1 Introduction

The strongly perturbed dynamical environment in the proximity of small asteroids has been a great challenge for the mission design, navigation, and control. To have a thorough understanding of the dynamical behavior, the spacecraft dynamics about asteroids have been studied broadly, including the orbital dynamics, attitude dynamics, and the gravitationally coupled orbit–attitude dynamics (full dynamics).

Many works have focused on the orbital dynamics near asteroids, such as Scheeres (1994, 2012a, b, 2014), Hu (2002), Hu and Scheeres (2004), Liu et al. (2011a, b), Russell (2012), Yu and Baoyin (2012a, b, 2013), Li et al. (2013), Jiang et al. (2014), Llanos et al. (2014), and so on. It has been found that the orbital motion is strongly perturbed by the non-spherical gravity of the asteroid, solar radiation pressure (SRP) and solar tide. Stable orbits against these multiple perturbations should be constructed carefully with analytical approaches and numerical tests (Scheeres 2012a, b).

Several works have studied the attitude dynamics of spacecraft around asteroids, such as Riverin and Misra (2002), Misra and Panchenko (2006), Kumar (2008), Wang and Xu (2013a, b), and Zhang and Zhao (2014, 2015). These results have shown that the attitude motion is perturbed strongly by the non-spherical mass distribution and rotation of the asteroid, and the attitude stability domain is modified remarkably in comparison with, even totally different from, the classical stability domain in a central gravity field.

In the close proximity of a small asteroid, due to the large ratio of the spacecraft dimension to the orbit radius, there is a significant gravitational coupling between the orbit and attitude motions of a large spacecraft, as shown by Wang and Xu (2014a). The magnitude of gravitational orbit–attitude coupling can be estimated by \(\varepsilon =\rho /{r_0 }\), where \(\rho \) is the spacecraft characteristic dimension and \(r_0 \) is the orbital radius (Sincarsin and Hughes 1983). For a spacecraft (\(\rho \sim 10\,\hbox {m})\) around Earth, \(\varepsilon \) is order of \(10^{-6}\). However, for a spacecraft around a small asteroid, \(\varepsilon \) can be several orders of magnitude larger due to the small size of the asteroid and the resulting small orbit radius. Therefore, the traditional spacecraft dynamics, in which the orbit and attitude motions are treated separately, will no longer maintain a high precision.

The gravitationally coupled orbit-attitude dynamics, which can be also called the full dynamics, has been proposed to take into account the gravitational orbit-attitude coupling with the spacecraft modeled as an extended rigid body. The gravitationally coupled orbit–attitude dynamics (full dynamics) of a rigid spacecraft has been studied qualitatively in a central gravity field (Wang et al. 1991, 1992; Sanyal 2004; Teixidó Román 2010), as well as in a \(J_{2}\)-truncated gravity field of a small spheroid asteroid (Wang and Xu 2013c, d; Wang et al. 2014a), and in the second degree and order gravity field of a uniformly rotating asteroid described with the harmonic coefficients \(C_{20}\) and \(C_{22}\) (Wang and Xu 2014b; Wang et al. 2014b). These studies have mainly focused on the general qualitative properties of the system, including the relative equilibria and their stability. The coupled orbit–attitude dynamics of a rigid spacecraft has been used in the control and navigation of the close-proximity operations near asteroids (Lee et al. 2014; Sanyal et al. 2014; Misra et al. 2015).

The gravitationally coupled orbit–attitude motions of small celestial bodies has been studied in the full two body problem (F2BP), i.e., two rigid bodies orbiting each other interacting through the mutual gravitational potential (Scheeres 2004, 2006, 2009; Fahnestock and Scheeres 2008; McMahon and Scheeres 2013; Woo et al. 2013). Notice that although the orbit–attitude coupling has been adopted in the dynamics of high area-to-mass ratio space debris (Früh et al. 2013) and the relative dynamics of spacecraft (Segal and Gurfil 2009), these couplings have different physical origins, and should not be confused with the gravitational orbit–attitude coupling.

Although the gravitationally coupled orbit–attitude dynamics has been proposed and studied, if from the viewpoint of orbital dynamics, the gravitational orbit–attitude coupling of the spacecraft actually causes another orbital perturbation to the Kepler two-body problem besides the non-spherical gravity of the asteroid, SRP and solar tide. This gravitational orbit–attitude coupling perturbation (GOACP) depends on the spacecraft attitude and has its origin in the fact that the gravity acting on a non-spherical extended body, the real case of the spacecraft, is actually different from that acting on a point mass, the model of the spacecraft in the orbital dynamics. In this paper, we will show that GOACP and the non-spherical terms of the asteroid’s gravity are both second-order terms of the gravitational force. Because of the dynamical sensitivity, GOACP needs to be considered for a large spacecraft.

Due to GOACP, the previous close-proximity orbital dynamics, in which only the non-spherical gravity is considered, will no longer have a high precision for a large spacecraft. If GOACP is neglected, as an unmodeled perturbation, it will degenerate the control accuracy, and fuel will be needed to null out its effect. If it is taken into account, the dynamics will be revealed better, and the fuel consumption will be more effective. That is, fuel can be saved by considering GOACP.

Therefore, we intend to take into account GOACP in the close-proximity orbital dynamics besides the non-spherical gravity. Since GOACP depends on the spacecraft attitude, we assume that the spacecraft is controlled ideally to a given attitude with respect to the asteroid, and then we focus on the orbital motion perturbed by the non-spherical gravity and GOACP with the given attitude. This new orbital model can be called the attitude-restricted orbital dynamics, where restricted means that the orbital motion is studied as a restricted problem at a given attitude.

The traditional spacecraft dynamics and the attitude-restricted orbital dynamics are two different approximations of the exact spacecraft motion. In the former, the spacecraft is first treated as a point mass in the orbital dynamics and the attitude motion is treated as a restricted problem on the predetermined orbit, whereas, in the latter, the spacecraft attitude is first given and the orbital motion is treated as a restricted problem at the predetermined attitude. The traditional dynamics is applicable when the spacecraft is small in comparison with the orbital radius and GOACP is negligible, such as around Earth, planets, and other large celestial bodies. The attitude-restricted orbital dynamics is applicable near a small asteroid or comet when GOACP is significant and the attitude can be controlled ideally.

Besides providing a more precise orbital model, the attitude-restricted orbital dynamics is also reasonable from the viewpoint of space engineering. Notice that the gravity gradient torque caused by the asteroid is small and can be well stabilized by using the attitude control system. In space engineering, a strong control of the spacecraft attitude against exterior perturbations can be achieved at the cost of electricity, which can be generated by solar panels, whereas the orbital control is much weaker, and the natural orbital dynamics need to be fully utilized for the purpose of saving fuel, since the fuel is more limited than the electricity.

Because of GOACP, the phase space of attitude-restricted orbital dynamics will be different from and more complicated than that of previous close-proximity orbital dynamics. In the present paper, we will give a qualitative study of attitude-restricted orbital dynamics. Equations of motion will be obtained and equilibrium points will be investigated as well. These equilibrium points will not only give a basic picture of the phase space, but also provide potential applications for the asteroid close-proximity operations, such as the asteroid body-fixed hovering.

2 Statement of problem

As described by Fig. 1, the problem we studied here is the motion of a rigid spacecraft B around a small asteroid P. Body-fixed reference frames of the asteroid and spacecraft are given by \(S_{P}=\{{\varvec{u}}, {\varvec{v}}, {\varvec{w}}\}\) and \(S_{B}=\{{\varvec{i}}, {\varvec{j}}, {\varvec{k}}\}\) with O and C as their origins, respectively. The origin of \(S_{P}\) is fixed at the asteroid mass center, and the coordinate axes are chosen to be aligned along the principal moments of inertia. The moments of inertia of the asteroid are assumed to satisfy

$$\begin{aligned} I_{P,zz} >I_{P,yy} ,I_{P,zz} >I_{P,xx}. \end{aligned}$$
(1)
Fig. 1
figure 1

The spacecraft moving around a small asteroid

Full size image

The second degree and order gravity field of the asteroid can be represented by the harmonic coefficients \(C_{20}\) and \(C_{22}\) with other harmonic coefficients vanished, since the origin of \(S_{P}\) is fixed at the mass center and the coordinate axes are aligned along the principal moments of inertia. \(C_{20}\) and \(C_{22}\) can be defined by (Hu and Scheeres 2004)

$$\begin{aligned} C_{20} =-\frac{1}{2Ma_e^2 }\left( {2I_{P,zz} -I_{P,xx} -I_{P,yy} } \right) <0, \quad C_{22} =\frac{1}{4Ma_e^2 }\left( {I_{P,yy} -I_{P,xx} } \right) , \end{aligned}$$
(2)

where M and \(a_e \) are the asteroid’s mass and mean equatorial radius, respectively.

It is assumed that the asteroid is rotating uniformly around its maximum-moment principal axis, i.e., \({\varvec{w}}\)-axis, with the angular velocity \(\omega _T \). Also \(S_{B}\) is attached to the mass center of the spacecraft and coincides with the principal axes frame.

