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.     

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.     

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.     

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′′.     

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.     

Dynamics of a gyrostat satellite with the vector of gyrostatic moment in the principal plane of inertia

Dynamics of a gyrostat satellite moving along a circular orbit in a central Newtonian field of force is investigated. In a particular case, when the gyrostatic moment vector lies in one of the satellite’s principal central planes of inertia, all positions of equilibrium are determined, and the conditions of their existence are analyzed. Also determined are bifurcation values of dimensionless parameters, at which the number of equilibrium positions changes. As a result of analysis of the generalized energy integral, for each equilibrium orientation the sufficient conditions of stability are derived. Evolution of the regions where the sufficient conditions of stability are valid is investigated under variation of the system’s dimensionless parameters.     

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.     

航天飞行器相对运动动力学及其控制方法

采用本地轨道从标系对两邻飞船间相对运动动力学展开研究，指出在相对运动中也存在平衡状态，用相平面方法分析了其稳定性，基于此分析，综合出相对运动控制方法，即距离速率控制方法，受控运动轨迹是一条稳定的稳态直线，进而建立了全方位距离速率控制方法。最后以系绳卫星系统和飞船安全为例完成了计算机模拟。     

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.     

Nanosatellite <Emphasis Type="Italic">REFLECTOR</Emphasis>: Choice of parameters of the attitude control system

We consider methodological and applied problems of choosing parameters and developing a passive gravity-gradient attitude control system for the REFLECTOR nanosatellite. The attitude control system consists of appropriately distributed satellite mass and additional masses forming the required tensor of inertia, as well as of a set of hysteresis rods manufactured from a magnetically soft material. The combination of these elements allows one, independent of the satellite’s initial angular motion, to maintain its three-axial orientation in the orbital coordinate system with preset performance in precision and fast response. The satellite’s dynamics is investigated using asymptotic and numerical methods. The problems of arrangement of the additional masses and hysteresis rods are discussed, as well as their interaction with magnetizable elements.     

Theory of physical libration of the Moon with the liquid core: Forced librations

For a two-layer model of the Moon that consists of a solid nonspherical mantle and an ellipsoidal homogeneous liquid core, a theory of forced librations under the effect of gravitational Earth’s moments has been developed. The motion of the Moon over its orbit has been described by the high-accuracy theory of DE/LE-4 orbital motion. Tables have been constructed that present forced librations of the Moon caused by the second harmonic of its force function, in the neighborhood of its motion according to the generalized Cassini laws. Disturbances of the first-order with respect to dynamic compressions of the Moon and its core are obtained in analytical form for Andoyer variables and Poincare variables and for the projection of the angular velocity vector of Moon’s mantle rotation and the Poincare coordinate system (relative to which core’s liquid accomplishes simple motion) on its major central axes of inertia, as well as for the classical variables in the Moon libration theory, etc. Constructed tables of the forced librations theory give the amplitudes and periods of librations and combinations of arguments of the orbital motion theory that correspond to libration parameters. The interpretation of basic variations has been given and a comparison with the previous theories has been carried out, in particular with the modern empirical theory constructed based on the laser observation data.     

太阳同步轨道卫星热控分系统分析及优化

定义地心日照轨道坐标系,并在此坐标系下简化卫星与地球相对位置的复杂计算,以及卫星轨道外热流分析过程中相关角度的计算,使轨道外热流的分析仿真更加快速、简洁。以正六棱柱形卫星为例,建立热网络模型,对其表面在一圈轨道内所受的轨道外热流进行仿真,并结合仿真结果计算进出地影区时卫星内部的温度。在此基础上,建立以热控分系统多层隔热材料质量最小化为目标的优化问题;在满足高低温工况卫星内部温度在-10～＋35℃范围内的约束下,对多层隔热材料厚度和散热窗大小进行了优化。     

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.     

