Gravity-gradient stabilization of gyrostat satellites with rotor axes in principal planes (original) (raw)

Attitude stability of artificial satellites subject to gravity gradient torque

Celestial Mechanics and Dynamical Astronomy, 2009

The stability of the rotational motion of artificial satellites is analyzed considering perturbations due to the gravity gradient torque, using a canonical formulation, and Andoyer's variables to describe the rotational motion. The stability criteria employed requires the reduction of the Hamiltonian to a normal form around the stable equilibrium points. These points are determined through a numerical study of the Hamilton's equations of motion and linear study of their stability. Subsequently a canonical linear transformation is used to diagonalize the matrix associated to the linear part of the system resulting in a normalized quadratic Hamiltonian. A semi-analytic process of normalization based on Lie-Hori algorithm is applied to obtain the Hamiltonian normalized up to the fourth order. Lyapunov stability of the equilibrium point is performed using Kovalev and Savchenko's theorem. This semi-analytical approach was applied considering some data sets of hypothetical satellites, and only a few cases of stable motion were observed. This work can directly be useful for the satellite maintenance under the attitude stability requirements scenario.

A Study for Semi-Passive Gravity Gradient Stabilization of Small Satellites

1987

This paper gives the results of a dynamical analysis of the Globesat gravity gradient stabilized satellite in a 500 km circular orbit. The linearized equations of motion are developed and the stability of the satellite is investigated. The satellite is equipped with magnetic torquers for the purpose of providing attitude correction torques. These correction torques can be used to effect large changes in orientation and for producing small impulses for damping residual librational motions. The analysis shows that such a satellite can be captured into a gravity gradient stabilized mode, and that residual motions can be damped to small steady state values.

Bifurcation and Chaos in an Apparent-Type Gyrostat Satellite

Nonlinear Dynamics, 2005

Attitude dynamics of an asymmetrical apparent gyrostat satellite has been considered. Hamiltonian approach and Routhian are used to prove that the dynamics of the system consists of two separate parts, an integrable and a non-integrable. The integrable part shows torque free motion of gyrostat, while the non-integrable part shows the effect of rotation about the earth and asphericity of the satellite's inertia ellipsoid. Using these results, theoretically when the non-integrable part is eliminated, we are able to design a satellite with exactly regular motion. But from the engineering point of view the remaining errors of manufacturing process of the mechanical parts cause that the non-integrable part can not be eliminated, completely. So this case can not be achieved practically. Using Serret-Andoyer canonical variable the Hamiltonian transformed to a more appropriate form. In this new form the effect of the gravity, asphericity, rotational motion and spin of the rotor are explicitly distinguished. The results lead us to another way of control of chaos. To suppress the chaotic zones in the phase space, higher rotational kinetic energy can be used. Increasing the parameter related to the spin of the rotor causes the system's phase space to pass through a heteroclinic bifurcation process and for the sufficiently large magnitude of the parameter the heteroclinic structure can be eliminated. Local bifurcation of the phase space of the integrable part and global heteroclinic bifurcation of whole system's phase space are presented. The results are examined by the second order Poincaré surface of section method as a qualitative, and the Lyapunov characteristic exponents as a quantitative criterion.

Attitude stability analyses for small artificial satellites

Journal of Physics: Conference Series, 2013

The objective of this paper is to analyze the stability of the rotational motion of a symmetrical spacecraft, in a circular orbit. The equilibrium points and regions of stability are established when components of the gravity gradient torque acting on the spacecraft are included in the equations of rotational motion, which are described by the Andoyer's variables. The nonlinear stability of the equilibrium points of the rotational motion is analysed here by the Kovalev-Savchenko theorem. With the application of the Kovalev-Savchenko theorem, it is possible to verify if they remain stable under the influence of the terms of higher order of the normal Hamiltonian. In this paper, numerical simulations are made for a small hypothetical artificial satellite. Several stable equilibrium points were determined and regions around these points have been established by variations in the orbital inclination and in the spacecraft principal moment of inertia. The present analysis can directly contribute in the maintenance of the spacecraft's attitude.

