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.     

The use of refined model of aerodynamic moment in the problem of reconstruction of the <Emphasis Type="Italic">Foton</Emphasis> satellite rotational motion

A new mathematical model of the uncontrolled rotational motion of the Foton satellite is presented. The model is based on the Euler dynamic equations of rigid body motion and takes into account the action upon the satellite of four external mechanical moments: gravitational, restoring aerodynamic, moment with constant components in the satellite-fixed coordinate system, and moment arising due to interaction of the Earth’s magnetic field with the satellite’s proper magnetic moment. To calculate the aerodynamic moment a special geometrical model of the outer satellite shell is used. Detailed form of the formulas giving above-mentioned moments in the equations of satellite motion is agreed with the form of the considered motion. Model testing is performed by determining with its help the rotational motion of the Foton M-2 satellite (it was in orbit from May 31, 2005 to June 16, 2005) using the data of the onboard measurements of the Earth’s magnetic field strength. The use of the new model has led to a relatively small improvement in the accuracy of the motion determination, but allowed us to obtain physically real estimates of some parameters.     

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.     

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.     

Determination of aerodynamic forces on satellites by theory and wind tunnel experiments

Theoretical and experimental aspects of satellite aerodynamics in free molecular flow have been studied. A FORTRAN-program package has been developed to calculate free molecular aerodynamic force and moment coefficients of general and special convex shapes, using an approximation of the surfaces by plane triangular elements.As a special case for concave bodies, the drag of a satellite of TD1A-similar shape has been investigated by an approximate theory and by wind tunnel experiments. Comparisons are made between experimental results, the approximate theory and exact theory.     

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.     

Estimation of residual microaccelerations on board an artificial earth satellite in the monoaxial solar orientation mode

The mode of monoaxial solar orientation of a designed artificial Earth satellite (AES), intended for microgravitational investigations, is studied. In this mode the normal line to the plane of satellite’s solar batteries is permanently directed at the Sun, the absolute angular velocity of a satellite is virtually equal to zero. The mode is implemented by means of an electromechanical system of powered flywheels or gyrodynes. The calculation of the level of microaccelerations arising on board in such a mode, was carried out by mathematical modeling of satellite motion with respect to the center of masses under an effect of gravitational and restoring aerodynamic moments, as well as of the moment produced by the gyrosystem. Two versions of a law for controlling the characteristic angular momentum of a gyrosystem are considered. The first version provides only attenuation of satellite’s perturbed motion in the vicinity of the position of rest with the required velocity. The second version restricts, in addition, the increase in the accumulated angular momentum of a gyrosystem by controlling the angle of rotation of the satellite around the normal to the light-sensitive side of the solar batteries. Both control law versions are shown to maintain the monoaxial orientation mode to a required accuracy and provide a very low level of quasistatic microaccelerations on board the satellite.     

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.     

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.     

Evolution of Motion of a Binary Planet

A model of a binary planet, consisting of a material point of small mass and a deformable viscoelastic sphere, is suggested. The center of mass of the binary planet moves in the gravitational field of a central body in the plane, which contains planets forming the binary planet. A deformable spherical planet rotates around the axis orthogonal to the plane of planetary motion. Planet deformations are described by the linear theory of viscoelasticity. It is shown that with an appropriate approximation of the gravitational potential, there is a class of quasicircular orbits, when the eccentricities of an orbit of the center of mass of a binary planet and an orbit, describing mutual planet motion, are equal to zero. The further evolution of motion is investigated in this class of orbits with the use of the canonical Poincare–Andoyer variables. Corresponding averaged equations are found, and phase pictures are constructed; the stability of stationary solutions is investigated on the basis of equations in variations. For the Solar system planets with their satellites, forming binary planets, the application of the presented model allows us to conclude that satellites sooner or later will fall on the corresponding planets.     

