Satellite dynamics under the influence of gravitational and aerodynamic torques. A study of stability of equilibrium positions

The dynamics of the rotational motion of a satellite, moving in the central Newtonian force field under the influence of gravitational and aerodynamic torques, is investigated. The paper proposes a method for determining all equilibrium positions (equilibrium orientations) of a satellite in the orbital coordinate system for specified values of aerodynamic torque and the major central moments of inertia; the sufficient conditions for their existence are obtained. For each equilibrium orientation the sufficient stability conditions are obtained using the generalized energy integral as the Lyapunov function. The detailed numerical analysis of the regions where the stability conditions of the equilibrium positions are satisfied is carried out depending on four dimensionless parameters of the problem. It is shown that, in the general case, the number of satellite’s equilibrium positions, for which the sufficient stability conditions are satisfied, varies from 4 to 2 with an increase in the value of the aerodynamic torque magnitude.     

Gravity-oriented satellite dynamics subject to gravitational and active damping torques

The dynamics of the rotational motion of a satellite moving in the central Newtonian field of force over a circular orbit under the effect of gravitational and active damping torques, which depend on the satellite angular velocity projections, has been investigated. The paper proposes a method of determining all equilibrium positions (equilibrium orientations) of a satellite in the orbital coordinate system for specified values of damping coefficients and principal central moments of inertia. The conditions of their existence have been obtained. For a zero equilibrium position where the axes of the satellite-centered coordinate system coincide with the axes of the orbital coordinate system, the necessary and sufficient conditions for asymptotic stability are obtained using the Routh–Hurwitz criterion. A detailed analysis of the regions where the conditions of the asymptotic stability of a zero equilibrium position are fulfilled have been obtained depending on three dimensionless parameters of the problem, and the numerical study of the process of attenuation of satellite’s spatial oscillations for various damping coefficients has been carried out. It has been shown that there is a wide range of damping parameters from which, by choosing the necessary values, one can provide the asymptotic stability of satellite’s zero equilibrium position in the orbital coordinate system.     

Dynamics of an axisymmetric gyrostat satellite under the action of gravitational moment

Dynamics of an axisymmetric gyrostat satellite moving in the central Newtonian force field along a circular orbit is investigated. All equilibrium positions of the gyrostat satellite in the orbital coordinate system are determined, and the sufficient stability conditions of equilibrium positions are derived.     

Dynamics of an axisymmetric satellite under the action of gravitational and aerodynamic torques

Dynamics of attitude motion of an axisymmetric satellite moving in a circular orbit under the action of gravitational and aerodynamic torques is investigated. All equilibrium positions of the satellite in the orbital coordinate system are determined numerically, and sufficient conditions of stability of the equilibrium positions are derived.     

The Dynamics of a Satellite-Gyrostat with a Single Nonzero Component of the Vector of Gyrostatic Moment

The dynamics of a satellite-gyrostat moving in the central Newtonian force field along a circular orbit is studied. In the particular case when the vector of gyrostatic moment is parallel to one of the satellite’s principal central axes of inertia, all the equilibrium states are determined. For each equilibrium orientation, sufficient conditions of stability are obtained as a result of the analysis of the generalized energy integral, and necessary conditions of stability are determined as a result of analysis of the linearized equations of motion. The evolution of regions of validity for the conditions of stability of equilibrium positions are studied in detail depending on the parameters of the problem. All the bifurcation values of the parameters at which qualitative changes of the regions of stability take place are determined.__________Translated from Kosmicheskie Issledovaniya, Vol. 43, No. 4, 2005, pp. 283–294.Original Russian Text Copyright © 2005 by Sarychev, Mirer, Degtyarev.     

Electrodynamic stabilization of earth-orbiting satellites in equatorial orbits

A satellite with electrodynamic stabilization system is considered. Based on the method of Lyapunov functions, sufficient conditions of the asymptotic stability of direct equilibrium position of this satellite in the orbital coordinate system under perturbing action of a gravitational moment are obtained. These conditions allow one to ensure a rational choice of parametric control coefficients depending on parameters of the satellite and its orbit.     

