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- Periodic Attitudes of Libration Point Spacecrafts in the Earth-Moon System
- On the perturbed restricted three-body problem
- Lagrange point
- Methods in Astrodynamics and Celestial Mechanics, Volume 17

Professor, Universitat de Barcelona. Astrodynamics Celestial Mechanics. Celestial Mechanics and Dynamical Astronomy 56 4 , ,

## Periodic Attitudes of Libration Point Spacecrafts in the Earth-Moon System

The attitude motion of a rigid spacecraft is studied in the Earth-Moon circular restricted three-body problem. Firstly, the equilibrium attitude and its stability as functions of the moments of inertia are discussed when the spacecraft is assumed at the libration points.

Then, periodic attitudes of a spacecraft with mass distribution given in the stable regions are obtained. Regarding space mission applications, the Sun orientation is discussed, and the orbit-attitude resonances are constructed for spacecrafts working on the libration point orbits by means of a continuation procedure.

In recent decades, the dynamics about the libration points in the circular restricted three-body problem CRTBP has been studied extensively [ 1 — 17 ], and so far, several libration point missions have been conducted [ 18 — 24 ]. As we know, both orbital and attitude motions of a spacecraft are required to be considered in practical missions. However, the attitude motion near the libration point has attracted few attentions. In the environment of CRTBP, understanding the natural attitude motion may offer options that lessen the control effort for the attitude control system.

In the proper circumstances, the natural dynamical revolution may assist, or even replace, the attitude control action [ 25 ]. Thus, to design the spacecraft control system better, an understanding of the coupling orbital and attitude motion is of significance.

Kane and Marsh [ 26 ] considered the attitude stability of an axial symmetric satellite which is located at the libration points. The satellite is spinning about its axis of symmetry that is normal to the primary bodies' orbital plane. Later, Robinson [ 27 ] studied the attitude stability of a dumbbell satellite located at the equilateral points.

The same author [ 28 ] also considered the attitude stability of an arbitrary-shaped rigid satellite which is located at a Lagrangian point, and the author constructed a linear stability diagram about the stable attitude of the satellite. Abad et al. Brucker and Gurfil [ 30 ] identified the dynamics and stability of a rigid body spacecraft at the collinear Lagrange points using Poincare maps.

It is more probable that a spacecraft is not exactly placed at a Lagrangian point but located on a periodic orbit. Recently, several authors have investigated the attitude motion of a spacecraft on planar or three-dimensional periodic orbits. Using the Euler angle pitch, roll, and yaw, Wong et al.

The stability corresponding to different mass distributions is shown at Sun-Earth Lagrangian L2 and L4, and the effects of orbital motion on its attitude dynamics in the vicinity of a collinear Lagrangian point are investigated.

Knutson et al. Meng et al. Bucci et al. Colagrossi and Lavagna [ 36 ] investigated the coupling between orbit-attitude dynamics and the flexibility of the structure in a slightly different way than the one presented in this paper. More recently, to construct a basis for understanding attitude motion within a multibody problem with application to spacecraft flight dynamics, a review about attitude dynamics in the CRTBP was made by Guzzetti and Howell [ 25 ].

Recently, using numerical techniques, Lei et al. Using the same method, in this present work, we aim to construct the attitude motions of a rigid spacecraft moving at and around the libration points in the CRTBP. Euler angles in terms of the rotation sequence that describes the relationship between the body-fixed frame with respect to the orbital frame are taken to define the attitude of a rigid spacecraft. The equilibrium attitudes and the periodic attitude motion around them are obtained for rigid spacecrafts working at the Earth-Moon libration points.

Based on these solutions, periodic attitude motions along the libration point orbit are discussed. Periodic motions of the orbit and attitude can be a novel contribution to the exploration of the CRTBP, and it could serve as a stepping stone for more dynamically complex environment. Considering that the solar-sail spacecraft has been studied extensively [ 38 — 41 ], the Sun orientation is also investigated for the special type of spacecraft effectively using radiation pressure on a reflective sail for propulsion.

