci. 52, 685±692, 2004. Hughes, P.C. Spacecraft Attitude Dynamics, pp. 281±298. John Wiley, New York, 1986. Kinoshita, H. Stationary motions of an axisymmetric body around a spherical body and their stability. Publ. Astron. Soc. Jpn. 22, 383±403, 1970.

36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Kinoshita, H. Stationary motions of a triaxial body and their stability. Publ. Astron. Soc. Jpn. 24, 409±417, 1972a. Kinoshita, H. First-order perturbations of the two finite body problem. Publ. Astron. Soc. Jpn. 24, 423±457, 1972b. Koon, W.-S., Marsden, J.E., Ross, S.D. et al. Geometric mechanics and the dynamics of asteroid pairs. Ann. N. Y. Acad. Sci. 1017, 11±38, 2004. Kumar, K.D. Attitude dynamics and control of satellites orbiting rotating asteroids. Acta Mech. 198, 99±118, 2008. Maciejewski, A.J. Reduction, relative equilibria and potential in the two rigid bodies problem. Celest. Mech. Dyn. Astron. 63, 1±28, 1995. Maciejewski, A.J. A simple model of the rotational motion of a rigid satellite around an oblate planet. Acta Astronom. 47, 387±398, 1997. Marsden, J.E. Lectures on Mechanics, London Mathematical Society Lecture Note Series 174, Cambridge University Press, Cambridge, 1992. McMahon, J.W., Scheeres, D.J. Dynamic limits on planar libration-orbit coupling around an oblate primary. Celest. Mech. Dyn. Astron. 115(4), 365±396, 2013. Misra, A.K., Panchenko, Y. Attitude dynamics of satellites orbiting an asteroid. J. Astronaut. Sci. 54(3&4), 369±381, 2006. Mondéjar, F., Vigueras, A. The Hamiltonian dynamics of the two gyrostats problem. Celest. Mech. Dyn. Astron. 73, 303±312, 1999. Mondéjar, F., Vigueras, A., Ferrer, S. Symmetries, reduction and relative equilibria for a gyrostat in the three-body problem. Celest. Mech. Dyn. Astron. 81, 45±50, 2001.

37

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Riverin, J.L., Misra, A.K. Attitude dynamics of satellites orbiting small bodies. AIAA/AAS Astrodynamics Specialist Conference and Exhibit, AIAA 2002±4520, Monterey, California, 5±8 August 2002. San-Juan, J.F., Abad, A., Scheeres, D.J. et al. A first order analytical solution for spacecraft motion about (433) Eros. AIAA/AAS Astrodynamics Specialist Conference and Exhibit, AIAA 2002-4543, Monterey, California, 5±8 August 2002. Scheeres, D.J. Dynamics about uniformly rotating triaxial ellipsoids: Applications to asteroids. Icarus 110, 225±238, 1994. Scheeres, D.J. Stability of relative equilibria in the full two-body problem. Ann. N. Y. Acad. Sci. 1017, 81±94, 2004. Scheeres, D.J. Relative equilibria for general gravity fields in the sphere-restricted full 2-body problem. Celest. Mech. Dyn. Astron. 94, 317±349, 2006. Scheeres, D.J. Stability of the planar full 2-body problem. Celest. Mech. Dyn. Astron. 104, 103±128, 2009. Scheeres, D.J. Orbit mechanics about asteroids and comets. J. Guid. Control Dyn. 35(3), 987±997, 2012. Scheeres, D.J., Hu, W. Secular motion in a 2nd degree and order-gravity field with no rotation. Celest. Mech. Dyn. Astron. 79, 183±200, 2001. Scheeres, D.J., Ostro, S.J., Hudson, R.S. et al. Orbits close to asteroid 4769 Castalia. Icarus 121, 67±87, 1996. Scheeres, D.J., Ostro, S.J., Hudson, R.S. et al. Dynamics of orbits close to asteroid 4179 Toutatis. Icarus 132, 53±79, 1998.

38

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Scheeres, D.J., Williams, B.G., Miller, J.K. Evaluation of the dynamic environment of an asteroid: Applications to 433 Eros. J. Guid. Control Dyn. 23(3), 466±475, 2000. Vera, J.A. Eulerian equilibria of a gyrostat in Newtonian interaction with two rigid bodies. SIAM J. Appl. Dyn. Syst. 7, 1378±1396, 2008. Vera, J.A. Dynamics of a triaxial gyrostat at a Lagrangian equilibrium of a binary asteroid. Astrophys. Space Sci. 323, 375±382, 2009. Vera, J.A. On the dynamics of a gyrostat on Lagrangian equilibria in the three body problem. Multibody Syst. Dyn. 23(3), 263±291, 2010. Vera, J.A., Vigueras, A. Reduction, relative equilibria and stability for a gyrostat in the n-body problem. Monografías del Seminario Matemático García de Galdeano 31, 257±271, 2004. Vera, J.A., Vigueras, A. Hamiltonian dynamics of a gyrostat in the n-body problem: relative equilibria. Celest. Mech. Dyn. Astron. 94, 289±315, 2006. Wang, Y., Xu, S. Analysis of gravity-gradient-perturbed attitude dynamics on a stationary orbit around an asteroid via dynamical systems theory. AIAA/AAS Astrodynamics Specialist Conference, AIAA 2012-5059, Minneapolis, Minnesota, 13±16 August 2012, 2012a. Wang, Y., Xu, S. Hamiltonian structures of dynamics of a gyrostat in a gravitational field. Nonlinear Dyn. 70(1), 231±247, 2012b. Wang, Y., Xu, S. Gravity gradient torque of spacecraft orbiting asteroids. Aircr. Eng. Aerosp. Tec. 85(1), 72±81, 2013a. Wang, Y., Xu, S. Equilibrium attitude and stability of a spacecraft on a stationary orbit around an asteroid. Acta Astronaut. 84, 99±108, 2013b. Wang, Y., Xu, S. Attitude stability of a spacecraft on a stationary orbit around an asteroid

39

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

subjected to gravity gradient torque. Celest. Mech. Dyn. Astron. 115(4), 333±352, 2013c. Wang, L.-S., Krishnaprasad, P.S., Maddocks, J.H. Hamiltonian dynamics of a rigid body in a central gravitational field. Celest. Mech. Dyn. Astron. 50, 349±386, 1991. Wang, L.-S., Lian, K.-Y., Chen, P.-T. Steady motions of gyrostat satellites and their stability. IEEE T. Automat. Contr. 40(10), 1732±1743, 1995.

40

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

List of the figure captions Fig. 1. The spacecraft on a stationary orbit around the asteroid. Fig. 2. The domain of the harmonic coefficients is divided into region I, region II, region III and region IV according to the sign of Aast , Bast and Dast in the case of ae Rs upper side is the negative side for Aast , Bast and Dast ). Fig. 3. The nonlinear stability domain in the

V x -V z plane in the region I of Fig. 2.

Fig. 4. The nonlinear stability domain in the

V x -V z plane in the region II of Fig. 2.

Fig. 5. The nonlinear stability domain in the

V x -V z plane in the region III of Fig. 2.

41

0.8 (right

Highlights: Ź Harmonic coefficients C20 and C22 of the gravity field are considered. Ź Non-canonical Hamiltonian structure of the problem is derived. Ź Equilibrium attitude is determined from a global point of view. Ź Nonlinear stability conditions are obtained with energy-Casimir method. Ź Nonlinear stability domain is different from the case in a central gravity field.