Gravitational Orbit Attitude Coupling Research Articles

Out-of-plane equilibrium points and invariant manifolds about an asteroid with gravitational orbit—attitude coupling perturbation

Out-of-plane equilibrium points and invariant manifolds about an asteroid with gravitational orbit—attitude coupling perturbation

Non-equatorial equilibrium points around an asteroid with gravitational orbit-attitude coupling perturbation

A recently proposed orbital dynamics model in the close proximity of an asteroid, which is called “attitude-restricted orbital dynamics”, includes the perturbation caused by the spacecraft’s gravitational orbit–attitude coupling. This orbital model improves the precision of classical point-mass orbital model with only the non-spherical gravity. Equatorial equilibrium points have been investigated in the previous paper. In this paper, the in-plane non-equatorial equilibrium points, which are outside the asteroid’s equatorial plane but within its longitudinal principal plane, are further studied for a uniformly-rotating asteroid. These non-equatorial equilibrium points are more diverse than those in the classical point-mass orbital dynamics without gravitational orbit–attitude coupling perturbation (GOACP). Two families of them have been found. The equatorial equilibrium points studied before and the non-equatorial ones studied here give a complete map of equilibrium points in the asteroid’s principal planes. Compared with the classical point-mass orbital dynamics without GOACP, the equatorial equilibrium points have extended the longitude range of equilibrium points around an asteroid, while the non-equatorial ones studied here will extend the latitude range. These equatorial and non-equatorial equilibrium points provide natural hovering positions for the asteroid close-proximity operations.

Trajectory Optimization for Asteroid Landing Considering Gravitational Orbit-Attitude Coupling

This paper proposes a trajectory optimization method considering gravitational orbit-attitude coupling (GOAC) for asteroid landing. First, by modeling the spacecraft as a rigid body rather than a point mass, and using the polyhedral method to describe the irregular gravitational field of the asteroid, GOAC is fully embedded into the 6 degree-of-freedom (DOF) dynamic model of the spacecraft. Second, attitude control, which is assumed to consume electric energy, is introduced into the optimization, the indirect method is adopted to formulate the 6-DOF fuel-optimal control problem, and the boundary conditions are modified to expand the search space of the problem. Third, a two-phase homotopic approach is proposed to realize a smooth continuation from the 6-DOF energy-optimal problem to the fuel-optimal problem. Finally, a numerical example is presented to demonstrate the feasibility of the proposed method, and the comparison results show that the obtained 6-DOF optimal trajectory can guarantee the fuel-optimality of the landing mission under GOAC.

Trajectory Tracking Near Small Bodies Using Only Attitude Control

Gravitational orbit–attitude coupling shows a noticeable influence on the motion of a rigid spacecraft in close proximity to small solar system bodies. The gravity gradient moment changes the attitude of the spacecraft, which in turn alters the total gravity force on the spacecraft and has an impact on the orbital motion. In this paper, the polyhedron model of small bodies is adopted and a distributed point-mass model of the spacecraft is developed to reflect this coupling phenomenon more accurately. In particular, taking advantage of the gravitational orbit–attitude coupling, a trajectory tracking scheme for a rigid body using only attitude control is proposed. By changing the attitude of the spacecraft, the gravitational orbit–attitude coupling generates a control force to make the spacecraft track the reference trajectory. This reference trajectory is generated based on a point-mass model. Simulation results in landing and orbiting around asteroid Bennu show the feasibility of this tracking approach. A significant improvement is seen for the coupling-based controlled trajectory over natural motions. The tracking scheme reduces the frequency of orbit control maneuvers and provides a redundant control in the event of thruster failures. This approach can be used as a reference for future small-body missions.

Relative equilibria and stability of a dumbbell spacecraft about an asteroid

The gravitational orbit-attitude coupling becomes pronounced in spacecraft dynamics about small asteroids, due to the large ratio of the spacecraft's dimension to the orbital radius. In this study, relative equilibria and their stability of the gravitationally coupled orbit-attitude dynamics (full dynamics) of a dumbbell spacecraft about an irregular-shaped asteroid are investigated. The shape of the asteroid is represented by a homogeneous polyhedron, and its non-spherical gravity is calculated accordingly. By using geometric mechanics, the Hamiltonian structure and equations of motion of the full dynamics are derived. Then, the orbit-attitude equilibria about asteroids 4769 Castalia, 433 Eros, and 6489 Golevka are calculated numerically, respectively, and the drift of orbit-attitude equilibria with respect to the dumbbell's varying length is also discussed. By comparing with the equilibria of a point mass, it has been found that a classical point-mass equilibrium can generate three orbit-attitude equilibria located nearby. With the increase of the dumbbell's length, the effects of orbit-attitude coupling become stronger, and the orbit-attitude equilibria diverge further from the classical point-mass equilibria. The spectral stability of the orbit-attitude equilibria is determined numerically and analyzed. It has been found that the gravitational orbit-attitude coupling is likely to make the equilibria unstable in our simulations. The dumbbell is a good approximation of a spacecraft consisting of two modules that are connected by trusses, and its orbit-attitude equilibria have potential applications in asteroid proximity operations, such as body-fixed hovering.