This paper is organized as follows. In Section 3 , the equilibrium attitude and its stability corresponding to different shaped rigid bodies are constructed when the spacecraft is located at a given libration point. And, the periodic attitude motion around the stable equilibrium attitude is obtained. Then, the Sun orientation and the orbit-attitude resonances of a spacecraft along a Lyapunov orbit around libration point L4 are shown in Section 4.

Section 5 concludes this work. In this paper, we consider a spacecraft moving in the CRTBP, which describes the motion of a spacecraft with infinitesimal mass under the gravitational attraction of two massive bodies with mass and with mass , moving in circular orbits around their barycentre. Without loss of generality, we assume that , and in this case, we call as the massive primary and as the secondary.

Usually, the motion of is described in a barycentric rotating coordinate system see Figure 1 , in which the origin coincides with the mass centre of the primaries, the - axis is directed from towards , the - axis is parallel to the angular momentum vector of the primaries, and the - axis is determined by the right-handed coordinate frame.

For consideration of computation accuracy, the variables are normalized by taking the distance between the primaries, their total mass, and orbital period divided by as the units of length, mass, and time, respectively. Under this normalized system, the mass of the secondary is , and the mass of the primary is. In the barycentric coordinate frame, the primary and secondary bodies are located at and , respectively. In this paper, we only consider the planar orbital motion of the spacecraft.

The differential equations, governing the motion of the spacecraft, can be written as [ 42 ] where is the effective potential of the CRTBP. The distance of the spacecraft from the primaries is and given by. It is known that 1 possesses a conserved quantity, known as Jacobi integral, and five Lagrangian libration points exist in the model specified by 1. Three of them lying on the - axis are denoted as collinear libration points L1, L2, and L3.

The remaining points denoted by L4 and L5 form equilateral triangles with the primary bodies. Now, we reconsider that the spacecraft is not a point mass but a finite-size rigid body moving nearby a libration point in the CRTBP.

It means that the infinitesimal point mass is replaced by a finite-size one. The orbital motion of the spacecraft can be described in the reference frame refer to Figure 1. The attitude motion is described by the orientation difference between the orbital reference frame and the body-fixed reference frame see Figure 1 which is aligned with the principal axes of the body.

The relationship between the body-fixed frame with respect to the orbital frame is measured in the 3—2—1 pitch-roll-yaw rotation sequence with the Euler angles. Since that the dimensions of the spacecraft are too small compared with its distance to each of the primaries, the attitude has negligible influence upon the orbital motion so that we can assume that the orbital motions are decoupled with the attitude motions.

For the spacecraft moving nearby a libration point, the gravitational gradient torques exerted by the primary bodies are the dominant torques acting on a spacecraft. The moments of inertia of the spacecraft about the body principal axes , , and are denoted by , , and , respectively. In 5 , , , and or are the components of the unit vector along which is the distance from the - th primary body to the centre mass of the spacecraft expressed in body-fixed frame.

Considering the angular velocity of a spacecraft in the inertial frame, there are six state variables to describe its attitude motion in this paper. With the same nondimensional unit as the orbital motion, the kinematic differential equations are described as [ 31 ] where are the inertia ratios. The relationship of , , and can be transformed by [ 31 , 37 ]. Considering the Earth-Moon system as a case study in the part, the attitude motions of spacecrafts at L2 and L4 are studied.

Considering that a spacecraft is located at libration point L2, it is known that the equilibrium attitude has zero Euler angles, i. By denoting the state relative to the equilibrium attitude as , the Euler angles of the spacecraft can be expressed by , , and. Around the equilibrium attitude, 6 can be linearized as follows [ 31 ]; where. By analysing the characteristic equations of linear equation 10 , the stability conditions can be written as [ 25 , 28 ] where , and.

Then, the stability diagram on the plane for spacecrafts located at L2 is shown in Figure 2 a , and coloured regions I and II represent that the equilibrium attitude is stable with the corresponding mass distribution. The stable domains are distributed in the first and third quadrants, similar to the conclusions given by Guzzetti and Howell, Robinson, and Wong et al.