Stability of triangle libration points in generalized restricted circular three-body problem

Equilibrium positions of a small-mass body are considered with respect to a precessing dumbbell. The dumbbell represents two rigidly fixed spherical gravitating bodies. Such a system can be considered as a model of a binary asteroid. Stability of relative equilibrium positions with equal distances from the small mass to the attracting centers is studied. By analogy with the classical restricted three-body problem, these positions are referred to as triangle libration points. It is shown that the character of stability of these libration points is determined by three constant parameters: nutation angle and angular velocity of precession, as well as the ratio of masses at the ends of the dumbbell. Stability conditions are derived in the linear approximation, and the regions of stability and instability in the space of problem parameters are constructed. The paper is a continuation of [1].     

On the stability of spinning satellites

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 2×2 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.     

Determination of attitude motion of the Foton M-2 satellite using the data of onboard measurements of its angular velocity

We present the resutls of a prompt determination of the uncontrolled attitude motion of the Foton M-2 satellite, which was in orbit from May 31 to June 16, 2005. The data of onboard measurements of the angular velocity vector were used for this determination. The measurement sessions were carried out once a day, each lasting 83 min. Upon terminating a session, the data were transmitted to the ground to be processed using the least squares method and integrating the equations of motion of the satellite with respect to its center of mass. As a result of processing, the initial conditions of motion during a session were estimated, as well as parameters of the mathematical model used. The satellite’s actual motion is determined for 12 such sessions. The results obtained in flight completely described the satellite’s motion. This motion, having begun with a small angular velocity, gradually became faster, and in two days became close to the regular Euler precession of an axisymmetric solid body. On June 14, 2005 the angular velocity of the satellite with respect to its longitudinal axis was approximately 1.3 degrees per second, and the angular velocity projection onto a plane perpendicular to this axis had a magnitude of about 0.11 degrees per second. The results obtained are consistent with more precise results obtained later by processing the data on the Earth’s magnetic field measured on the same satellite, and they complement the latter in determination of the motion in the concluding segment of the flight, when no magnetic measurements were performed.     

On plane oscillations of a pendulum with variable length suspended on the surface of a planet’s satellite

The problem of planar oscillations of a pendulum with variable length suspended on the Moon’s surface is considered. It is assumed that the Earth and Moon (or, in the general case, a planet and its satellite, or an asteroid and a spacecraft) revolve around the common center of mass in unperturbed elliptical Keplerian orbits. We discuss how the change in length of a pendulum can be used to compensate its oscillations. We wrote equations of motion, indicated a rule for the change in length of a pendulum, at which it has equilibrium positions relative to the coordinate system rotating together with the Moon and Earth. We study the necessary conditions for the stability of these motions. Chaotic dynamics of the pendulum is studied numerically and analytically.     

Stability and chaos of a damped satellite partially filled with liquid

The stability and chaotic motions of a damped satellite partially filled with liquid which is subjected to external disturbance are investigated in this paper. With linearization analysis, the stability of the two non-trivial equilibrium points is studied. The homoclinic and heteroclinic orbits are found by using the undetermined coefficient method, and the convergence of the series expansions of these two types of orbits is proved. It analytically demonstrates that there exist homoclinic orbits of the Si’lnikov type that join the two non-trivial equilibrium points to themselves, and therefore smale horseshoes and the horseshoe chaos occur for this system via the Si’lnikov criterion. In addition, there also exists a heteroclinic orbit connecting the two non-trivial equilibrium points. Numerical simulations are also given, which verify the analytical results. The system can be chaotic through period-doubling bifurcations as the amplitude of the external disturbance varies, and backward period-doubling bifurcations as the angular momentum of the rotor varies.     

