Periodic motions of a nearly dynamically symmetric satellite in the neighborhood of hyperboloidal precession

The motion of a satellite close to a dynamically symmetric solid body in a Newtonian gravitational field over a circular orbit is studied. The system of differential equations describing the body’s motion is close to a system with cyclic coordinate. New classes of periodic motions are constructed in the neighborhood of a known partial solution to an unperturbed problem, hyperboloidal precession of a dynamically symmetric satellite. In the resonance case, when the ratio of one frequency of small oscillations of a reduced system with two degrees of freedom in the neighborhood of a stable equilibrium position to the frequency of cyclic coordinate variation is close to an integer number, there exist one or three families of periodic motions that are analytic in terms of fractional powers of a small parameter. A study of stability of these motions was performed with the help of KAM (Kolmogorov-Arnold-Moser) theoty. Faling the described resonance there exists a unique family of periodic motions that is analytic in terms of integer powers of a small parameter. The check-up of stability of these motrons was carried out. We distinguished the cases of parametric resonance, resonances of the third and fourth orders, and a non-resonant case. In the resonance cases our study relies on well-known results on stability of Hamiltonian systems during resonances [1]. In the non-resonant case we use the KAM theory [2].     

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.     

Biaxial Mode of Rotation of a Satellite in the Orbit Plane

A mode of motion of a satellite with respect to its center of mass is studied, which is called the biaxial rotation in the orbit plane. In this mode of rotation, an elongated and nearly dynamically symmetric satellite rotates around the longitudinal axis, which, in turn, rotates around the normal to the plane of an orbit; the angular velocity of rotation around the longitudinal axis is several times larger than the orbital angular velocity, deviations of this axis from the orbit plane are small. Such a rotation is convenient in the case when it is required to secure a sufficiently uniform illumination of the satellite's surface by the Sun at a comparatively small angular velocity of the satellite. The investigation consists of the numerical integration of equations of the satellite's motion, which take into account gravitational and restoring aerodynamic moments, as well as the evolution of the orbit. At high orbits, the mode of the biaxial rotation is conserved for an appreciable length of time, and at low orbits it is destroyed due to the impact of the aerodynamic moment. The orbit altitudes and the method of constructing the initial conditions of motion that guarantee a sufficiently prolonged period of existence of this mode are specified.     

Asymptotic study of a complete magnetic attitude control cycle providing a single-axis orientation

The angular motion of an axisymmetrical satellite equipped with the active magnetic attitude control system is examined. Attitude control system has to ensure necessary orientation of the axis of symmetry in the inertial space. It implements the following strategy: coarse reorientation of the axis of symmetry with nutation damping or “-Bdot” without initial detumbling; spinning-up about the axis of symmetry to achieve the property of a gyro; fine reorientation of the axis in the inertial space. Dynamics of the satellite is analytically studied using averaging technique on the complete control loop consisting of five algorithms. Solutions of the equations of motion are obtained in terms of quadratures for most cases or even in closed-form. The latter allowed to study the dependence of motion parameters including time-response with respect to the orbit inclination and other parameters for all algorithms.     

One method of gravitational attitude control of a rotating satellite

The mode of spinning up a low-orbit satellite in the plane of its orbit is studied. In this mode, the satellite rotates around its longitudinal axis (principal central axis of the minimum moment of inertia), which executes small oscillations with respect to the normal to the orbit plane; the angular velocity of the rotation around the longitudinal axis is several tenths of a degree per second. Gravitational and restoring aerodynamic moments were taken into account in the equations of satellite’s motion, as well as a dissipative moment from eddy currents induced in the shell of the satellite by the Earth’s magnetic field. A small parameter characterizing deviation of the satellite from a dynamically symmetric shape and nongravitational external moments are introduced into the equations. A two-dimensional integral surface of the equations of motion, describing quasistationary rotations of the satellite close to cylindrical precession of the corresponding symmetrical satellite in a gravitational field, has been studied by the method of small parameter and numerically. We propose to consider such quasistationary rotations as unperturbed motions of the satellite in the spin-up mode.     