The stability domain in the first quadrant is a triangular region which represents , and in the third quadrant, the stability domain represents. In this part, we take a spacecraft with mass distribution in region I as an example. The linear solution of attitude motion around the equilibrium attitude can be written as [ 31 , 37 ] where and are amplitudes in the yaw and roll directions, is the amplitude of the pitch motion, and , , and are the corresponding initial phase angles.

In 13 , , , and are the frequencies of attitude motion, which are determined by and the constant coefficients are determined by. In this case, there are three types of basic periodic attitudes emanating from the equilibrium attitude.

Figure 3 represents the configurations of these types of periodic attitudes with deg, deg, and deg, respectively. Based on these linear solutions, we construct the corresponding periodic attitudes see Figure 4 in the CRTBP model by means of numerical techniques.

The state of attitude is denoted by , and the corresponding period is T. If only an initial state of attitude is known, we can obtain the periodic attitude motion by integrating 6 over one period. To generate a family of periodic solution, the well-known continuation scheme called pseudo-arclength continuation [ 37 , 43 ] is used in this investigation.

Based on these linear solutions in Figure 4 , the corresponding attitude families are obtained shown in the Euler angle space see the first family in Figure 5 c. As it is shown in Figure 6 , three families of periodic attitudes have been constructed. The attitude periods of the families are extended in the range of [ Meanwhile, we obtained the stability of these periodic attitude motions. The stability of the periodic attitude can be reflected by its monodromy matrix, which corresponds to the state transition matrix over one period [ 37 , 44 ].

Three reciprocal pairs of eigenvalues [ 45 ] exist in the periodic attitude motion which is an autonomous and Hamiltonian system. Among the three pairs of eigenvalues, there are two pairs of eigenvalues which are denoted by and with and , and the other pair of eigenvalues is unitary, i.

Similar to the stability index defined for the periodic orbits in the CRTBP [ 46 — 48 ], two stability indices in attitude motion are defined as and. If both conditions and are satisfied, the periodic attitude is stable; otherwise, it is unstable.

The tangent bifurcations occur when or , and the period-doubling bifurcations take place when or. For the first attitude family, two pairs of eigenvalues and are shown in Figure 5 a , and two stability indices and are shown in Figure 5 b.

From Figure 5 , we can conclude that the periodic attitudes corresponding to the first attitude family are always stable. The Euler angles along the evolution of the first attitude family are shown in Figure 5 c. Similarly, we obtain the equilibrium attitude of the rigid spacecraft fixed at libration point L4 with. And the Euler angles of the spacecraft can be expressed by , , and by denoting the state relative to the equilibrium attitude as. Then, 6 could be linearized about the equilibrium attitude at L4 as follows [ 31 ]: where with.

By analysing the characteristic equations of 16 , the stability conditions can be written as [ 25 , 28 ] where and. Then, the stability diagram on the plane for spacecrafts located at L4 is shown in Figure 2 b , and coloured regions III and IV represent that the equilibrium attitude is stable with the corresponding mass distribution.

In this part, we take a spacecraft of the mass distribution in region III with and as an example. The linear solution of attitude motion around the equilibrium attitude can also be written in a similar form to 13 , where , , and are the amplitudes and , , and are the initial phase angles. Different with the case of L2, the frequencies , , and here are determined by and the constant coefficients are determined by.

## On the perturbed restricted three-body problem

Scientific Research An Academic Publisher. Three-Body Problem formulated by Newton provided route to the analysis of closed form analytical solution. This solution remains elusive even today, as one has never been found for the three-body problem. Euler developed the restricted problem using a rotating frame in the s and located collinear points. Along with Euler, Lagrange considered this form of the three-body problem and calculated the locations of equilateral points, often known as libration or Lagrange points.