Stability of motion of a space vehicle under constantly acting disturbances

The attitude stability of an Earth-orbiting satellite experiencing aerodynamic torque is studied. This is accomplished by applying the theory of total stability (or, stability under constantly acting disturbances) to the equations of motion. The satellite is gravity-gradient stabilized and a damping torque is incorporated. The aerodynamic torque results from the presence of two flat panels attached to the cylindrical body of the vehicle. This perturbing torque is treated as an additive disturbance in Euler's equations of motion. A Lyapunov function is constructed and then used in an appropriate theorem on stability under constantly acting disturbances. Explicit expressions are thereby found for bounds on a hypothetical initial condition (a rotation from the Earth-pointing equilibrium) and on the aerodynamic torque, so that if these disturbances are less than their respective bounds then the resulting attitude motion of the satellite will not exceed a pre-assigned value. These bounds depend on that pre-assigned value, and on the physical parameters of the satellite. The design implications of these bounds are then discussed.     

Uncontrolled motion of the <Emphasis Type="Italic">Foton M-2</Emphasis> satellite and quasistatic microaccelerations on its board

The results of determining the uncontrolled rotational motion of the Foton M-2 satellite (in orbit from May 31 to June 16, 2005) are presented. The determination was made using the data of onboard measurements of the Earth’s magnetic field strength. Segments 270 min long (three orbits) were selected from these data covering the first two thirds of the flight. On each such segment the data were processed jointly by the least squares method using integration of the equations of attitude motion of the satellite. In processing, the initial conditions of motion and parameters of the used mathematical model were estimated. The thus obtained results gave a complete overview of the satellite motion. This motion, having started with a small angular velocity, gradually accelerated, and in two days became close to the regular Euler precession of an axisymmetric solid body. On June 09, 2005 (the last day of measurements) the angular velocity of the satellite relative to its lengthwise axis was about 1.1 deg/s, while the projection of the angular velocity onto a plane perpendicular to this axis had an absolute value of about 0.11 deg/s. Deviations of the lengthwise axis from a normal to the orbit plane did not exceed 60°. Based on the results of determination of the rotational motion of the satellite, calculations of quasi-static microaccelerations on its board are performed.     

Optimum parameters of a gravitational satellite-stabilizer system

Dynamics of a satellite-stabilizer system is studied. It is supposed that there is a viscous friction in a hinge connecting two bodies, but there is no elasticity. The attitude motion in a plane of circular orbit is considered, and parameters are determined, at which natural oscillations near a stable equilibrium position in the orbital coordinate system are damped out most rapidly. The rate of transient processes is estimated by a value of the degree of stability of linearized equations of motion. The optimization of the degree of stability is sequentially performed in dimensionless damping coefficient (the first stage) and in inertial system parameters (the second stage). The result of the first stage is the partition of system parameter space into the regions, in each of which the maximum of the degree of stability is reached on a particular configuration of roots of the characteristic equation. It is shown at the second stage that the global maximum is reached at two points of parameter space, where one of system bodies degenerates into a plate, and the characteristic equation has four equal real roots.     

TETHERED SATELLITE SYSTEM ANALYSIS (1) — TWO-DIMENSIONAL CASE AND REGULAR DYNAMICS

The study on tethered satellite system (TSS) in two-dimensional in-planar motion is restricted in that the tether is assumed to be massless. The equations of motion are given in a spherical coordinate system to describe the magnitude (tether length) and direction angle of the position vector between the satellites. A length rate control algorithm is adopted, and the controlled motion of the directional angle by the algorithm will have a stable equilibrium state. The equilibrium state is a fixed point if the orbit of the base-satellite is circular, and a limit cycle if the orbit is elliptic. The value and stability of the equilibrium state are determined by the parameters of the control algorithm, and the bifurcation analysis is also given. Two typical TSS missions have been simulated.     

A specific case of stability of cylindrical precession of a satellite

In a central Newtonian gravitational field, the motion of a dynamically symmetrical satellite along an elliptical orbit of arbitrary eccentricity is considered. The particular motion of the satellite is known when its axis of symmetry is perpendicular to the orbit plane, and the satellite rotates about this axis with a constant angular velocity (cylindrical precession). A nonlinear analysis of stability of this motion has been performed under the assumption that the geometry of the satellite mass corresponds to a thin plate. At small values of orbit eccentricity e the analysis is analytical, while numerical analysis is used for arbitrary values of e.     

