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.     

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.     

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.     

Determination of attitude motion of the <Emphasis Type="Italic">Foton M-3</Emphasis> satellite according to the data of onboard measurements of the Earth’s magnetic field

The results of reconstruction of rotational motion of the Foton M-3 satellite during its uncontrolled flight in September 2007 are presented. The reconstruction was performed by processing the data of onboard measurements of the Earth’s magnetic field obtained by the DIMAC instruments. The measurements were carried out continuously throughout the flight, but the processing technique dealt with the data portions covering time intervals of a few orbital revolutions. The data obtained on each such interval were processed jointly by the least squares method with using integration of the equations of satellite motion relative to its center of mass. When processing, the initial conditions of motion and the used mathematical model’s parameters were estimated. The results of processing 16 data sets gave us complete information about the satellite motion. This motion, which began at a low angular velocity, had gradually accelerated and in five days became close to the regular Euler precession of an axisymmetric solid body. At the end of uncontrolled flight the angular velocity of the satellite relative to its lengthwise axis was 0.5 deg/s; the angular velocity projection onto the plane perpendicular to this axis had a magnitude of about 0.18 deg/s.     

Uncontrollable Rotational Motion of the Mir Orbital Station

The results of determining the rotational motion of the Mir orbital station are presented for four long segments of its unmanned uncontrolled flight in 1999–2000. The determination was carried out using the data of onboard measurements of the Earth's magnetic field intensity. These data, taken for a time interval of several hours, were jointly processed by the least squares method with the help of integration of the equations of station motion relative to its center of mass. As a result of this processing, the initial conditions of motion and the parameters of the mathematical model used were evaluated. The technique of processing is verified using the telemetry data on angular velocity of the station and its attitude parameters. Two types of motion were applied on the investigated segments. One of them (three segments) presents a rotation around the axis of the minimum moment of inertia. This axis executes small oscillations with respect to a normal to the orbit plane. Such a motion was used for the first time on domestic manned orbital complexes. The second type of motion begins with a biaxial rotation which, in a few weeks, goes over into a motion very similar to the rotation around the normal to the orbit plane, but around the axis of the maximum moment of inertia.     

Analysis of Low-Frequency Microaccelerations onboard the Foton-11Satellite

The method and the results of investigating the low-frequency component of microaccelerations onboard the Foton-11satellite are presented. The investigation was based on the processing of data of the angular velocity measurements made by the German system QSAM, as well as the data of measurements of microaccelerations performed by the QSAM system and by the French accelerometer BETA. The processing was carried out in the following manner. A low-frequency (frequencies less than 0.01 Hz) component was selected from the data of measurements of each component of the angular velocity vector or of the microacceleration, and an approximation was constructed of the obtained vector function by a similar function that was calculated along the solutions to the differential equations of motion of the satellite with respect to its center of mass. The construction was carried out by the least squares method. The initial conditions of the satellite motion, its aerodynamic parameters, and constant biases in the measurement data were used as fitting parameters. The time intervals on which the approximation was constructed were from one to five hours long. The processing of the measurements performed with three different instruments produced sufficiently close results. It turned out to be that the rotational motion of the satellite during nearly the entire flight was close to the regular Eulerian precession of the axially symmetric rigid body. The angular velocity of the satellite with respect to its longitudinal axis was about 1 deg/s, while the projection of the angular velocity onto the plane perpendicular to this axis had an absolute value of about 0.2 deg/s. The magnitude of the quasistatic component of microaccelerations in the locations of the accelerometers QSAM and BETA did not exceed 5 × 10^–5–10^–4m/s²for the considered motion of the satellite.     

The Mode of Gravitational Orientation for the International Space Station

The results of the determination of the uncontrolled attitude motion of the International Space Station during its unmanned flight in 1999 are presented. The data of onboard measurements of three components of the angular velocity are used for this determination. These data covering an interval of slightly less than one orbit were jointly processed by the least squares method, by integrating the equations of motion of the station relative to its center of mass. As a result of this processing, the initial conditions of the motion and the parameters of the mathematical model used were estimated. The actual motion of the station has been determined for 20 such intervals during April–November. Throughout these intervals, the station rotated about the axis of the minimum moment of inertia, the latter executing small oscillations relative to the local vertical. Such a mode, known as the mode of gravitational orientation of a rotating satellite or the mode of generalized gravitational orientation, was planned before the flight. The measurements were made to verify it. The quasistatic component of the microaccelerations aboard the station is estimated for this mode.     