The attitude of the spacecraft is described with respect to \(S_{P}\) by \({\varvec{A}}\)

$$\begin{aligned} \varvec{A}=\left[ {\varvec{\alpha }} ,{\varvec{\beta }} ,{\varvec{\gamma }}\right] ^{T}, \end{aligned}$$
(3)

where vectors \({\varvec{\alpha }}\), \({\varvec{\beta }}\), and \({\varvec{\gamma }}\) are coordinates of \({\varvec{u}}, {\varvec{v}}\), and \({\varvec{w}}\) expressed in \(S_{B}\), respectively. \({\varvec{A}}\) is also the coordinate transformation matrix from \(S_{B}\) to \(S_{P}\). The position vector of the spacecraft with respect to the asteroid mass center O expressed in \(S_{P}\) is given by \(\varvec{r}=[x,y,z]^{T}\). \(\bar{\varvec{r}}=[\bar{{x}},\bar{{y}},\bar{{z}}]^{T}\) is the unit vector along \({\varvec{r}}\). The mass of the spacecraft is denoted by m, and the inertia tensor \({\varvec{I}}\) is denoted by

$$\begin{aligned} \varvec{I}=diag\left\{ {I_{xx} ,I_{yy} ,I_{zz} } \right\} , \end{aligned}$$
(4)

with the moments of inertia \(I_{xx} \), \(I_{yy} \), and \(I_{zz} \).

3 Equations of motion and perturbations

The equations of orbital motion expressed in the asteroid frame that is uniformly rotating around the maximum-moment principal axis is given by (Scheeres 2012b)

$$\begin{aligned} \ddot{\varvec{r}}+2{\varvec{\omega }}_T \times \dot{\varvec{r}}+{\varvec{\omega }}_T \times {\varvec{\omega }}_T \times \varvec{r}={\varvec{\alpha }}_{s/c}, \end{aligned}$$
(5)

where \({\varvec{\alpha }}_{s/c} \) is the acceleration of the spacecraft and \({\varvec{\omega }}_T =[0,0,\omega _T ]^{T}\).

In the attitude-restricted orbital dynamics, the asteroid’s non-spherical gravity and GOACP are considered. The perturbations caused by SRP and solar tide are neglected. According to Wang et al. (2014b), \({\varvec{\alpha }}_{s/c} \) expressed in the asteroid frame can be obtained through a coordinate transformation from \(S_{B}\) to \(S_{P}\)

$$\begin{aligned} {\varvec{\alpha }}_{s/c} ={\varvec{\alpha }}_{Kepler} +{\varvec{\alpha }}_{{\textit{NSG}}} +{\varvec{\alpha }}_{{\textit{OAC}}}. \end{aligned}$$
(6)

In Eq. (6), \({\varvec{\alpha }}_{Kepler} \) is the Kepler two-body acceleration

$$\begin{aligned} {\varvec{\alpha }}_{Kepler} =-\frac{\mu }{r^{2}}\bar{\varvec{r}}, \end{aligned}$$
(7)

where \(\mu =\textit{GM}\), and G is the gravitational constant; \({\varvec{\alpha }} _{\textit{NSG}}\) is the perturbation acceleration of the asteroid’s non-spherical gravity truncated on the second-order

$$\begin{aligned} {\varvec{\alpha }}_{\textit{NSG}} =\frac{3\mu }{2r^{4}}\left\{ {\left[ {\tau _0 \left( {1-5\bar{{z}}^{2}} \right) -10\tau _2 \left( {\bar{{x}}^{2}-\bar{{y}}^{2}} \right) } \right] \bar{\varvec{r}}+2\tau _0 \bar{{z}}\varvec{e}_3 +4\tau _2 \left( {\bar{{x}}\varvec{e}_1 -\bar{{y}}\varvec{e}_2 } \right) } \right\} , \end{aligned}$$
(8)

where \(\tau _0 =a_e^2 C_{20} \), \(\tau _2 =a_e^2 C_{22} \), \(\varvec{e}_1 =[ {1,0,0} ]^{T}\), \(\varvec{e}_2 =[ {0,1,0} ]^{T}\), and \(\varvec{e}_3 =[ {0,0,1} ]^{T}\); \({\varvec{\alpha }}_{\textit{OAC}}\) is the acceleration caused by GOACP truncated on the second-order

$$\begin{aligned} {\varvec{\alpha }}_{\textit{OAC}} =\frac{3\mu }{2r^{4}}\left\{ {\left[ {5\bar{\varvec{r}}^{T}\varvec{A}\left( {\frac{\varvec{I}}{m}} \right) \varvec{A}^{T}\bar{\varvec{r}}-tr\left( {\frac{\varvec{I}}{m}} \right) } \right] \bar{\varvec{r}}-2\varvec{A}\left( {\frac{\varvec{I}}{m}} \right) \varvec{A}^{T}\bar{\varvec{r}}} \right\} . \end{aligned}$$
(9)

Here it deserves our special attention that, from the viewpoint of two bodies interacting through their mutual gravitational potential, the force model given by Eq. (6) is a second-order approximation of the full gravitational force. That is, \({\varvec{\alpha }}_{s/c} \) in Eq. (6) contains the zeroth-order term \({\varvec{\alpha }}_{Kepler} \), caused by the interaction between the zeroth-order mass distributions of the asteroid and spacecraft, i.e., their masses, the second-order term \({\varvec{\alpha }}_{\textit{NSG}}\), caused by the interaction between the zeroth-order mass distribution of the spacecraft and the second-order mass distribution of the asteroid, i.e., \(C_{20}\) and \(C_{22}\), and the second-order term \({\varvec{\alpha }}_{\textit{OAC}}\), caused by the interaction between the zeroth-order mass distribution of the asteroid and the second-order mass distribution of the spacecraft, i.e., the moments of inertia.

Since the first-order mass distribution parameters of the asteroid and spacecraft, i.e., their products of inertia, are all vanished in the principal axes frames \(S_{P}\) and \(S_{B}\), in \({\varvec{\alpha }}_{s/c}\), there is not a second-order term caused by the interaction between the first-order mass distributions of the bodies. In \({\varvec{\alpha }}_{s/c} \), there are not first-order terms either, which, if exist, would be caused by the interaction between the first-order mass distribution of one body and the zeroth-order mass distribution (mass) of the other body.

In the model, GOACP \({\varvec{\alpha }}_{\textit{OAC}}\), which is associated with the central gravity term (zeroth-order mass distribution) of the asteroid and the second-order mass distribution of the spacecraft, is the dominant term of the full GOACP. Here, the full GOACP can be defined as the difference of the gravitational force on an extended spacecraft versus that on a point mass spacecraft. Actually, there are higher-order terms of GOACP, which are associated with higher-order gravity terms of the asteroid or higher-order mass distribution parameters of the spacecraft, but their effect is much weaker than that of \({\varvec{\alpha }} _{\textit{OAC}}\).

As stated above, from the viewpoint of two bodies interacting through the mutual potential, the non-spherical gravity \({\varvec{\alpha }} _{\textit{NSG}}\) and GOACP \({\varvec{\alpha }}_{\textit{OAC}}\) are actually both second-order terms. The difference between them is which body is treated as a point mass and which body is treated as an extended body with the second-order mass distribution: in \({\varvec{\alpha }}_{\textit{NSG}}\), the spacecraft and asteroid are treated as a point mass and an extended body, respectively, whereas, in \({\varvec{\alpha }}_{\textit{OAC}}\), it is the opposite case.

As shown by Eqs. (8) and (9), the ratio of GOACP to the asteroid non-spherical gravity is order of \((\rho /a_{e})^{2}\), determined by the relative dimension of the spacecraft with respect to the asteroid. Usually, \(\rho \) is smaller than \(a_{e}\), therefore in the previous close-proximity orbital dynamics only \({\varvec{\alpha }}_{\textit{NSG}}\) is considered and \({\varvec{\alpha }}_{\textit{OAC}}\) is neglected. However, as \(\rho \) increases, GOACP will increase at the rate of \(\rho ^{2}\). In this paper, as stated before, what we intend is to include \({\varvec{\alpha }}_{\textit{OAC}}\) for large spacecraft.

Notice that if SRP and solar tide are considered, because of the asteroid’s rotation, the solar position is changing with time in the asteroid body-fixed frame, and the existence of equilibrium points may be much more complicated. In this study, we will focus on the equilibrium points under perturbations of only the non-spherical gravity and GOACP. The effects of SRP and solar tide can be assessed numerically in future studies.

GOACP \({\varvec{\alpha }}_{\textit{OAC}}\) depends on the parameter \(\varvec{I}/m\) of the spacecraft, which can be described by three parameters: the mass distribution parameters \(\sigma _x \) and \(\sigma _y \), and the characteristic dimension \(\rho \). \(\sigma _x \) and \(\sigma _y \) are defined as

$$\begin{aligned} \sigma _x =\left( {\frac{I_{zz} -I_{yy} }{I_{xx} }} \right) ,\quad \sigma _y =\left( {\frac{I_{zz} -I_{xx} }{I_{yy} }} \right) , \end{aligned}$$
(10)

which have the following range

$$\begin{aligned} -1\le \sigma _y \le 1,\quad -1\le \sigma _x \le 1. \end{aligned}$$
(11)

Bounds of \(\sigma _y \) and \(\sigma _x \) are determined by the properties of the moments of inertia of a rigid body: \(I_{xx} +I_{yy} >I_{zz} \), \(I_{xx} +I_{zz} >I_{yy} \), and \(I_{yy} +I_{zz} >I_{xx} \).

The characteristic dimension \(\rho \), which is an estimation but not the real value, is defined by (Wang et al. 2014a)

$$\begin{aligned} \frac{1}{2}\rho ^{2}=\frac{I_{xx} }{m}. \end{aligned}$$
(12)

As shown above, \({I_{xx} }/m\) describes the spacecraft characteristic dimension, and \(\sigma _x \) and \(\sigma _y \) describe its non-spherical mass distribution. These are two basic elements of the gravitational orbit–attitude coupling, which will be more significant with a larger ratio of the characteristic dimension to the orbital radius or with a more non-spherical mass distribution.