A study of evolution of spin-up mode of a satellite in the plane of its orbit

We investigate the mode of spinning up a low-orbit satellite in the plane of its orbit. In this mode the satellite rotates around its principal central axis of the minimum moment of inertia which executes small oscillations with respect to the normal to the orbit plane; the angular velocity of the rotation around this axis several times exceeds the mean orbital motion. Gravitational and restoring aerodynamic moments are taken into account in the satellite’s equations of motion. A small parameter characterizing deviation of the satellite from a dynamically symmetric shape is introduced into the equations. A two-dimensional integral surface of the equations of motion, describing quasi-steady-state rotations of the satellite close to cylindrical precession of the corresponding symmetrical satellite in a gravitational field, has been studied by the method of small parameter and numerically. Such quasi-steady-state rotations are suggested to be considered as unperturbed motions of the satellite in the spin-up mode. Investigation of the integral surface is reduced to numerical solution of a periodic boundary value problem of a certain auxiliary system of differential equations and to calculation of quasi-steady-state rotations by the two-cycle method. A possibility is demonstrated to construct quasi-steady rotations by way of minimization of a special quadratic functional.     

Secular evolution of rotary motion of a charged satellite in a decaying orbit

An electrostatically charged Earth satellite whose orbit is decaying due to the Earths oblateness is considered. Secular perturbations of the orbit are taken into account: they are caused by the second zonal harmonic of the geopotential. These perturbations represent deviations of the longitude of the ascending node and perigee argument, the orbit form being invariable and the orbit inclination to the equatorial plane being constant. The attitude rotary motion of the satellite under the action of perturbing moments of the gravitational and Lorentz forces is studied. The magnetic field of the Earth is taken in a quadrupole approximation. The evolution of the satellites rotary motion is investigated on the basis of new differential equations in s-parameters specially constructed for this purpose. Using the method of averaging, basic regularities of the secular evolution of rotary motion of a screened satellite are revealed. It is found that the rotary motion of a charged satellite essentially depends on the quadrupole component of the geomagnetic potential.__________Translated from Kosmicheskie Issledovaniya, Vol. 43, No. 2, 2005, pp. 111–125.Original Russian Text Copyright © 2005 by Tikhonov.     

Variation of the Satellite Orbit Altitude by the Light Pressure Force

The possibility of the uncontrolled increase of the altitude of an almost circular satellite orbit by the force of the light pressure is investigated. The satellite is equipped with a damper and a system of mirrors (solar batteries can serve as such a system). The flight of the satellite takes place in the mode of a single-axis gravitational orientation, the axis of its minimum principal central moment of inertia makes a small angle with the local vertical and the motion of the satellite around this axis constitutes forced oscillations under the impact of the moment of force of the light pressure. The form of the oscillations and the initial orbit are chosen so that the transverse component of the force of the light pressure acting upon the satellite be positive and the semimajor axis of the orbit would continuously increase. As this takes place, the orbit remains almost circular. We investigate the evolution of the orbit over an extended time interval by the method which employs separate integration of the equations of the orbital and rotational motions of the satellite. The method includes outer and inner cycles. The outer cycle involves the numerical integration of the averaged equations of motion of the satellite center of mass. The inner cycle serves to calculate the right-hand sides of these equations. It amounts to constructing an asymptotically stable periodic motion of the satellite in the mode of a single-axis gravitational orientation for current values of the orbit elements and to averaging the equations of the orbital motion along it. It is demonstrated that the monotone increase of the semimajor axis takes place during the first 15 years of motion. In actuality, the semimajor axis oscillates with a period of about 60 years. The eccentricity and inclination of the orbit remain close to their initial values.     

Closed form perturbation solution of a fast rotating triaxial satellite under gravity-gradient torque

The attitude dynamics of a fast rotating triaxial satellite under gravity-gradient is revisited. The essentially unique reduction of the Euler-Poinsot Hamiltonian, which can be performed in different sets of variables, provides a suitable set of canonical variables that expedites the perturbation approach. Two canonical transformations reduce the perturbed problem to its secular terms. The secular Hamiltonian and the transformation equations of the averaging are computed in closed form of the triaxiality coefficient, thus being valid for any triaxial body. The solution depends on Jacobi elliptic functions and integrals, and applies to non-resonant rotations under the assumption that the rotation rate is much higher than the orbital or precessional motion.     