An Analysis of the Low-Frequency Component in Measurements of Angular Velocity and Microacceleration onboard the Foton-12 Satellite

We compare the results of two methods used to determine the angular velocity of the Foton-12 satellite and the low-frequency component of microaccelerations onboard it. The first method is based on reconstruction of the satellite's rotational motion using the data of onboard measurements of the strength of the Earth's magnetic field. The motion (time dependence of the orientation parameters and angular velocity) was found from the condition of best approximation of the measurement data by the functions calculated along the solutions to equations of attitude motion of the satellite. The solutions found were used to calculate the quasistatic component of microaccelerations at certain points of the satellite, in particular, at the point of location of an accelerometer of the QSAM system. Filtration of the low-frequency component of the angular velocity and microacceleration from the data of measurements by a sensor of angular velocity and by the accelerometer of this system served as a second method. The filtration was made using the discrete Fourier series. A spectral analysis of the functions representing the results of determining the angular velocity and microacceleration by both methods is performed. Comparing the frequencies and amplitudes of the harmonic component of these functions allowed us to estimate the accuracy of measurements made by the QSAM system in the low-frequency range.     

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.     

Evolution of the Orbit of an Earth Satellite with a Solar Sail

The efficiency of using the light pressure of solar radiation for increasing the semimajor axis of the orbit of an Earth Satellite carrying a solar sail is estimated. The orbit is nearly circular and has an altitude of about 900 km. The satellite is in the mode of single-axis solar orientation: it rotates at an angular velocity of 1 deg/s around the axis of symmetry, which traces the direction to the Sun. This mode is maintained by the solar sail, which serves in this case as a solar stabilizer. The following method of increasing the semimajor axis of the orbit (which is equivalent to increasing the total energy of the satellite's orbital motion) is considered. On those sections of the orbit, where the angle between the light pressure force acting upon the sail and the vector of geocentric velocity of the satellite does not exceed a specified limit, the sail is functioning as a solar stabilizer. On those sections of the orbit, where the above-indicated angle exceeds this limit, the sail is furled by way of turning the edges of the petals towards the Sun. Such a control increases the semimajor axis by more than 150 km for three months of flight. In this case, the accuracy of solar orientation decreases insignificantly.     

Modes of uncontrolled rotational motion of the <Emphasis Type="Italic">Progress M-29M</Emphasis> spacecraft

We have reconstructed the uncontrolled rotational motion of the Progress M-29M transport cargo spacecraft in the single-axis solar orientation mode (the so-called sunward spin) and in the mode of the gravitational orientation of a rotating satellite. The modes were implemented on April 3–7, 2016 as a part of preparation for experiments with the DAKON convection sensor onboard the Progress spacecraft. The reconstruction was performed by integral statistical techniques using the measurements of the spacecraft’s angular velocity and electric current from its solar arrays. The measurement data obtained in a certain time interval have been jointly processed using the least-squares method by integrating the equations of the spacecraft’s motion relative to the center of mass. As a result of processing, the initial conditions of motion and parameters of the mathematical model have been estimated. The motion in the sunward spin mode is the rotation of the spacecraft with an angular velocity of 2.2 deg/s about the normal to the plane of solar arrays; the normal is oriented toward the Sun or forms a small angle with this direction. The duration of the mode is several orbit passes. The reconstruction has been performed over time intervals of up to 1 h. As a result, the actual rotational motion of the spacecraft relative to the Earth–Sun direction was obtained. In the gravitational orientation mode, the spacecraft was rotated about its longitudinal axis with an angular velocity of 0.1–0.2 deg/s; the longitudinal axis executed small oscillated relative to the local vertical. The reconstruction of motion relative to the orbital coordinate system was performed in time intervals of up to 7 h using only the angularvelocity measurements. The measurements of the electric current from solar arrays were used for verification.     