4 Equilibrium points

Equilibrium points means that the spacecraft is staying stationary in the asteroid body-fixed frame. That is to say, the spacecraft is on a stationary orbit. Since the asteroid is rotating uniformly around its \({\varvec{w}}\)-axis, the trajectory of the spacecraft in the inertial space is a circle perpendicular to the asteroid’s rotational axis.

4.1 Equilibrium and stability conditions

4.1.1 Equilibrium conditions

Equilibrium conditions can be easily obtained by setting \(\ddot{\varvec{r}}=\mathbf{0}\) and \(\dot{\varvec{r}}=\mathbf{0}\) in the equations of motion (5)

$$\begin{aligned} {\varvec{\omega }}_T \times {\varvec{\omega }}_T \times \varvec{r}={\varvec{\alpha }}_{s/c}, \end{aligned}$$
(13)

which actually means that the gravitational force acting on the spacecraft balances the centrifugal force of the circular orbital motion. By using Eqs. (6)–(9), the equilibrium condition (13) can be written as

$$\begin{aligned} {\varvec{\omega }}_T \times {\varvec{\omega }}_T \times \varvec{r}= & {} -\frac{\mu }{{r}^{2}}\bar{\varvec{r}}+\frac{3\mu }{2r^{4}}\left\{ {\left[ {\tau _0 \left( {1-5\bar{{z}}^{2}} \right) -10\tau _2 \left( {\bar{{x}}^{2}-\bar{{y}}^{2}} \right) } \right] \bar{\varvec{r}}+2\tau _0 \bar{{z}}\varvec{e}_3 +4\tau _2 \left( {\bar{{x}}\varvec{e}_1 -\bar{{y}}\varvec{e}_2 } \right) } \right\} \nonumber \\&+\,\frac{3\mu }{2r^{4}}\left\{ {\left[ {5\bar{\varvec{r}}^{T}\varvec{A}\left( {\frac{\varvec{I}}{m}} \right) \varvec{A}^{T}\bar{\varvec{r}}-tr\left( {\frac{\varvec{I}}{m}} \right) } \right] \bar{\varvec{r}}-2\varvec{A}\left( {\frac{\varvec{I}}{m}} \right) \varvec{A}^{T}\bar{\varvec{r}}} \right\} . \end{aligned}$$
(14)

4.1.2 Stability conditions

The stability of equilibrium points is an important qualitative property. To study the stability, we need to linearize the orbital dynamics Eq. (5) at the equilibrium point \(\varvec{r}_e \). Equation (5) can be rewritten as

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\left[ {{\begin{array}{c} {\dot{\varvec{r}}} \\ \varvec{r} \\ \end{array} }} \right] =\left[ {{\begin{array}{cc} {-2\omega _T \hat{\varvec{e}}_3 }&{}\quad {-\omega _T^2 \hat{\varvec{e}}_3 \hat{\varvec{e}}_3 } \\ \mathbf{E}&{}\quad \mathbf{0} \\ \end{array} }} \right] \left[ {{\begin{array}{c} {\dot{\varvec{r}}} \\ \varvec{r} \\ \end{array} }} \right] +\left[ {{\begin{array}{c} {{\varvec{\alpha }}_{s/c} } \\ \mathbf{0} \\ \end{array} }} \right] , \end{aligned}$$
(15)

where \(\mathbf{E}\) is the \(3\times 3\) identity matrix, and the hat map \(\wedge \) is

$$\begin{aligned} \hat{\varvec{e}}_3 =\left[ {{\begin{array}{ccc} 0&{}\quad {-1}&{}\quad 0 \\ 1&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0 \\ \end{array} }} \right] . \end{aligned}$$
(16)

\(\dot{\varvec{r}}\) and \(\varvec{r}\) can be written in terms of variations \(\delta \dot{\varvec{r}}\) and \(\delta \varvec{r}\) as

$$\begin{aligned} \dot{\varvec{r}}=\mathbf{0}+\delta \dot{\varvec{r}},\quad \varvec{r}=\varvec{r}_e +\delta \varvec{r}. \end{aligned}$$
(17)

Then, the linearized equation near \(\varvec{r}_e \) is given by

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\left[ {{\begin{array}{c} {\mathbf{0}+\delta \dot{\varvec{r}}} \\ {\varvec{r}_e +\delta \varvec{r}} \\ \end{array} }} \right] =\left[ {{\begin{array}{cc} {-2\omega _T \hat{\varvec{e}}_3 }&{}\quad {-\omega _T^2 \hat{\varvec{e}}_3 \hat{\varvec{e}}_3 } \\ \mathbf{E}&{}\quad \mathbf{0} \\ \end{array} }} \right] \left[ {{\begin{array}{c} {\mathbf{0}+\delta \dot{\varvec{r}}} \\ {\varvec{r}_e +\delta \varvec{r}} \\ \end{array} }} \right] +\left[ {{\begin{array}{c} {{{\varvec{\alpha }}_{s/c} } \big |_e + {\frac{\partial {\varvec{\alpha }}_{s/c} }{\partial \varvec{r}}} \bigg |_e \delta \varvec{r}} \\ \mathbf{0} \\ \end{array} }} \right] . \end{aligned}$$
(18)

According to equilibrium conditions at \(\varvec{r}_e \)

$$\begin{aligned} \left[ {{\begin{array}{c} \mathbf{0} \\ \mathbf{0} \\ \end{array} }} \right] =\left[ {{\begin{array}{cc} {-2\omega _T \hat{\varvec{e}}_3 }&{}\quad {-\omega _T^2 \hat{\varvec{e}}_3 \hat{\varvec{e}}_3 } \\ \mathbf{E}&{}\quad \mathbf{0} \\ \end{array} }} \right] \left[ {{\begin{array}{c} \mathbf{0} \\ {\varvec{r}_e } \\ \end{array} }} \right] +\left[ {{\begin{array}{c} {{{\varvec{\alpha }}_{s/c} } \big |_e } \\ \mathbf{0} \\ \end{array} }} \right] , \end{aligned}$$
(19)

the variation equation near \(\varvec{r}_e \) can be obtained

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\left[ {{\begin{array}{c} {\delta \dot{\varvec{r}}} \\ {\delta \varvec{r}} \\ \end{array} }} \right] =\left[ {{\begin{array}{cc} {-2\omega _T \hat{\varvec{e}}_3 }&{}\quad {-\omega _T^2 \hat{\varvec{e}}_3 \hat{\varvec{e}}_3 +{\frac{\partial {\varvec{\alpha }}_{s/c} }{\partial \varvec{r}}} \bigg |_e } \\ \mathbf{E}&{}\quad \mathbf{0} \\ \end{array} }} \right] \left[ {{\begin{array}{c} {\delta \dot{\varvec{r}}} \\ {\delta \varvec{r}} \\ \end{array} }} \right] . \end{aligned}$$
(20)

The linear stability is determined by the system matrix of the linearized system

$$\begin{aligned} \varvec{D}\left( {\varvec{r}_e } \right) =\left[ {{\begin{array}{cc} {-2\omega _T \hat{\varvec{e}}_3 }&{}\quad {-\omega _T^2 \hat{\varvec{e}}_3 \hat{\varvec{e}}_3 +{\frac{\partial {\varvec{\alpha }}_{s/c} }{\partial \varvec{r}}} \bigg |_e } \\ \mathbf{E}&{}\quad \mathbf{0} \\ \end{array} }} \right] , \end{aligned}$$
(21)

all the eigenvalues of which are required to have a non-positive real part for the linear stability. Since the system is conservative, there is only even terms in the characteristic polynomial of \(\varvec{D}({\varvec{r}_e })\), and the eigenvalues will be symmetrical with respect to both the real and imaginary axes. Then, the linear stability requires all the eigenvalues of \(\varvec{D}( {\varvec{r}_e } )\) to be purely imaginary.

Notice that \(\varvec{D}( {\varvec{r}_e })\) is determined by the angular velocity \(\omega _T \) and the gradient of \({\varvec{\alpha }} _{s/c} \). According to Eqs. (6)–(9), the gradient of \({\varvec{\alpha }}_{s/c} \) can be derived as

$$\begin{aligned} \frac{\partial {\varvec{\alpha }}_{s/c} }{\partial \varvec{r}}= & {} -\frac{\mu }{r^{3}}\left( {\mathbf{E}-3\bar{\varvec{r}}\bar{\varvec{r}}^{T}} \right) \nonumber \\&-\,\frac{3\mu }{2r^{5}}\left\{ {5\bar{\varvec{r}}^{T}\varvec{A} \left( {\frac{\varvec{I}}{m}}\right) \varvec{A}^{T}\bar{\varvec{r}}- tr\left( {\frac{\varvec{I}}{m}} \right) \!+\!\tau _0 \left( {1-5\bar{{z}}^{2}} \right) -10\tau _2 \left( {\bar{{x}}^{2}-\bar{{y}}^{2}} \right) } \right\} \left\{ {7\bar{\varvec{r}}\bar{\varvec{r}}^{T}-\mathbf{E}} \right\} \nonumber \\&-\,\frac{3\mu }{r^{5}}\left\{ {\left[ {tr\left( {\frac{\varvec{I}}{m}} \right) -\tau _0 } \right] \bar{\varvec{r}}\bar{\varvec{r}}^{T}+\varvec{A}\left( {\frac{\varvec{I}}{m}} \right) \varvec{A}^{T}-\tau _0 \varvec{e}_3 \varvec{e}_3 ^{T}-2\tau _2 \left( {\varvec{e}_1 \varvec{e}_1^T -\varvec{e}_2 \varvec{e}_2^T } \right) } \right\} \nonumber \\&-\,\frac{15\mu }{r^{5}}\left\{ {-\varvec{A}\left( {\frac{\varvec{I}}{m}} \right) \varvec{A}^{T}\bar{\varvec{r}}\bar{\varvec{r}}^{T}-\bar{\varvec{r}}\bar{\varvec{r}}^{T}\varvec{A}\left( {\frac{\varvec{I}}{m}} \right) \varvec{A}^{T}+\tau _0 \bar{{z}}\left( {\varvec{e}_3 \bar{\varvec{r}}^{T}+\bar{\varvec{r}}\varvec{e}_3^T } \right) } \right. \nonumber \\&+\,\left. {2\tau _2 \left[ {\bar{{x}}\left( {\varvec{e}_1 \bar{\varvec{r}}^{T}+\bar{\varvec{r}}\varvec{e}_1^T } \right) - \bar{{y}}\left( {\varvec{e}_2 \bar{\varvec{r}}^{T}+\bar{\varvec{r}} \varvec{e}_2^T } \right) } \right] } \right\} . \end{aligned}$$
(22)