星载转台活动电缆扭转阻力矩测试与分析

星载转台活动电缆扭转过程中产生的阻力矩可能导致电缆绝缘层磨损破裂,阻力矩的稳定性还会影响转台控制的精度。鉴于此,文章对活动电缆扭转阻力矩进行测试分析,定量分析阻力矩影响因素,探究不同工况下阻力矩的变化规律,考查不同运动状态下阻力矩的稳定性。试验结果表明,小线束、分布式活动电缆布线方案满足型号阻力矩要求,是后续星载转台电缆布线设计中的可选方案。     

Single-Axis Solar Orientation of a Satellite of the Earth

The possibility of using the mode of single-axis solar orientation is considered for a satellite placed into a nearly circular orbit with an altitude of 900 km and bearing a solar sail. The satellite (together with the sail) has an axisymmetric structure, its symmetry axis being the principal central axis of the maximum moment of inertia. The center of the sail pressure lies on this axis and is displaced with respect to the satellite's center of mass. The symmetry axis of the satellite is set to the Sun so that its center of mass would be located between the Sun and the pressure center and would rotate around this axis with an angular velocity of a few degrees per second. The satellite's axis of symmetry makes a slow precession under the action of the gravitational moment and the moment of light pressure forces. Though the maximum magnitudes of these moments are comparable, the moment of the light pressure forces dominates and controls the precession in such a way that the symmetry axis orientation to the Sun remains unchanged.     

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.     

动态环路法磁矩测量技术研究

文章提出了一种新的航天器磁矩测量方法——动态环路法。首先,利用高斯电势级数公式建立了航天器的磁性偶极-四极模型。然后,针对模型中的8个磁矩分量,基于动态环路法的基本测量原理,设计了5组磁通感应线圈;根据8个磁矩分量的磁感应强度分布以及5组线圈的具体形状和位置,给出了各磁矩分量的磁通量表达式和利用积分法计算各个磁矩分量的公式。最后,对在推导过程中由于简化带来的近似误差和利用积分法计算公式理论计算误差进行了初步分析。当线圈间隔L为半径r的5%时,近似误差和积分法理论计算误差分别不超过2%和0.17%。结果表明,该方法不但能够获得航天器的磁矩大小并计算出其磁心坐标,而且还具有测量过程简单、测量速度较快以及测量精度高的优点。     

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.     

Attitude stabilization of a satellite by magnetic coils

Stabilization problem for a satellite is considered. The only measurement is of the geomagnetic field in the satellite coordinates. The control is carried out by a magnetic moment of current coils (magnetorquers) mounted on the satellite body. The stabilizer constructed in this work solves the problems of magnetic and gravitational stabilization. Qualitative analysis and results of numerical simulation are presented. The results of simulation show that the proposed stabilization system is reliable, and has an appropriate accuracy and does not need powerful sources of energy, and therefore can be used for attitude control of small satellites.     

A small parameter method in problems of maneuvering space vehicles with aerodynamic efficiency

The motion of a satellite with aerodynamic efficiency along a low near-circular orbit is considered in the paper. The controls of bank angle γ and lift coefficient C_y are used as control functions. The introduction of a small parameter ($\text{? = (}\text{?}{}_0\text{· S · g}{}_0\text{2G}\text{)}$) makes it possible to integrate an adjoint system of equations and to obtain an approximate solution to the complete problem in the class of piecewise-constant control functions. Maximum values for the coordinates of heading angle η and lateral derivation from the plane of a reference orbit ?, which are connected with orbit plane angle by the relation cos i = cos ? · cos h, are used as criteria of maneuvering capability for a satellite with aerodynamic efficiency. Optimal programs for bank angle and incidence variation are derived and the influence of lift-to-drag ratio on the vehicle maneuvering capabilities has been estimated.It is shown that the process of the optimal motion is a special kind of gravitational skipping similar to the Keplerian motion but with continuous descent.     

Regularization of the Limit Problem of Satellite Librations in a Keplerian Orbit Taking Light Pressure into Account

The planar librations of a satellite whose center of mass moves along an elliptic orbit are considered. It is assumed that not only the gravitational moment but also the forces of light pressure act upon the satellite. Account is taken of the fact that the right-hand sides of the differential equations are nonanalytic functions of the phase variables. When e 1, e being the orbit's eccentricity, the deformations of solutions are considered for the case of a satellite moving along a highly elongated orbit. Such transformation of the initial system of differential equations is carried out so that the new system becomes regular up to the value e = 1. A limit problem corresponding to the case e = 1 is considered. When the azimuth angle of the light source coincides with the direction to the pericenter, the dynamical system is reversible. In this case, the known families of the periodic solutions to the problem can be continued up to the limit case.     

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.