Experimental investigation of the modes of operation of uncontrolled attitude motion of the Progress spacecraft

Results of in-flight tests of three modes of uncontrolled attitude motion of the Progress spacecraft are described. These proposed modes of experiments related to microgravity are as follows: (1) triaxial gravitational orientation, (2) gravitational orientation of the rotating satellite, and (3) spin-up in the plane of the orbit around the axis of the maximum moment of inertia. The tests were carried out from May 24 to June 1, 2004 onboard the spacecraft Progress M1-11. The actual motion of this spacecraft with respect to its center of mass, in the above-mentioned modes, was determined by telemetric information about an electric current tapped off from solar batteries. The values of the current obtained during a time interval of several hours were processed jointly using the least squares method by integration of the equations of the spacecraft’s attitude motion. The processing resulted in estimation of the initial conditions of motion and of the parameters of mathematical models used. For the obtained motions the quasi-static component of microaccelerations was computed at a point onboard, where installation of experimental equipment is possible.     

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.     

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.     

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.     

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.     

Low-Frequency Microaccelerations onboard the Foton-11 Satellite

We analyze the microacceleration measurements carried out onboard the Foton-11 satellite with the three-component accelerometer BETA. The microaccelerations were recorded virtually throughout the entire orbital flight of the Foton-11 satellite. The data obtained were analyzed in the following way. First they were used to determine the actual rotational motion of the satellite for several arbitrarily selected time intervals 4 h long. This problem was solved by constructing the approximation of the microacceleretation low-frequency component (previously determined from the data) by its calculated analog computed along the solutions to differential equations of rotational motion of the satellite. The approximation was made by the least squares method. As a result, those mathematical model parameters and the solutions to equations of motion were found that gave the best consistency of the microacceleretation low-frequency component and its calculated analog. Then the spectral analysis of the low-frequency component and its calculated analog was made. It was shown that, although basic harmonics of these functions coincided sufficiently well, some harmonics of the low-frequency component failed to be interpreted in terms of the satellite's rotational motion.     

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.     

卫星姿态的几何确定方法初探

利用卫星模拟遥测数据确定的卫星姿态是卫星发射中的重要参数。从几何姿态确定原理入手,研究如何利用遥测信息确定自旋卫星的姿态,列写了卫星周期、自旋角速度、两面角及太阳角的计算公式,分九种情况列写了卫星姿态的计算方法及公式,并对文中方法在海上测控任务中的应用情况进行了分析。     

Determination of the Attitude Motion of a Satellite Based on a Kinematical Model of Motion and Using Data of Measurements of Its Angular Velocity and the Earth’s Magnetic Field Strength

An integral statistical procedure of determination of the attitude motion of a satellite using the data of onboard measurements of angular velocity vectors and the strength of the Earth’s magnetic field (EMF) is suggested. The procedure uses only the equations of kinematics of a solid body and is applicable to determining both controlled and uncontrollable motions of a satellite at any external mechanical moments acting upon it. When applying this procedure, the data of measurements of both types, accumulated during a certain interval of time, are processed jointly. The data of measuring the angular velocity are smoothed by discrete Fourier series, and these series are substituted into kinematical Poisson equations for elements of the matrix of transition from a satellite-fixed coordinate system to the orbital coordinate system. The equations thus obtained represent a kinematical model of the satellite motion. The solution to these equations (which approximate the actual motion of a satellite) is found from the condition of the best (in the sense of the least squares method) fit of the data of measuring the EMF strength vector to its calculated values. The results of testing the suggested procedure using the data of measurements of the angular velocity vectors onboard the Foton-12 satellite and measurements of EMF strengths are presented.__________Translated from Kosmicheskie Issledovaniya, Vol. 43, No. 4, 2005, pp. 295–305.Original Russian Text Copyright © 2005 by Abrashkin, Volkov, Voronov, Egorov, Kazakova, Pankratov, Sazonov, Semkin.