We can see that the formula of \(\frac{\partial {\varvec{\alpha }}_{s/c} }{\partial \varvec{r}}\) is tedious and it is difficult to obtain the explicit formula of the characteristic polynomial of \(\varvec{D}({\varvec{r}_e })\). The stability depends on not only the asteroid parameters: \(\mu \), \(\omega _T \), \(\tau _0 \), and \(\tau _2 \), but also the spacecraft parameters: \(\varvec{A}\), \(\varvec{I}\), and m.

In the calculation of off-axis equilibrium points, we will determine the stability by numerical calculations and give some discussions. It will need lots of work to investigate the stability thoroughly with respect to all the system parameters. The stability analysis of the six degree-of-freedom rigid body motion has been carried out in several previous works, such as Wang et al. (1991, 1992, 2014b), Wang and Xu (2013d). A similar method can be used in the stability analysis of this problem. However, should keep in mind that in this problem the three degree-of-freedom translational motion is studied, but not the six degree-of-freedom rigid body motion.

4.2 Previous results

In previous studies on the close-proximity orbital dynamics about asteroids, such as Hu (2002), only the perturbation acceleration of the asteroid’s non-spherical gravity was considered, i.e., \({\varvec{\alpha }} _{s/c} ={\varvec{\alpha }}_{Kepler} +{\varvec{\alpha }}_{\textit{NSG}}\). Then, the equilibrium condition is

$$\begin{aligned}&{\varvec{\omega }}_T \times {\varvec{\omega }}_T \times \varvec{r}\nonumber \\&\quad =-\frac{\mu }{r^{2}}\bar{\varvec{r}}+\frac{3\mu }{2r^{4}}\left\{ {\left[ {\tau _0 \left( {1-5\bar{{z}}^{2}} \right) -10\tau _2 \left( {\bar{{x}}^{2}-\bar{{y}}^{2}} \right) } \right] \bar{\varvec{r}}+2\tau _0 \bar{{z}}\varvec{e}_3 +4\tau _2 \left( {\bar{{x}}\varvec{e}_1 -\bar{{y}}\varvec{e}_2 } \right) } \right\} .\qquad \quad \end{aligned}$$
(23)

Previous results based on Eq. (23) have shown that within the equatorial plane of the asteroid there exist two kinds of equilibrium points, which lie on the \({\varvec{u}}\)-axes and on the \({\varvec{v}}\)-axis. It is impossible to have equilibrium points at other longitudes.

The radii of these stationary orbits on the \({\varvec{u}}\)- and \({\varvec{v}}\)-axes satisfy the following equations, respectively

$$\begin{aligned} \omega _T^2 r= & {} \frac{\mu }{r^{2}}-\frac{3\mu }{2r^{4}}\left( {\tau _0 -6\tau _2 } \right) , \end{aligned}$$
(24)
$$\begin{aligned} \omega _T^2 r= & {} \frac{\mu }{r^{2}}-\frac{3\mu }{2r^{4}}\left( {\tau _0 +6\tau _2 } \right) . \end{aligned}$$
(25)

The stationary orbits lying on the longer-axis (\({\varvec{u}}\)-axis) are always unstable, while those lying on the shorter-axis (\({\varvec{v}}\)-axis) are stable under the following approximated condition (Hu 2002)

$$\begin{aligned} \left( {\frac{\mu }{\omega _T^2 }} \right) ^{2/3}+\tau _0 -162\tau _2 >0. \end{aligned}$$
(26)

4.3 Equilibrium points within equatorial plane

Because of GOACP \({\varvec{\alpha }}_{\textit{OAC}}\), the equilibrium points determined by Eq. (14) in the attitude-restricted orbital dynamics will be different from the previous classical equilibrium points without \({\varvec{\alpha }}_{\textit{OAC}}\) determined by Eq. (23).

GOACP \({\varvec{\alpha }}_{\textit{OAC}}\) introduces several new parameters into the system, including the spacecraft attitude \(\varvec{A}\) and parameter \(\varvec{I}/m\), which includes \(\sigma _x \), \(\sigma _y \), and \(\rho \). The system is more complicated than the previous classical system Eq. (23), and the types and locations of equilibrium points will be more diverse.

In the present paper, we will study the equilibrium points within the asteroid equatorial plane, that is \(\bar{{z}}=0\) in the equilibrium condition (14). Then, we can have the new equilibrium condition within the equatorial plane as

$$\begin{aligned} \omega _T^2 \varvec{r}= & {} \frac{\mu }{r^{2}}\bar{\varvec{r}}- \frac{3\mu }{2r^{4}}\left\{ {\left[ {\tau _0 -10\tau _2 \left( {\bar{{x}}^{2}-\bar{{y}}^{2}} \right) +5\bar{\varvec{r}}^{T}\varvec{A}\left( {\varvec{I}/m} \right) \varvec{A}^{T}\bar{\varvec{r}}-tr({\varvec{I}/m})} \right] \bar{\varvec{r}}} \right. \nonumber \\&+\,\left. {4\tau _2 \left( {\bar{{x}}\varvec{e}_1 -\bar{{y}}\varvec{e}_2 } \right) -2\varvec{A}\left( {\varvec{I}/m} \right) \varvec{A}^{T} \bar{\varvec{r}}} \right\} , \end{aligned}$$
(27)

which requires that \(4\tau _2 ({\bar{{x}}\varvec{e}_1 -\bar{{y}}\varvec{e}_2 } )-2\varvec{A}( {\varvec{I}/m})\varvec{A}^{T}\bar{\varvec{r}}\) is parallel to the position vector \(\bar{\varvec{r}}\).

4.4 On-axis equilibrium points

First we focus on the on-axis equilibrium points. We assume that the equilibrium point is located on the equatorial principal axes, i.e., on the \({\varvec{u}}\)- or \({\varvec{v}}\)-axis.

4.4.1 On \({\varvec{u}}\)-axis

If the spacecraft is located on the \({\varvec{u}}\)-axis, we have \(\bar{{x}}=\pm 1\), \(\bar{{y}}=0\), and \(\bar{\varvec{r}}=[ {\pm 1,0,0} ]=\pm \varvec{e}_1 \) in Eq. (27), and then the new equilibrium condition is given by

$$\begin{aligned} \omega _T^2 r\varvec{e}_1 =\frac{\mu }{r^{2}}\varvec{e}_1 -\frac{3\mu }{2r^{4}}\left\{ {\left[ {\tau _0 -6\tau _2 +5\varvec{e}_1 ^{T}\varvec{A}({\varvec{I}/m} )\varvec{A}^{T}\varvec{e}_1 -tr({\varvec{I}/m} )} \right] \varvec{e}_1 -2\varvec{A}({\varvec{I}/m} )\varvec{A}^{T}\varvec{e}_1 } \right\} .\nonumber \\ \end{aligned}$$
(28)

This requires that \(\varvec{e}_1 \) is the eigenvector of the matrix \(\varvec{A}({\varvec{I}/m})\varvec{A}^{T}\), which is actually the inertia tensor of the spacecraft expressed in the asteroid frame \(S_{P}\). That is to say, the \({\varvec{u}}\)-axis of the asteroid lies on the principal axis of the spacecraft.