Hamiltonian structure and relative equilibria of an axisymmetric rigid body in a second degree and order gravity field: cylindrical and generalized hyperbolic equilibria

The full dynamics of an axisymmetric rigid body in a uniformly rotating second degree and order gravity field are investigated, where orbit and attitude motions of the body are coupled through the gravity. Compared with the classical orbital dynamics with the body considered as a point mass, the full dynamics is a higher-precision model in close proximity of the central body where the gravitational orbit–attitude coupling is significant, such as a spacecraft about a small asteroid or an irregular-shaped natural satellite about a planet. The full dynamics are modeled by using the non-canonical Hamiltonian structure, in terms of variables expressed in the frame fixed with the central body. A Poisson reduction is carried out by means of the axial symmetry of the body, and a reduced system with lower dimension, as well as its non-canonical Hamiltonian structure and equations of motion, is obtained through the reduction process. With the second-order potential, three types of relative equilibria are found to be possible: cylindrical equilibria, generalized hyperbolic equilibria, and conic equilibria, which are counterparts to cylindrical equilibria, hyperbolic equilibria, and conic equilibria of an axisymmetric rigid body in a spherical gravity field, respectively. The geometrical properties and existence of the cylindrical equilibria and generalized hyperbolic equilibria are investigated in detail. It has been found that compared with the classical results in a spherical gravity field, the relative equilibria in this study are more complicated and diverse. The most significant difference is that the non-spherical gravity field enables the existence of non-Lagrangian hyperbolic equilibria, called generalized hyperbolic, which cannot exist in a spherical gravity.

Orbital perturbation due to orbit-attitude coupling near asteroids

PurposeThe purpose of this paper is to assess the orbital perturbation caused by the gravitational orbit–attitude coupling of spacecraft in the proximity of asteroids.Design/methodology/approachThe gravitational orbit–attitude coupling perturbation (GOACP), which has been neglected before in the close-proximity orbital dynamics about asteroids, is investigated and compared with other orbital perturbations. The GOACP has its origin in the fact that the gravity acting on a non-spherical extended body is actually different from that acting on a point mass located at the body’s center of mass, which is the approximated model in the orbital dynamics. Besides, a case study of a tethered satellite system is given by numerical simulations.FindingsIt is found that the ratio of GOACP to the asteroid’s non-spherical gravity is the order of ρ/ae, where ρ is the spacecraft’s characteristic dimension and ae is the asteroid’s mean radius. It can also be seen that as ρ increases, GOACP will also increase but the solar radiation pressure (SRP) will decrease due to the decreasing area-to-mass ratio. The GOACP will be more significant than SRP at small orbital radii for a large-sized spacecraft. Based on the results by analyses and simulations, it can be concluded that GOACP needs to be considered in the orbital dynamics for a large-sized spacecraft in the proximity of a small asteroid.Practical implicationsThis study is of great importance for the future asteroids missions for scientific explorations and near-Earth objects mitigation.Originality/valueThe GOACP, which has been neglected before, is revealed and studied.

Gravitational orbit–attitude coupling dynamics of a large solar power satellite

The gravitational coupling of the orbit and attitude motions of a large flexible solar power satellite (SPS) is studied in this paper. The orbital and attitude dynamical equations of a SPS with flexible vibration are firstly derived. Then, the gravitational force and torque are expanded to the fourth order of a Taylor series in the small size/orbital ratio. The gravitational forces and torques generated by the Sun, the Moon and the oblateness of the Earth are also investigated. The simulation examples are finally presented, and the results have demonstrated that the higher order gravitational force and torque of the Earth have great influences on the orbital motion and ground pointing precision of the SPS.