Determination of parameters of attitude motion of a small communication satellite using the data of measurements of current of solar panels

A communication satellite (small spacecraft) injected into a geosynchronous orbit is considered. Flywheel engines are used to control the rotational spacecraft motion. The spacecraft after the emergency situation has passed into a state of uncontrolled rotation. In this case, no direct telemetric information about parameters of its rotational motion was accessible. As a result, the problem arose to determine the rotational satellite motion according to the available indirect information: current taken from the solar panels. Telemetric measurements of solar panel current obtained on the time interval of a few hours were simultaneously processed by the least squares method integrating the equations of rotational satellite motion. We present the results of processing 10 intervals of the measurement data allowing one to determine the real rotational spacecraft motion and to estimate the total angular momentum of flywheel engines.     

基于改进遗传算法的移动目标成像侦测任务规划问题研究

基于先验信息调用成像侦察卫星监控陆地或海洋移动目标动态信息是卫星成像侦察面临的新 课题。在已知移动目标位置等先验信息基础上,动态构造目标可变潜在区域及其运动预测模 型,利用STK辅助构造其候选成像观测活动集合;在此集合及目标运动预测模型基础上 对动态可变区域成像卫星调度问题进行建模,并设计了一种基于模拟退火算法及遗传算法的 改进遗传算法对问题进行求解,得到近最优的移动目标成像侦测方案。最后通过实例及算法 对比验证了规划模型及算法对解决该类问题的合理性和有效性。
     

Averaging of the equation of motion of a satellite with respect to its center of mass in degenerate cases: 1. Oscillations in the absolute frame of reference

The problem of attitude oscillations of a satellite with a small dynamic asymmetry in the plane of the orbit leads to a system degenerate to the fifth order from the point of view of the method of averaging. An explicit expression for the dominant term is obtained by integration in the complex plane. The recurrence procedure of calculating the higher approximations of the method of averaging is considered, as well as an approach to the analysis of the structure of derived expressions.     

Modeling of the aerodynamic moment acting upon a satellite

We consider the issues of modeling the moments of aerodynamic forces acting upon a satellite with gravitational system of stabilization. It is assumed that satellite orbits are almost circular with heights in the range 550–750 km. Simplified analytical expressions are suggested for the aerodynamic moment in the case when a satellite moves in the regime of gravitational orientation. Accuracy of the obtained expressions is estimated to be compared with that of expressions derived under the assumption of constant coefficient of frontal resistance. An analysis is made of short-periodic variations of the atmosphere density occurring due to orbital motion of a satellite. It is demonstrated that these variations can result in a substantial change of the aerodynamic moment, and their approximation by a truncated Fourier series is suggested. Estimates of the accuracy of the suggested approximation are given.     

Translational–Rotational Motion of a Viscoelastic Sphere in Gravitational Field of an Attracting Center and a Satellite

We study the translational–rotational motion of a planet modeled by a viscoelastic sphere in the gravitational fields of an immovable attracting center and a satellite modeled as material points. The satellite and the planet move with respect to their common center of mass that, in turn, moves with respect to the attracting center. The exact system of equations of motion of the considered mechanical system is deduced from the D'Alembert–Lagrange variational principle. The method of separation of motions is applied to the obtained system of equations and an approximate system of ordinary differential equations is deduced which describes the translational–rotational motion of the planet and its satellite, taking into account the perturbations caused by elasticity and dissipation. An analysis of the deformed state of the viscoelastic planet under the action of gravitational forces and forces of inertia is carried out. It is demonstrated that in the steady-state motion, when energy dissipation vanishes, the planet's center of mass and the satellite move along circular orbits with respect to the attracting center, being located on a single line with it. The viscoelastic planet in its steady-state motion is immovable in the orbital frame of reference. It is demonstrated that this steady-state motion is unstable.     