Without loss of generality, we assume that the \({\varvec{u}}\)-axis lies on the \({\varvec{i}}\)-axis of the spacecraft and have the same positive direction. Other cases can be converted into this case by changing the arrangement of axes of the spacecraft’s body frame. In this case, the relative attitude of the spacecraft with respect to the asteroid is just a single axis rotation around the \({\varvec{u}}\)-axis, i.e.,

$$\begin{aligned} \varvec{A}=\left[ {{\begin{array}{ccc} 1&{}\quad 0&{}\quad 0 \\ 0&{}\quad {\cos \phi }&{}\quad {-\sin \phi } \\ 0&{}\quad {\sin \phi }&{}\quad {\cos \phi } \\ \end{array} }} \right] , \end{aligned}$$
(29)

where \(\phi \) is the rotational angle. Therefore, we have

$$\begin{aligned} \varvec{A}({\varvec{I}/m})\varvec{A}^{T}= \left[ {{\begin{array}{ccc} {{I_{xx} }/m}&{}\quad 0&{}\quad 0 \\ 0&{}\quad {\cos ^{2}\phi {I_{yy} }/m+\sin ^{2}\phi {I_{zz} }/m}&{}\quad {\sin \phi \cos \phi \left( {{I_{yy} }/m-{I_{zz} }/m} \right) }\\ 0&{}\quad {\sin \phi \cos \phi \left( {{I_{yy} }/m-{I_{zz} }/m} \right) }&{}\quad {\sin ^{2}\phi {I_{yy} }/m+\cos ^{2}\phi {I_{zz} }/m} \\ \end{array} }} \right] .\nonumber \\ \end{aligned}$$
(30)

Then, the equilibrium condition (28) can be written as

$$\begin{aligned} \omega _T^2 r=\frac{\mu }{r^{2}}-\frac{3\mu }{2r^{4}}\left[ {2\frac{I_{xx} }{m}-\frac{I_{yy} }{m}-\frac{I_{zz} }{m}+\tau _0 -6\tau _2 } \right] . \end{aligned}$$
(31)

The orbital radius of the equilibrium point, which is the same with the orbital radius of the relative equilibrium in the gravitationally coupled orbit–attitude (full) dynamics (Wang et al. 2014b; Wang and Xu 2014b), will be determined by Eq. (31). At the relative equilibrium in Wang et al. (2014b), the spacecraft has the same attitude with the asteroid, i.e., \(\phi =0\). Here, the rotational angle \(\phi \) does not affect the location of the equilibrium point. This is because in the orbital dynamics Eq. (5) the gravitational force has been truncated up to the second order, and at the equilibrium point the rotational angle \(\phi \) only affects the higher-order terms.

Through the comparison between Eqs. (24) and (31), we can easily see the effect of GOACP \({\varvec{\alpha }}_{\textit{OAC}} \). Different from the previous orbital dynamics without \({\varvec{\alpha }}_{\textit{OAC}} \) in Eq. (24), the orbital motion of the spacecraft in Eq. (31) is affected by its moments of inertia. This effect can be considered as an equivalent change of the oblateness and ellipticity of the asteroid from the viewpoint of orbital dynamics without \({\varvec{\alpha }}_{\textit{OAC}}\). Notice that in the case of \(I_{xx} =I_{yy} =I_{zz} \), i.e., the spacecraft is a homogeneous sphere under the second-order approximation, the effects of moments of inertia are vanished. In this case, Eq. (31) will be reduced to the previous result without \({\varvec{\alpha }}_{\textit{OAC}}\) in Eq. (24). Besides, for a larger \(\rho \), GOACP is more significant. These results are consistent with the physical origin of the gravitational orbit–attitude coupling.

Because of the symmetry, if the \({\varvec{u}}\)-axis lies on the \({\varvec{i}}\)-axis of spacecraft but has the opposite direction, we can also have the orbital radius given by Eq. (31).

If the \({\varvec{u}}\)-axes lies on the \({\varvec{j}}\)- and \({\varvec{k}}\)-axes of the spacecraft, with the same method, we can obtain the similar equilibrium conditions, respectively

$$\begin{aligned} \omega _T^2 r= & {} \frac{\mu }{r^{2}}-\frac{3\mu }{2r^{4}}\left[ {2\frac{I_{yy} }{m}-\frac{I_{xx} }{m}-\frac{I_{zz} }{m}+\tau _0 -6\tau _2 } \right] , \end{aligned}$$
(32)
$$\begin{aligned} \omega _T^2 r= & {} \frac{\mu }{r^{2}}-\frac{3\mu }{2r^{4}}\left[ {2\frac{I_{zz} }{m}-\frac{I_{xx} }{m}-\frac{I_{yy} }{m}+\tau _0 -6\tau _2 } \right] . \end{aligned}$$
(33)

4.4.2 On \({\varvec{v}}\)-axis

If we assume that the spacecraft is located on the \({\varvec{v}}\)-axis, we will have \(\bar{{x}}=0\), \(\bar{{y}}=\pm 1\), and \(\bar{\varvec{r}}=[ {0,\pm 1,0}]=\pm \varvec{e}_2 \) in the equilibrium condition (27), and then the corresponding new equilibrium condition is given by

$$\begin{aligned} \omega _T^2 r\varvec{e}_2 =\frac{\mu }{r^{2}}\varvec{e}_2 -\frac{3\mu }{2r^{4}}\left\{ {\left[ {\tau _0 +6\tau _2 +5\varvec{e}_2 ^{T}\varvec{A}({\varvec{I}/m} )\varvec{A}^{T}\varvec{e}_2 -tr({\varvec{I}/m} )} \right] \varvec{e}_2 -2\varvec{A}({\varvec{I}/m} )\varvec{A}^{T}\varvec{e}_2 } \right\} .\nonumber \\ \end{aligned}$$
(34)

This requires that \(\varvec{e}_2 \) is the eigenvector of \(\varvec{A}({\varvec{I}/m})\varvec{A}^{T}\). That is to say, the \({\varvec{v}}\)-axis of the asteroid lies on the principal axis of the spacecraft.

With a similar method with the cases on the \({\varvec{u}}\)-axis, we can obtain the equations determining the orbital radius of equilibrium points as

$$\begin{aligned} \omega _T^2 r= & {} \frac{\mu }{r^{2}}-\frac{3\mu }{2r^{4}}\left[ {2\frac{I_{xx} }{m}-\frac{I_{yy} }{m}-\frac{I_{zz} }{m}+\tau _0 +6\tau _2 } \right] , \end{aligned}$$
(35)
$$\begin{aligned} \omega _T^2 r= & {} \frac{\mu }{r^{2}}-\frac{3\mu }{2r^{4}}\left[ {2\frac{I_{yy} }{m}-\frac{I_{xx} }{m}-\frac{I_{zz} }{m}+\tau _0 +6\tau _2 } \right] , \end{aligned}$$
(36)
$$\begin{aligned} \omega _T^2 r= & {} \frac{\mu }{r^{2}}-\frac{3\mu }{2r^{4}}\left[ {2\frac{I_{zz} }{m}-\frac{I_{xx} }{m}-\frac{I_{yy} }{m}+\tau _0 +6\tau _2 } \right] , \end{aligned}$$
(37)

which correspond to the cases that the \({\varvec{v}}\)-axis lies on the \({\varvec{i}}\)-, \({\varvec{j}}\)-, and \({\varvec{k}}\)-axes of the spacecraft, respectively. The differences between Eqs. (35)–(37) and the previous orbital dynamics Eq. (25) are due to GOACP \({\varvec{\alpha }} _{\textit{OAC}}\).

As stated above, the on-axis equilibrium points can be divided into four groups, which are on the +\({\varvec{u}}\)-, \(-{\varvec{u}}\)-, \(+{\varvec{v}}-\), and \(-{\varvec{v}}\)-axes of the asteroid, respectively. In each group, there exist three different cases that the equatorial principal axis of the asteroid lies on the \({\varvec{i}}\)-, \({\varvec{j}}\)-, and \({\varvec{k}}\)-axes of the spacecraft, respectively. Although the longitudes of equilibrium points are the same with those in the previous orbital dynamics without \({\varvec{\alpha }}_{\textit{OAC}}\), the orbital radii are different due to GOACP \({\varvec{\alpha }}_{\textit{OAC}} \), as shown by Eqs. (31)–(33) and (35)–(37).

4.5 Off-axis equilibrium points

In this subsection, we obtain the off-axis equilibrium points, which are located off the \({\varvec{u}}\)- and \({\varvec{v}}\)-axes of the asteroid. These off-axis equilibrium points cannot exist in the previous orbital dynamics without \({\varvec{\alpha }}_{\textit{OAC}}\), but can exist in the attitude-restricted orbital dynamics due to GOACP.

The spacecraft can stay at a stable equilibrium point without orbital control, and stay at an unstable equilibrium point with small orbital control without needing to null out the unbalanced gravity. Therefore, the off-axis equilibrium points can be useful in the asteroid close-proximity operations, such as the asteroid body-fixed hovering.

4.5.1 Off-axis equilibrium conditions

As for these off-axis equilibrium points, we have \(\bar{{x}}\ne 0\), \(\bar{{y}}\ne 0\), and \(\bar{\varvec{r}}=[ {\bar{{x}},\bar{{y}},0} ]\) in the equilibrium condition (27), which cannot be simplified further.