Hou, L. Motions around the collinear libration points in the elliptic restricted three-body problem are studied. Literal expansions of the Lissajous orbits and the halo orbits are obtained. These expansions depend on two amplitude parameters and the orbital eccentricity of the two primaries. Numerical simulations are done to check the validity of these literal series and to compare them with the results in the circular restricted three-body problem. According to the properties of these literal expansions, three kinds of symmetric periodic orbits around the collinear libration points are discussed. First, the linear model is discussed.

## Lagrange point

Methods in Astrodynamics and Celestial Mechanics is a collection of technical papers presented at the Astrodynamics Specialist Conference held in Monterey, California, on September , , under the auspices of the American Institute of Aeronautics and Astronautics and Institute of Navigation. The conference provided a forum for tackling some of the most interesting applications of the methods of celestial mechanics to problems of space engineering. Comprised of 19 chapters, this volume first treats the promising area of motion around equilibrium configurations. Following a discussion on limiting orbits at the equilateral centers of libration, the reader is introduced to the asymptotic expansion technique and its application to trajectories.

Normally, the two objects exert an unbalanced gravitational force at a point, altering the orbit of whatever is at that point. At the Lagrange points, the gravitational forces of the two large bodies and the centrifugal force balance each other. This can make Lagrange points an excellent location for satellites, as few orbit corrections are needed to maintain the desired orbit.

*Orbital maneuvers between the Lagrangian points and the primaries in the Earth-Sun system. Almeida Prado.*

### Methods in Astrodynamics and Celestial Mechanics, Volume 17

Hou, L. Motions around the collinear libration points in the elliptic restricted three-body problem are studied. Literal expansions of the Lissajous orbits and the halo orbits are obtained. These expansions depend on two amplitude parameters and the orbital eccentricity of the two primaries. Numerical simulations are done to check the validity of these literal series and to compare them with the results in the circular restricted three-body problem.

The three-body problem has a special relevance, particularly in astrophysics and astrodynamics. In general, the three-body problem is classified into two types:. Thus, the general problem has some applications in celestial mechanics such as the dynamics of triple star systems and only a very few in space dynamics and solar system dynamics , whereas the restricted problem plays an important role in studying the motion of artificial satellites. It can be used also to evaluate the motion of the planets, minor planets and comets. The restricted problem gives an accurate description not only regarding the motion of the Moon but also with respect to the motion of other natural satellites. Furthermore, the restricted problem has many applications not only in celestial mechanics but also in physics, mathematics and quantum mechanics, to name a few.

Description The contents of this book represent the latest and some of the most interesting applications of the methods of celestial mechanics to problems of space engineering. This area of research involves advanced dynamical and astronomical theories and the application of such theories to the selection of new trajectories, the analysis of proposals for future new space experiments, and the setting of design parameters for proposed new spacecraft of the future. This book should prove useful for the solution of immediate engineering problems as well as for future concepts. Skip to main content. Description Description The contents of this book represent the latest and some of the most interesting applications of the methods of celestial mechanics to problems of space engineering.

#### 1st Edition

The attitude motion of a rigid spacecraft is studied in the Earth-Moon circular restricted three-body problem. Firstly, the equilibrium attitude and its stability as functions of the moments of inertia are discussed when the spacecraft is assumed at the libration points. Then, periodic attitudes of a spacecraft with mass distribution given in the stable regions are obtained. Regarding space mission applications, the Sun orientation is discussed, and the orbit-attitude resonances are constructed for spacecrafts working on the libration point orbits by means of a continuation procedure. In recent decades, the dynamics about the libration points in the circular restricted three-body problem CRTBP has been studied extensively [ 1 — 17 ], and so far, several libration point missions have been conducted [ 18 — 24 ]. As we know, both orbital and attitude motions of a spacecraft are required to be considered in practical missions. However, the attitude motion near the libration point has attracted few attentions.

Last update: 25 July Record number: Veuillez activer JavaScript. Por favor, active JavaScript. Bitte aktivieren Sie JavaScript. Si prega di abilitare JavaScript. The Astrodynamics Network. Fourteen partners composed the network, eight of them university departments and research institutes, and the remaining six aerospace companies and space agencies.

Hou, L.