Implementation of a semianalytic satellite theory with recovery of short period terms

First order averaging is applied to the artificial satellite problem to obtain the averaged orbit which includes the secular, long and medium period effects of the oblateness of the Earth and the third body perturbations of the moon and sun. Perturbation theory is then used to recover the short period effects due to J₂, the moon, and sun. The perturbation analysis is carried out by means of Lie series and is developed through the first order. Optimization of the resulting short period series was then accomplished in several steps: first all separate algebraic coefficients were precalculated and stored; then all redundant SIN/COS calls were eliminated; next all repetition of numeric and algebraic coefficients were precalculated in pairs; application of the distributive principle allowed a significant reduction in additions and multiplications; finally trigonometric identities were used to further reduce the SIN/COS computations. The result of this optimization along with an interpolator for the averaged equations of motion results in a computer program which requires only $\text{1}\text{6}$ the CPU time (with no loss in accuracy) of the original non-optimized test program.     

General dynamics of a large class of flexible satellite systems

The paper presents a general formulation for librational dynamics of satellites with an arbitrary number, type and orientation of deploying flexible appendages. In particular, the case of beam-type flexible appendages deploying from a satellite in an arbitrary orbit is considered. The governing nonlinear, nonautonomous and coupled equations for vibration of the appendages and libration of the satellite are integrated numerically. Several cases of practical importance are considered making the system progressively more general and hence complex: (i) planar case representing pitch and appendage oscillations in the orbital plane; (ii) general attitude motion with planar vibrations of flexible members; (iii) above two cases together with the out-of-plane component of vibrations. Results show that under critical combinations of the system parameters the combined effect of flexibility and deployment can be substantial.     

Realization of modes of satellite attitude motion with a small level of microaccelerations using electromechanical executive devices

Quasi-static microaccelerations are estimated for a satellite specially designed to perform space experiments in the field of microgravity. Three modes of attitude motion of the spacecraft are considered: passive gravitational orientation, orbital orientation, and semi-passive gravitational orientation. In these modes the lengthwise axis of the satellite is directed along the local vertical, while solar arrays lie in the orbit plane. The second and third modes are maintained using electromechanical executive devices: flywheel engines or gyrodynes. Estimations of residual microaccelerations are performed with the help of mathematical modeling of satellite’s attitude motion under the action of gravitational and aerodynamic moments, as well as the moment produced by the gyro system. It is demonstrated that all modes ensure rather low level of quasi-static microaccelerations on the satellite and provide for a fairly narrow region of variation for the vector of residual microacceleration. The semi-passive gravitational orientation ensures also a limited proper angular momentum of the gyro system.     

Evolution of Rotational Motion of a Symmetrical Satellite with Viscoelastic Rods

We study the motion of a symmetrical satellite with a pair of flexible viscoelastic rods in a central Newtonian gravitational field. A restricted problem formulation is considered, when the satellite's center of mass moves along a fixed circular orbit. A small parameter is introduced which is inversely proportional to the stiffness of flexible elements. Another small parameter is equal to the ratio of the squared orbital angular velocity and the squared magnitude of the initial angular velocity of the satellite. In order to describe the satellite rotational motion relative to the center of mass, we use the canonical Andoyer variables. In the undisturbed formulation of the problem, i.e., at = 0 and = 0, these variables are the action–angle variables. Equations describing the evolution of motion are derived by an asymptotic method which combines the method of separating motions for systems with an infinite number of degrees of freedom and the Krylov–Bogolyubov method for systems with fast and slow variables. The manifolds of stationary motions are found, and their stability is investigated on the basis of equations in variations. Phase portraits are constructed which describe the rotational motion of a satellite at the stage of slow dissipative evolution.     

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.     

Periodic motions of a satellite-gyrostat relative to its center of mass under the action of gravitational torque

We investigated periodic motions of the axis of symmetry of a model satellite of the Earth, which are similar to the motions of the longitudinal axes of the Mir orbital station in 1999–2001 and the Foton-M3 satellite in 2007. The motions of these spacecraft represented weakly disturbed regular Euler precession with the angular momentum vector of motion relative to the center of mass close to the orbital plane. The direction of this vector during the motion was not practically changed. The model satellite represents an axisymmetric gyrostat with gyrostatic moment directed along the axis of symmetry. The satellite moves in a circular orbit and undergoes the action of the gravitational torque. The motion of the axis of symmetry of this satellite relative to the absolute space is described by fourth-order differential equations with periodic coefficients. The periodic solutions to this system with special symmetry properties are constructed using analytical and numerical methods.