Equation (27) requires that \(4\tau _2 ({\bar{{x}}\varvec{e}_1 -\bar{{y}}\varvec{e}_2 })-2\varvec{A}( {\varvec{I}/m})\varvec{A}^{T}\bar{\varvec{r}}\) is parallel to \(\bar{\varvec{r}}\). Notice that \(\bar{\varvec{r}}\) is within the equatorial plane spanned by \(\varvec{e}_1 \) and \(\varvec{e}_2 \), therefore \(\varvec{A}({\varvec{I}/m} )\varvec{A}^{T}\bar{\varvec{r}}\) should also be within the equatorial plane. Then, we can know that the equatorial plane is the principal plane of the inertia tensor \(\varvec{A}( {\varvec{I}/m})\varvec{A}^{T}\) that is expressed in \(S_{P}\). That is to say, one of the \({\varvec{i}}\)–\({\varvec{j}}\) plane, \({\varvec{j}}\)–\({\varvec{k}}\) plane, and \({\varvec{i}}\)–\({\varvec{k}}\) plane of the spacecraft is within the \(\varvec{e}_1\)–\(\varvec{e}_2 \) plane of the asteroid. Without loss of generality, we assume that the \({\varvec{i}}\)–\({\varvec{j}}\) plane is within the \(e_1\)–\(e_2 \) plane, and the unit normal vector \({\varvec{k}}\) of the \({\varvec{i}}\)–\({\varvec{j}}\) plane has the same direction with \(\varvec{e}_3 \). Other cases can be converted into this case by changing the arrangement of axes of \(S_{B}\). That is to say, the relative attitude of the spacecraft with respect to the asteroid is just a single axis rotation around the \({\varvec{w}}\)-axis, i.e.,

$$\begin{aligned} \varvec{A}=\left[ {{\begin{array}{ccc} {\cos \psi }&{}\quad {-\sin \psi }&{}\quad 0 \\ {\sin \psi }&{}\quad {\cos \psi }&{}\quad 0 \\ 0&{}\quad 0&{}\quad 1 \\ \end{array} }} \right] , \end{aligned}$$
(38)

where \(\psi \) is the rotational angle. Therefore, we have

$$\begin{aligned}&\varvec{A}({\varvec{I}/m})\varvec{A}^{T}=\left[ {{\begin{array}{ccc} {\cos ^{2}\psi {I_{xx} }/m+\sin ^{2}\psi {I_{yy} }/m}&{}\quad {\sin \psi \cos \psi \left( {{I_{xx} }/m-{I_{yy} }/m} \right) }&{}\quad 0 \\ {\sin \psi \cos \psi \left( {{I_{xx} }/m-{I_{yy} }/m} \right) }&{}\quad {\sin ^{2}\psi {I_{xx} }/m+\cos ^{2}\psi {I_{yy} }/m}&{}\quad 0 \\ 0&{}\quad 0&{}\quad {{I_{zz} }/m} \\ \end{array} }} \right] ,\nonumber \\ \end{aligned}$$
(39)
$$\begin{aligned}&4\tau _2 \left( {\bar{{x}}\varvec{e}_1 -\bar{{y}}\varvec{e}_2 } \right) -2\varvec{A}( {\varvec{I}/m})\varvec{A}^{T}\bar{\varvec{r}} \nonumber \\&\quad =2\left[ {{\begin{array}{c} {-\left( {\cos ^{2}\psi {I_{xx} }/m+\sin ^{2}\psi {I_{yy} }/m-2\tau _2 } \right) \bar{{x}}-\sin \psi \cos \psi \left( {{I_{xx} }/m-{I_{yy} }/m} \right) \bar{{y}}} \\ {-\sin \psi \cos \psi \left( {{I_{xx} }/m-{I_{yy} }/m} \right) \bar{{x}}-\left( {\sin ^{2}\psi {I_{xx} }/m+\cos ^{2}\psi {I_{yy} }/m+2\tau _2 } \right) \bar{{y}}} \\ 0 \\ \end{array} }} \right] .\nonumber \\ \end{aligned}$$
(40)

Then, the condition that \(4\tau _2 ({\bar{{x}}\varvec{e}_1 -\bar{{y}}\varvec{e}_2 })-2\varvec{A}( {\varvec{I}/m})\varvec{A}^{T}\bar{\varvec{r}}\) is parallel to \(\bar{\varvec{r}}\) is equivalent to

$$\begin{aligned} \frac{\left( {\cos ^{2}\psi {I_{xx} }/m+\sin ^{2}\psi {I_{yy} }/m-2\tau _2 } \right) \bar{{x}}+\sin \psi \cos \psi \left( {{I_{xx} }/m-{I_{yy} }/m} \right) \bar{{y}}}{\sin \psi \cos \psi \left( {{I_{xx} }/m-{I_{yy} }/m} \right) \bar{{x}}+\left( {\sin ^{2}\psi {I_{xx} }/m+\cos ^{2}\psi {I_{yy} }/m+2\tau _2 } \right) \bar{{y}}}=\frac{\bar{{x}}}{\bar{{y}}}, \end{aligned}$$
(41)

which can be further simplified as

$$\begin{aligned} \frac{\left( {\cos ^{2}\psi +\sin ^{2}\psi {I_{yy} }/{I_{xx} }-{4\tau _2 }/{\rho ^{2}}} \right) \bar{{x}}+\sin \psi \cos \psi \left( {1-{I_{yy} }/{I_{xx} }} \right) \bar{{y}}}{\sin \psi \cos \psi \left( {1-{I_{yy} }/{I_{xx} }} \right) \bar{{x}}+\left( {\sin ^{2}\psi +\cos ^{2}\psi {I_{yy} }/{I_{xx} }+{4\tau _2 }/{\rho ^{2}}} \right) \bar{{y}}}=\frac{\bar{{x}}}{\bar{{y}}}. \end{aligned}$$
(42)

We can see that the condition (42) is affected by the mass distribution parameter \({I_{yy} }/{I_{xx} }\), the characteristic dimension \(\rho \) and the attitude angle \(\psi \) of the spacecraft. These parameters are all introduced into the system by GOACP \({\varvec{\alpha }} _{\textit{OAC}}\). If \({\varvec{\alpha }}_{\textit{OAC}} \) is neglected, Eq. (41) will be \({\bar{{x}}}/{\bar{{y}}}=-{\bar{{x}}}/{\bar{{y}}}\), which implies that \(\bar{{x}}=0\) or \(\bar{{y}}=0\). This is the reason why the off-axis equilibrium point cannot exist in the previous orbital dynamics without \({\varvec{\alpha }}_{\textit{OAC}}\).

After some rearrangements, Eq. (42) can be simplified as

$$\begin{aligned}&\sin \psi \cos \psi \left( {1-{I_{yy} }/{I_{xx} }} \right) \left( {\bar{{x}}^{2}-\bar{{y}}^{2}} \right) \nonumber \\&\quad +\,\left[ {\left( {\sin ^{2}\psi -\cos ^{2}\psi } \right) \left( {1-{I_{yy} }/{I_{xx} }} \right) +{8C_{22} a_e^2 }/{\rho ^{2}}} \right] \bar{{x}}\bar{{y}}=0. \end{aligned}$$
(43)

By using the relation \(\bar{{x}}^{2}+\bar{{y}}^{2}=1\), Eq. (43) can be written further as

$$\begin{aligned} \bar{{x}}^{4}-\bar{{x}}^{2}+c=0, \end{aligned}$$
(44)

where

$$\begin{aligned} c=\frac{\sin ^{2}\psi \cos ^{2}\psi \left( {1-{I_{yy} }/{I_{xx} }} \right) ^{2}}{4\sin \psi ^{2}\cos \psi ^{2}\left( {1-{I_{yy} }/{I_{xx} }} \right) ^{2}+\left[ {\left( {\sin ^{2}\psi -\cos ^{2}\psi } \right) \left( {1-{I_{yy} }/{I_{xx} }} \right) +{8C_{22} a_e^2 }/{\rho ^{2}}} \right] ^{2}}>0.\nonumber \\ \end{aligned}$$
(45)

Therefore, we have

$$\begin{aligned} \bar{{x}}^{2}=\frac{1\pm \sqrt{1-4c}}{2}, \end{aligned}$$
(46)

which contains four solutions: if the terms \(\sin \psi \cos \psi ({1-{I_{yy} }/{I_{xx} }})\) and \(({\sin ^{2}\psi -\cos ^{2}\psi })\times ( {1-{I_{yy} }/{I_{xx} }} )+{8C_{22} a_e^2 }/{\rho ^{2}}\) in Eq. (43) have the same sign, \(\bar{{x}}^{2}-\bar{{y}}^{2}\) and \(\bar{{x}}\bar{{y}}\) will have the opposite signs, and then

$$\begin{aligned}&(1)\;\;\bar{{x}}=\sqrt{\frac{1+\sqrt{1-4c}}{2}},\quad \bar{{y}}=-\sqrt{\frac{1-\sqrt{1-4c}}{2}}; \nonumber \\&(2)\;\;\bar{{x}}=-\sqrt{\frac{1+\sqrt{1-4c}}{2}},\quad \bar{{y}}=\sqrt{\frac{1-\sqrt{1-4c}}{2}}; \nonumber \\&(3)\;\;\bar{{x}}=\sqrt{\frac{1-\sqrt{1-4c}}{2}},\quad \bar{{y}}=\sqrt{\frac{1+\sqrt{1-4c}}{2}}; \\&(4)\;\;\bar{{x}}=-\sqrt{\frac{1-\sqrt{1-4c}}{2}},\quad \bar{{y}}=-\sqrt{\frac{1+\sqrt{1-4c}}{2}}; \nonumber \end{aligned}$$
(47)

If the terms \(\sin \psi \cos \psi ({1-{I_{yy} }/{I_{xx} }} )\) and \(( {\sin ^{2}\psi -\cos ^{2}\psi })( {1-{I_{yy}}/{I_{xx} }})+{8C_{22} a_e^2 }/{\rho ^{2}}\) in Eq. (43) have the opposite signs, \(\bar{{x}}^{2}-\bar{{y}}^{2}\) and \(\bar{{x}}\bar{{y}}\) will have the same sign, and then

$$\begin{aligned}&(1)\;\;\bar{{x}}=\sqrt{\frac{1+\sqrt{1-4c}}{2}},\quad \bar{{y}}=\sqrt{\frac{1-\sqrt{1-4c}}{2}}; \nonumber \\&(2)\;\;\bar{{x}}=-\sqrt{\frac{1+\sqrt{1-4c}}{2}},\quad \bar{{y}}=-\sqrt{\frac{1-\sqrt{1-4c}}{2}}; \nonumber \\&(3)\;\;\bar{{x}}=\sqrt{\frac{1-\sqrt{1-4c}}{2}},\quad \bar{{y}}=-\sqrt{\frac{1+\sqrt{1-4c}}{2}}; \\&(4)\;\;\bar{{x}}=-\sqrt{\frac{1-\sqrt{1-4c}}{2}},\quad \bar{{y}}=\sqrt{\frac{1+\sqrt{1-4c}}{2}}. \nonumber \end{aligned}$$
(48)

These four solutions are located in four quadrants of the equatorial plane of the asteroid, i.e., the \(\varvec{e}_1 -\varvec{e}_2 \) plane, respectively. Because of the symmetry of the system, the equilibrium points in the first and third quadrants are symmetrical with respect to the asteroid center, and also the equilibrium points in the second and fourth quadrants. That is, in Eqs. (47) and (48) the first and second solutions have the opposite signs, and also the third and fourth solutions.