Stability of large damped flexible spacecraft with stored angular momentum

Vibrational stability of large flexible structurally damped spacecraft carrying internal angular momentum and undergoing large rigid body rotations is analysed modeling the systems as elastic continua. Initially, analytical solutions to the motion of rigid gyrostats under torque-free conditions are developed. The solutions to the gyrostats modeled as axisymmetric and triaxial spacecraft carrying three and two constant speed momentum wheels, respectively, with spin axes aligned with body principal axes are shown to be complicated. These represent extensions of solutions for simpler cases existing in the literature. Using these solutions and modal analysis, the vibrational equations are reduced to linear ordinary differential equations. Equations with periodically varying coefficients are analysed applying Floquet theory. Study of a few typical beam- and plate-like spacecraft configurations indicate that the introduction of a single reaction wheel into an axisymmetric satellite does not alter the stability criterion. However, introduction of constant speed rotors deteriorates vibrational stability. Effects of structural damping and vehicle inertia ratio are also studied.     

On the stability of stationary motions of a system of coaxial bodies

We consider the stability of stationary motions of a model of a spacecraft as a system of coaxial bodies with small asymmetry caused by the shift of the axes of dynamic symmetry of bodies relative to the axis of rotation. We determine the stationary motions of the system; their stability is studied with respect to both the projections of angular velocity and the position of the axis of rotation. The sufficient conditions for the stability of these stationary motions are obtained by constructing a Lyapunov function, and the necessary conditions are obtained by analyzing the corresponding linearized equations of perturbed motion.     

静止轨道卫星姿态偏置对天线方向图影响的分析

地球静止轨道（GEO）卫星通常具备滚动、俯仰姿态偏置能力,即卫星滚动或俯仰姿态控制目标角度不是0°,而是某一设定角度,以便卫星定点位置改变时也能确保天线方向图覆盖满足要求。文章对某GEO卫星姿态偏置后对天线方向图及卫星其它性能的影响进行了分析,给出了卫星姿态偏置情况下天线覆盖区域的计算方法,在计算时考虑了卫星轨道倾角的影响。分析结果表明：由于卫星天线方向图设计时均有一定的设计裕度,卫星定点位置发生变化时,通过进行姿态偏置,可使其天线方向图满足用户使用要求,卫星在轨测试及运行结果也验证了这一结论。     

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].     

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.     

空间连续型机器人位姿与构型的能量整形控制

针对空间连续型机器人(SCR)位置和姿态机动、连续型机械臂变形控制以及系统受到外界干扰的问题，基于连接与阻尼配置-无源性控制(IDA PBC)方法设计了控制器。通过能量整形得到空间连续型机器人期望的能量函数，再对系统注入阻尼使其渐近稳定。利用非线性干扰观测器估计系统受到的干扰，并对外界干扰进行补偿。仿真结果表明：设计的控制器可以使空间连续型机器人机动到指定的位置和姿态，同时控制连续型机械臂变形为期望的构型，并且具有主动抗干扰的能力。通过选择合适的能量整形系数和阻尼注入系数，可使系统以期望的动态性能到达指定的平衡点。     

Equilibrium attitude and stability of a spacecraft on a stationary orbit around an asteroid

The stationary orbits around an asteroid, if exist, can be used for communication and navigation purposes just as around the Earth. The equilibrium attitude and stability of a rigid spacecraft on a stationary orbit around a uniformly-rotating asteroid are studied. The linearized equations of attitude motion are obtained under the small motion assumption. Then, the equilibrium attitude is determined in both cases of a general and a symmetrical spacecraft. Due to the higher-order inertia integrals of the spacecraft, the equilibrium attitude is slightly away from zero Euler angles. Then necessary conditions of stability of this conservative system are analyzed based on the linearized equations of motion. The effects of different parameters, including the harmonic coefficients C₂₀ and C₂₂ of the asteroid and higher-order inertia integrals of the spacecraft, on the stability are assessed and compared. Due to the significantly non-spherical shape and rapid rotation of the asteroid, the effects of the harmonic coefficients C₂₀ and C₂₂ are very significant, while effects of the third- and fourth-order inertia integrals of the spacecraft can be neglected. Considering a spacecraft on a stationary orbit around an example asteroid, we show that the classical stability domain predicted by the Beletskii–DeBra–Delp method on a circular orbit in a central gravity field is modified due to the non-spherical mass distribution of the asteroid. Our results are confirmed by a numerical simulation.