Stability of Nonclassical Relative Equilibria of a Rigid Body in a J2 Gravity Field

AbstractThe gravitationally coupled orbit–attitude dynamics of a rigid body in a J2 gravity field is a generalization of the traditional point-mass J2 problem to take into account the gravitational orbit–attitude coupling of the considered body. Linear and nonlinear stability of nonclassical relative equilibria in the coupled orbit–attitude dynamics are studied with geometric mechanics in the present paper. Conditions of stability are obtained through the linear system matrix and projected Hessian matrix by using the energy–Casimir method. Linear and nonlinear stability regions are plotted in a wide range of system parameters. It is found that the stability regions are similar to those of classical relative equilibria while, at the same time, some differences do exist. For example, in some cases, the linear stability region contains not only the two regions that are analogues of the Lagrange region and DeBra–Delp region, but also a small irregular region in the third quadrant. Same as the case of classica...

Orbital dynamics and equilibrium points around an asteroid with gravitational orbit–attitude coupling perturbation

The strongly perturbed dynamical environment near asteroids has been a great challenge for the mission design. Besides the non-spherical gravity, solar radiation pressure, and solar tide, the orbital motion actually suffers from another perturbation caused by the gravitational orbit–attitude coupling of the spacecraft. This gravitational orbit–attitude coupling perturbation (GOACP) 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 approximation of the spacecraft in the orbital dynamics. We intend to take into account GOACP besides the non-spherical gravity to improve the previous close-proximity orbital dynamics. GOACP depends on the spacecraft attitude, which is assumed to be controlled ideally with respect to the asteroid in this study. 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. In the present paper, equilibrium points of the attitude-restricted orbital dynamics in the second degree and order gravity field of a uniformly rotating asteroid are investigated. Two kinds of equilibria are obtained: on and off the asteroid equatorial principal axis. These equilibria are different from and more diverse than those in the classical orbital dynamics without GOACP. In the case of a large spacecraft, the off-axis equilibrium points can exist at an arbitrary longitude in the equatorial plane. These results are useful for close-proximity operations, such as the asteroid body-fixed hovering.

Body-fixed orbit-attitude hovering control over an asteroid using non-canonical Hamiltonian structure

The orbit-attitude hovering means that both the position and attitude of the spacecraft are kept to be stationary in the asteroid body-fixed frame. The orbit-attitude hovering is discussed in the framework of the gravitationally coupled orbit-attitude dynamics, also called the full dynamics, in which the spacecraft is modeled as a rigid body to take into account the gravitational orbit-attitude coupling naturally. A feedback hovering control law is proposed by using the non-canonical Hamiltonian structure of the problem, which is consisted of two potential shapings and one energy dissipation. The first potential shaping is to create an artificial equilibrium at the desired hovering position-attitude. Then, the second potential shaping modifies the potential further so that the artificial equilibrium is a minimum of the modified Hamiltonian on the invariant manifold. Finally, the energy dissipation leads the motion to converge asymptotically to the minimum of the modified Hamiltonian, i.e., the artificial equilibrium for hovering. The feasibility of the hovering control law is verified through numerical simulations. The proposed hovering control law has a simple form and can be implemented by the spacecraft autonomously with little computation. This feature can be attributed to the utilization of the Hamiltonian structure and natural dynamical behaviors of the system in the control law design.

Stabilizing the coupled orbit–attitude dynamics of a rigid body in a J2 gravity field using Hamiltonian structure

The gravitationally coupled orbit–attitude dynamics of a rigid body in a J2 gravity field is a high-precision model for the spacecraft in the close proximity of a small spheroid celestial body, since the gravitational orbit–attitude coupling of the spacecraft is naturally taken into account in this model. A Hamiltonian structure-based feedback control law is proposed to stabilize relative equilibria of the rigid body in the J2 gravity field. The proposed stabilization control law is consisted of three parts: potential shaping, momentum control, and energy control. The potential shaping is used to modify the gravitational potential artificially so that the relative equilibrium is a minimum of the modified Hamiltonian. It is shown that the unstable relative equilibrium can always be stabilized in the Lyapunov sense by the potential shaping with sufficiently large feedback gains. Then the momentum control leads the motion to the same invariant manifold with the relative equilibrium, which is actually a momentum level set. The energy control introduces energy dissipation into the system and the motion will asymptotically converge to the minimum of the modified Hamiltonian on the invariant manifold, i.e., the relative equilibrium. The feasibility of the proposed stabilization control law is validated through numerical simulations. The control law can be implemented by the attitude control system and low-thrust engines onboard and the fuel consumption is reasonable. This Hamiltonian structure-based stabilization approach is applicable to non-canonical Hamiltonian systems commonly existing in the astrodynamics. Since the natural dynamical behaviors of the system is fully utilized, the proposed control law is very simple and is easy to implement autonomously with little computation in the space applications.