With Eqs. (45), (47), and (48), we can obtain the position vector \(\bar{\varvec{r}}=[ {\bar{{x}},\bar{{y}},0}]\), i.e., the longitude of the off-axis equilibrium point. Then according to Eq. (27), the orbit radius of the off-axis equilibrium point can be calculated by

$$\begin{aligned} \omega _T^2 r=\frac{\mu }{r^{2}}-\frac{3\mu }{2r^{4}}\left[ {\tau _0 -10\tau _2 \left( {\bar{{x}}^{2}-\bar{{y}}^{2}} \right) +5\bar{\varvec{r}}^{T}\varvec{A}( {\varvec{I}/m} )\varvec{A}^{T}\bar{\varvec{r}}-tr( {\varvec{I}/m} )+a} \right] , \end{aligned}$$
(49)

where \(a\bar{\varvec{r}}=4\tau _2 ( {\bar{{x}}\varvec{e}_1 -\bar{{y}}\varvec{e}_2 })-2\varvec{A}( {\varvec{I}/m})\varvec{A}^{T}\bar{\varvec{r}}\).

4.5.2 Calculation examples

We want to give some calculation examples of the off-axis equilibrium points and to investigate the effect of GOACP \({\varvec{\alpha }}_{\textit{OAC}}\).

The parameters of the asteroid are chosen to be

$$\begin{aligned} \mu =5\,\hbox {m}^{3}/\hbox {s}^{2},\quad C_{20} =-0.12,\quad C_{22} =0.01,\quad a_e =250\,\hbox {m},\quad \omega _T =2.9089\times 10^{-4}\,\hbox {s}^{-1}, \end{aligned}$$
(50)

which has a similar size with the target of the OSIRIS-REx mission, the asteroid 101955 Bennu (provisional designation 1999 \(\hbox {RQ}_{36}\)), but has larger values for \(C_{20}\) and \(C_{22}\) than the approximate values of Bennu, which are calculated under the assumption of filling its currently best determined shape model with a homogeneous bulk density. The period of the asteroid’s uniform rotation is 6 h.

The mass distribution of the spacecraft is set to be

$$\begin{aligned} I_{xx}{:}I_{yy}{:}I_{zz} =1.5{:}1{:}1.2, \end{aligned}$$
(51)

which is a common case in space engineering. As for \(\rho \), we choose six different values

$$\begin{aligned} \rho =2\,\hbox {m},\quad \rho =50\,\hbox {m},\quad \rho =100\,\hbox {m},\quad \rho =120\,\hbox {m},\quad \rho =140\;\hbox {m},\quad \rho =160\;\hbox {m}. \end{aligned}$$
(52)

The minimum \(\rho =2\;\hbox {m}\) corresponds to a general asteroid explorer, such as NEAR mission, OSIRIS-REx mission, and Hayabusa mission. The maximum \(\rho =160\;\hbox {m}\) corresponds to a much larger spacecraft or a space station in future asteroid deflection missions, such as a big gravity tractor, or the mother ship in the asteroid resource exploitation. These situations are extreme from the viewpoint of current deep space missions, but perhaps will be common in future asteroid missions.

By using Eqs. (47)–(49), the longitudes and orbital radii of off-axis equilibrium points can be calculated. We then plot the locus of equilibrium points in the equatorial plane of the asteroid with respect to the attitude angle \(\psi \) in six cases of different \(\rho \). The considered range of \(\psi \) is \(0\le \psi \le \pi \), since \(\pi \le \psi \le 2\pi \) is actually the same case with \(0\le \psi \le \pi \) because of the symmetry of the spacecraft inertia tensor. The stability of these off-axis equilibrium points are also calculated by using \(\varvec{D}( {\varvec{r}_e })\) in Eq. (21).

The loci of equilibrium points in the \({\varvec{u}}\)–\({\varvec{v}}\) plane in the six cases with respect to \(0\le \psi \le \pi \) are given in Figs. 2, 3, 4, 5, 6, and 7, respectively. In the loci, the stable equilibrium points are denoted by larger dots than the unstable ones. To distinguish between stable and unstable points better, we also indicate the stability using texts in the figures.

Fig. 2
figure 2

Loci of off-axis equilibrium points with respect to \(\psi \) in the case of \(\rho =2\;\hbox {m}\)

Full size image
Fig. 3
figure 3

Loci of off-axis equilibrium points with respect to \(\psi \) in the case of \(\rho =50\;\hbox {m}\)

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Fig. 4
figure 4

Loci of off-axis equilibrium points with respect to \(\psi \) in the case of \(\rho =100\;\hbox {m}\)

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Fig. 5
figure 5

Loci of off-axis equilibrium points with respect to \(\psi \) in the case of \(\rho =120\;\hbox {m}\)

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Fig. 6
figure 6

Loci of off-axis equilibrium points with respect to \(\psi \) in the case of \(\rho =140\;\hbox {m}\)

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Fig. 7
figure 7

Loci of off-axis equilibrium points with respect to \(\psi \) in the case of \(\rho =160\;\hbox {m}\)

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4.5.3 Discussion

According to Figs. 2, 3, 4, 5, 6, and 7, we can reach some conclusions about the locations and stability of the off-axis equilibrium points:

1. In these figures, the loci of equilibrium points and their stability have several symmetries. They are symmetrical with respect to the equatorial principal axes \({\varvec{u}}\)- and \({\varvec{v}}\)-axes. This is due to the symmetries of mass distributions of the asteroid and spacecraft. Besides, for a given value of \(\psi \), the equilibrium points and their stability are symmetrical with respect to the center of the asteroid, and the two symmetrical equilibrium points have the same parameters with each other. This symmetry has also been shown by Eqs. (47) and (48). Although in the figures the stability is indicated by using texts only for some equilibrium points, the stability of other equilibrium points is easy to know by these symmetries.

2. As \(\rho \) increases, the significance of GOACP \({\varvec{\alpha }}_{\textit{OAC}} \) also increases. Meanwhile, the shift of equilibrium point from the equatorial principal axes becomes larger, and the effect of GOACP on the stability of equilibrium points becomes more significant.

When \(\rho =2\;\hbox {m}\), GOACP \({\varvec{\alpha }}_{\textit{OAC}} \) is weak. The system is close to the previous orbital dynamics without \({\varvec{\alpha }}_{\textit{OAC}} \), and the equilibrium points are near to those in the previous model given by Eqs. (24) and (25), as shown by Fig. 2. The stability is also close to that in previous orbital dynamics determined by the asteroid parameters \(\mu \), \(\omega _T \), \(\tau _0 \) and \(\tau _2 \) as in Eq. (26): By using Eq. (26) we know that the classical equilibrium points on the \({\varvec{v}}\)-axis are stable, and those on the \({\varvec{u}}\)-axis are unstable; We also know that in Fig. 2 all the off-axis equilibrium points near the \({\varvec{v}}\)-axis are stable, whereas all those near the \({\varvec{u}}\)-axis are unstable.

In Figs. 3, 4, and 5, as \(\rho \) increases to \(120\;\hbox {m}\), the significance of GOACP \({\varvec{\alpha }}_{\textit{OAC}}\) increases. Consequently, the loci of equilibrium points expand from the vicinity of the classical equilibrium points without \({\varvec{\alpha }}_{\textit{OAC}}\), and the maximum shift of equilibrium points from the equatorial principal axes increases. Besides the locations, the stability property of equilibrium points also diverges from that in the previous dynamics without \({\varvec{\alpha }}_{\textit{OAC}}\): We can see that in the case of \(\rho =50\;\hbox {m}\) all equilibrium points near the \({\varvec{v}}\)-axis are stable and all the equilibrium points near the \({\varvec{u}}\)-axis are unstable, whereas when \(\rho \) increases to 100–120 m some equilibrium points near the \({\varvec{v}}\)-axis have become unstable, as shown by Figs. 4 and 5.

In Figs. 6 and 7, when \(\rho \) increases to 140–160 m, the loci of equilibrium points expand to two closed curves around the asteroid. That is to say, the equilibrium point can exist at an arbitrary longitude in the equatorial plane with an appropriate attitude angle \(\psi \). The equilibrium points on the outer closed curve are all unstable. On the inner closed curve, the equilibrium points near the \({\varvec{u}}\)-axis are stable and those near the \({\varvec{v}}\)-axis are unstable. Compared with Figs. 2, 3, 4, and 5, we can find that the locations of the stable equilibrium points have moved from near the v-axis to near the \({\varvec{u}}\)-axis.