On the stability of spinning satellites

Acta Astronautica, 2011

We study the directional stability of rigid and deformable spinning satellites in terms of two attitude angles. The linearized attitude motion of a free system about an assumed uniform-spin reference solution leads to a generic MGK system when the satellite is rigid or deformable. In terms of Lyapunov's stability theory, we investigate the stability with respect to a subset of the variables. For a rigid body, the MGK system is 6-dimensional, i.e., 3 rotational and 3 translational variables. When flexible parts are present the system can have any arbitrary dimension. The 2 Â 2 McIntyre-Myiagi stability matrix gives sufficient conditions for the attitude stability. A further development of this method has led to the Equivalent Rigid Body method. We propose an alternative practical method to establish sufficiency conditions for directional stability by using the Frobenius-Schur reduction formula. As practical applications we discuss a spinning satellite augmented with a spring-mass system and a rigid body appended with two cables and tip masses. In practice, the attitude stability must also be investigated when the spinning satellite is subject to a constant axial thrust. The generic format becomes MGKN as the thrust is a follower force. For a perfectly aligned thrust along the spin axis, Lyapunov's indirect method remains valid also when deformable parts are present. We illustrate this case with an apogee motor burn in the presence of slag. When the thrust is not on the spin axis or not pointing parallel to the spin axis, the uniform-spin reference motion does not exist and none of the previous methods is applicable. In this case, the linearization may be performed about the initial state. Even when the linearized system has bounded solutions, the non-linear system can be unstable in general. We illustrate this situation by an instability that actually happened in-flight during a station-keeping maneuver of ESA's GEOS-I satellite in 1979.

Passive magnetic stabilization of the rotational motion of the satellite in its inclined orbit

Applied Mathematical Sciences, 2015

The problem of perturbed rotational motion of the satellite is one of the most interesting, important and, at the same time, mathematically complex problems of celestial mechanics and space flight dynamics. Among existing stabilization systems, a passive magnetic stabilization systems have a special place, since they have an exceptional reliability and are easy to manufacture. In this paper the problem of passive magnetic stabilization of the rotational motion of the satellite is studied. It is assumed that passive magnetic system provides its orientation along the vector of the geomagnetic field strength H. The geomagnetic field is simulated by the direct dipole model, considering different orbits of inclination. In the considered model an effect of the gravitational torque is taken into account. Results of computational experiments are presented.

Study of Stability of Rotational Motion of Spacecraft with Canonical Variables

Mathematical Problems in Engineering, 2012

This work aims to analyze the stability of the rotational motion of artificial satellites in circular orbit with the influence of gravity gradient torque, using the Andoyer variables. The used method in this paper to analyze stability is the Kovalev-Savchenko theorem. This method requires the reduction of the Hamiltonian in its normal form up to fourth order by means of canonical transformations around equilibrium points. The coefficients of the normal Hamiltonian are indispensable in the study of nonlinear stability of its equilibrium points according to the three established conditions in the theorem. Some physical and orbital data of real satellites were used in the numerical simulations. In comparison with previous work, the results show a greater number of equilibrium points and an optimization in the algorithm to determine the normal form and stability analysis. The results of this paper can directly contribute in maintaining the attitude of artificial satellites.

Stability of the permanent rotations of an asymmetric gyrostat in a uniform Newtonian field

Applied Mathematics and Computation

The stability of the permanent rotations of a heavy gyrostat is analyzed by means of the Energy-Casimir method. Su cient and necessary conditions are established for some of the permanent rotations. The geometry of the gyrostat and the value of the gyrostatic moment are relevant in order to get stable permanent rotations. Moreover, the necessary conditions are also su cient, for some configurations of the gyrostat.