Space patrol: Variants of the optical barrier scheme

Different variants of the space patrol system to be designed for discovering and cataloging space objects hazardous for the Earth have been investigated. The basic idea of this system is to create an optical barrier using the telescopes deployed in a heliocentric orbit. Difficulties (as well as ways of overcoming them) of this program are analyzed, associated with form and position of the orbit of a space object relative to the patrol spacecraft, determination of orbit parameters, and mutual motion of space objects and the telescopes on spacecraft. The barrier’s schemes with scanning vertical or horizontal belts are considered. Some examples of observational conditions are presented for space objects crossing the barrier region: angular positions, velocities, distances, and numbers of days during which they are observed in the barrier region. The barrier’s characteristics are given for telescopes deployed in the orbits of the Earth and Venus.     

Generalized restricted circular three-body problem as a model for dynamics of binary asteroids

Exploration of the Solar System has recently revealed the existence of a large number of asteroids with satellites, which has stimulated interest in studying the dynamics of such systems. This paper is dedicated to the analysis of the relative motion of a binary asteroid. The conditions of existence of such a system (i.e., when its components do not run away) are derived in the Introduction. Then it is assumed that the satellite has no significant effect on the motion of the main asteroid, the latter being modeled as a dumbbell-like precessing solid body. The equations of motion of this system are a two-parameter generalization of the corresponding equations of the restricted circular three-body problem. It is demonstrated that in the system under consideration there exist steady-state motions in which the small asteroid is equidistant from attracting centers at the ends of the dumbbell (an analog to triangle libration points). The conditions of existence of such motions are derived, and the positions with respect to the dumbbell are analyzed in detail. Examination of the stability of the triangle libration points is reduced to investigation of a characteristic equation of the sixth degree. The stability conditions are derived in the case when the main asteroid executes near-planar motion.     

Amplitude—frequency analysis of Chandler oscillations of the Earth’s pole

An approximate nonlinear spectral-correlation model of fluctuations of the amplitude—frequency characteristics of the Chandler self-excited oscillations of the Earth’s pole is considered. The sensitivity of the model parameters to the asymmetry and anisotropy of fluctuation-dissipative moments of forces and to the effect of harmonic gravitation-tidal moments of forces is studied at Chandler frequency and frequencies close to it. The results of analytical and statistical modeling of the stability of the amplitude—frequency characteristics are presented. The influence of fluctuation disturbances of the white noise type on spectral-correlation characteristics of the oscillations is investigated.     

Determining Spacecraft Motion from Four Star Sensor Measurements

The BOKZ-M60 star sensor (a module that measures the coordinates of stars) has been designed for determining the parameters of the orientation of the intrinsic coordinate system relative to the inertial system from observations of stellar sky sections. The methods and results of processing of measurements by a set of four BOKZ-M60 sensors on the Resurs-P satellite no. 2 have been described. The time interval at which the satellite was in orbital orientation exceeds three orbital revolutions (19003 s). The joint processing of measurements by the four sensors conducted at the same time instants allowed the sensors to be associated with the universal coordinate system. With a root-mean-square error of less than 0.4′′ for each angle of rotation around its axes, this system is consistent with the model of the satellite’s rotational motion. The position of the universal system with respect to the instrumental coordinate system of the satellite was determined from the angular velocity measurements. Here, the root-mean-square errors for the values determined by the angles of rotation of the universal system around its axes were 0.044°, 0.051°, and 0.18°. The low-frequency (with frequencies less than 0.05 Hz) variations in the positions of intrinsic sensor coordinate systems relative to the universal system do not exceed 10′′. These are periodic variations with a fundamental frequency equal to the orbital frequency. The root-mean-square values of high-frequency components of these variations do not exceed 18′′.