Relative equilibria of full dynamics of a rigid body with gravitational orbit-attitude coupling in a uniformly rotating second degree and order gravity field

The motion of a rigid body in a uniformly rotating second degree and order gravity field is a good model for the gravitationally coupled orbit-attitude motion of a spacecraft in the close proximity of an asteroid. The relative equilibria of this full dynamics model are investigated using geometric mechanics from a global point of view. Two types of relative equilibria are found based on the equilibrium conditions: one is the Lagrangian relative equilibria, at which the circular orbit of the rigid body is in the equatorial plane of the central body; the other is the non-Lagrangian relative equilibria, at which the circular orbit is parallel to but not in the equatorial plane of central body. The existences of the Lagrangian and non-Lagrangian relative equilibria are discussed numerically with respect to the parameters of the gravity field and the rigid body. The effect of the gravitational orbit-attitude coupling is especially assessed. The existence region of the Lagrangian relative equilibria is given on the plane of the system parameters. Numerical results suggest that the negative C20 with a small absolute value and a negative C22 with a large absolute value favor the existence of the non-Lagrangian relative equilibria. The effect of the gravitational orbit-attitude coupling of the rigid body on the existence of the non-Lagrangian relative equilibria can be positive or negative, which depends on the harmonics C20 and C22, and the angular velocity of the rotation of the gravity field.

Stability of relative equilibria of the full spacecraft dynamics around an asteroid with orbit–attitude coupling

The full dynamics of spacecraft around an asteroid, in which the spacecraft is considered as a rigid body and the gravitational orbit–attitude coupling is taken into account, is of great value and interest in the precise theories of the motion. The spectral stability of the classical relative equilibria of the full spacecraft dynamics around an asteroid is studied with the method of geometric mechanics. The stability conditions are given explicitly based on the characteristic equation of the linear system matrix. It is found that the linearized system decouples into two entirely independent subsystems, which correspond to the motions within and outside the equatorial plane of the asteroid respectively. The system parameters are divided into three groups that describe the traditional stationary orbit stability, the significance of the orbit–attitude coupling and the mass distribution of the spacecraft respectively. The spectral stability of the relative equilibria is investigated numerically with respect to the three groups of system parameters. The relations between the full spacecraft dynamics and the traditional spacecraft dynamics, as well as the effect of the orbit–attitude coupling, are assessed. We find that when the orbit–attitude coupling is strong, the mass distribution of the spacecraft dominates the stability of the relative equilibria; whereas when the orbit–attitude coupling is weak, both the mass distribution and the traditional stationary orbit stability have significant effects on the stability. We also give a criterion to determine whether the orbit–attitude coupling needs to be considered.

On the nonlinear stability of relative equilibria of the full spacecraft dynamics around an asteroid

The full dynamics of a spacecraft around an asteroid, in which the gravitational orbit–attitude coupling is considered, has been shown to be of great value and interest. Nonlinear stability of the relative equilibria of the full dynamics of a rigid spacecraft around a uniformly rotating asteroid is studied with the method of geometric mechanics. The non-canonical Hamiltonian structure of the problem, i.e., Poisson tensor, Casimir functions and equations of motion, are given in the differential geometric method. A classical kind of relative equilibria of the spacecraft is determined from a global point of view, at which the mass center of the spacecraft is on a stationary orbit, and the attitude is constant with respect to the asteroid. The conditions of nonlinear stability of the relative equilibria are obtained with the energy-Casimir method through the semi-positive definiteness of the projected Hessian matrix of the variational Lagrangian. Finally, example asteroids with a wide range of parameters are considered, and the nonlinear stability criterion is calculated. However, it is found that the nonlinear stability condition cannot be satisfied by spacecraft with any mass distribution parameters. The nonlinear stability condition by us is only the sufficient condition, but not the necessary condition, for the nonlinear stability. It means that the energy-Casimir method cannot provide any information about nonlinear stability of the relative equilibria, and more powerful tools, which are the analogues of the Arnold’s theorem in the canonical Hamiltonian system with two degrees of freedom, are needed for a further investigation.

Gravitational orbit-attitude coupling for very large spacecraft

Motion equations for the gravitationally coupled orbit-attitude motion of a spacecraft are presented. The gravitational force and torque are expanded in a Taylor series in the small ratio (spacecraft size/orbital radius). A recursive definition for higher moments of inertia is introduced which permits terms up tofourth order to be retained. The expressions are fully nonlinear in the attitude variables. A quasi-sunpointing (QSP) passive attitude-control mode is used to assess the effects of higher moments of inertia and gravitational coupling. The attitude motion is detectably coupled to the orbital motion. However, the higher moments of inertia influence only the attitude motion.