Therefore, by using GOACP \({\varvec{\alpha }}_{\textit{OAC}}\), the spacecraft can stay at an arbitrary longitude around the asteroid without orbital control at the stable equilibrium points or with small orbital control at the unstable ones. This result is totally different from the previous orbital dynamics without \({\varvec{\alpha }}_{\textit{OAC}}\), and it provides many potential applications for the asteroid proximity-operations. For example, the asteroid body-fixed hovering at an arbitrary longitude can be achieved by using GOACP \({\varvec{\alpha }}_{\textit{OAC}}\).

Also notice that in Figs. 6 and 7 as \(\rho \) increases from 140 to 160 m, GOACP becomes more significant. Consequently, the separation between the two closed curves increases.

3. As shown by Figs. 2, 3, 4, and 5 when \(\rho \) is about and smaller than 120 m, the loci of equilibrium points with respect to \(\psi \) are four symmetrical closed curves around the classical equilibrium points without \({\varvec{\alpha }}_{\textit{OAC}}\). As \(\rho \) increases, the closed curves become larger. For a given value of \(\psi \), there exist four equilibrium points that are located on the four closed curves, respectively, and are symmetrical with respect to the asteroid center.

With the attitude angle \(\psi \) changing from 0 to \(\pi \), the equilibrium point will move along the closed curve for one cycle. When \(\psi \) changes from 0 to \(\pi /2\), the equilibrium point will move off the asteroid principal axis, and after reaching the maximum shift it will move back towards and finally return to the asteroid principal axis when \(\psi =\pi /2\). When \(\psi \) changes further from \(\pi /2\) to \(\pi \), the equilibrium point will move off the equatorial principal axis on the opposite direction.

The locus between \(\psi =\pi /2\) and \(\psi =\pi \) is symmetrical with that between \(\psi =0\) and \(\psi =\pi /2\) with respect to the asteroid principal axis. The equilibrium point will return to the starting point \(\psi =0\) on the asteroid principal axis when \(\psi =\pi \). The two equilibrium points on the asteroid principal axis when \(\psi =0(\pi )\) and \(\psi =\pi /2\), which are actually on-axis equilibrium points, correspond to the cases that the \({\varvec{i}}\)- and \({\varvec{j}}\)-axes of the spacecraft are parallel to the asteroid principal axis, respectively.

In the cases of \(\rho =100\;\hbox {m}\) and \(\rho =120\;\hbox {m}\), as the equilibrium point moves along the closed curve near the \({\varvec{v}}\)-axis, the stability property can be changed.

4. As shown by Figs. 5 and 6, when \(\rho \) increases from 120 to 140 m, the loci of equilibrium points have fundamental changes.

When \(\rho \) increases from 120 to 140 m, the starting point \(\psi =0\) and the ending point \(\psi =\pi \) of the two closed curves crossing the \({\varvec{u}}\)-axis in Fig. 5 “jump” from the \({\varvec{u}}\)- to +\({\varvec{v}}\)-axis and \(-{\varvec{v}}\)-axis in Fig. 6, and then these two closed curves in Fig. 5 become two open curves connecting the asteroid’s +\({\varvec{v}}\)- and \(-{\varvec{v}}\)-axes in Fig. 6. These two open curves form the outer closed curve around the asteroid, which means that the equilibrium point can exist at an arbitrary longitude with an appropriate attitude angle \(\psi \).

At the same time, the starting point \(\psi =0\) and the ending point \(\psi =\pi \) of the two closed curves crossing the \({\varvec{v}}\)-axis in Fig. 5 “jump” to +\({\varvec{u}}\)- and \(-{\varvec{u}}\)-axis in Fig. 6, and then these two closed curves crossing the \({\varvec{v}}\)-axis in Fig. 5 become two open curves connecting the asteroid’s +\({\varvec{u}}\)- and \(-{\varvec{u}}\)-axes in Fig. 6. These two open curves also form a closed curve, the inner one, which also means that the equilibrium point can exist at an arbitrary longitude with an appropriate attitude angle \(\psi \).

We can see that when the locations of equilibrium points have fundamental changes from Fig. 5 to Fig. 6, the stability property has followed the equilibrium points: the locations of the stable equilibrium points have moved from near the v-axis to near the \({\varvec{u}}\)-axis.

4.5.4 Attitude stabilization

As shown above, the asteroid body-fixed hovering at an arbitrary longitude can be achieved at the off-axis equilibrium points by using GOACP \({\varvec{\alpha }}_{\textit{OAC}}\).

However, in the attitude-restricted orbital dynamics, we have assumed that the spacecraft is controlled ideally to a given attitude with respect to the asteroid. At the off-axis equilibrium points, the spacecraft attitude is biased with respect to the nadir direction; therefore, the spacecraft will suffer from a constant gravity gradient torque caused by the asteroid gravity. The angular momentum caused by this constant gravity gradient torque needs to be absorbed consistently by the onboard attitude control system, i.e., the reaction wheels. When the angular momentum of the reaction wheels reaches saturation, fuel will be needed for the unloading. Therefore, it is necessary to assess the effect of the gravity gradient torque to see how much fuel will be needed to unload the angular momentum of the reaction wheels.

The second-order gravity gradient torque acting on the spacecraft expressed in its body-fixed frame \(S_{B}\) can be given by

$$\begin{aligned} \varvec{T}=\frac{3\mu }{{R}^{5}}\varvec{R}\times \varvec{IR}, \end{aligned}$$
(53)

where \(\varvec{R}=\varvec{A}^{T}\varvec{r}\) is the position vector with respect to the asteroid expressed in the spacecraft body-fixed frame \(S_{B}\). Since the relative attitude A with respect to the asteroid is a single axis rotation around the \({\varvec{w}}\)-axis, see Eq. (38), the position vector \(\varvec{R}\) is within the \({\varvec{i}}\)–\({\varvec{j}}\) plane of the spacecraft, and then the gravity gradient torque \(\varvec{T}\) has non-zero component only on the \({\varvec{k}}\)-axis of the spacecraft with its components on the \({\varvec{i}}\)- and \({\varvec{j}}\)-axes both equal to zero.

We have calculated the magnitude of gravity gradient torque per unit mass (kg) of the spacecraft for the off-axis equilibrium points in Figs. 2, 3, 4, 5, 6, and 7. Curves of magnitude of the gravity gradient torque per unit mass (kg) with respect to the attitude angle \(0\le \psi \le \pi \) in the cases of six different values of \(\rho \) are given in Fig. 8. For each value of \(\rho \), there are two curves corresponding to the two groups of off-axis equilibrium points that are not symmetrical with respect to the asteroid center.

Fig. 8
figure 8

Magnitude of gravity gradient torque per unit mass (kg) of spacecraft at off-axis equilibrium points

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We can find that as \(\rho \) increases, the magnitude of gravity gradient torque also increases. In the case of \(\rho =2\;\hbox {m}\) the magnitude of gravity gradient torque is too small to be seen in Fig. 8. In the case of \(\rho =160\;\hbox {m}\) the maximum magnitude of gravity gradient torque per unit mass (kg) of the spacecraft is about \(3\times 10^{-4}\;\hbox {N}\;\hbox {m/kg}\). In worst case, all the absorbed angular momentum by the reaction wheels needs to be unloaded by the thrusters. The average control acceleration by the thrusters to unload the angular momentum can be estimated as about \({3\times 10^{-4}}/\rho =\hbox {1.8750}\times 10^{-6}\;\hbox {m/s}^{2}\), which is practical. In a better case, before the wheels reach saturation, the angular momentum can be stored temporarily by the wheels and can be unloaded by the gravity gradient torque with the opposite direction in later mission operations. Therefore, the asteroid body-fixed hovering at the off-axis equilibrium points by using GOACP \({\varvec{\alpha }}_{\textit{OAC}}\) is practical in space engineering.

5 Conclusions

We have added GOACP into the close-proximity orbital dynamics about asteroids for a large spacecraft to improve the previous orbital models. By assuming that the spacecraft attitude is controlled ideally with respect to the asteroid, we have proposed a new orbital dynamics model called the attitude-restricted orbital dynamics.

Two kinds of equilibrium points within the asteroid equatorial plane are discovered: on- and off-axis. On-axis equilibrium points are located on the asteroid equatorial principal axes, and require one of the principal axes of the spacecraft to be parallel to the equatorial principal axis. The existence of off-axis equilibrium points, which cannot exist in previous orbital models, is attributed to GOACP. Off-axis equilibrium points within the equatorial plane require that one of the spacecraft principal planes is parallel to the equatorial plane. Locations of the off-axis equilibrium points depends on the relative attitude of spacecraft, and the loci of equilibrium points with respect to the attitude vary with the spacecraft characteristic dimension. In the case of an enough large characteristic dimension, the off-axis equilibrium points can exist at an arbitrary longitude in the equatorial plane with an appropriate attitude.

Our results have shown that, because of GOACP, the phase space of attitude-restricted orbital dynamics is much more complicated than that of the previous close-proximity orbital dynamics. The equilibrium points can give a basic picture of the phase space. More importantly, the spacecraft can stay at stable equilibrium points without orbital control and stay at unstable equilibrium points with small control without needing to null out the unbalanced gravity. Therefore, the equilibrium points are useful for the asteroid close-proximity operations. The asteroid body-fixed hovering at an arbitrary longitude can be achieved at the off-axis equilibrium points by using GOACP without or with small orbital control.

The location and attitude of the spacecraft at equilibrium points depends on its mass distribution. In the future, it is of great interest to study how to modify the mass distribution (shape) of the spacecraft, such as by using deployable appendices, to affect its position and attitude at the equilibrium points.