Determination of <Emphasis Type="Italic">Foton M-2</Emphasis> satellite attitude motion by the data of microacceleration measurements

The results of reconstruction of uncontrolled attitude motion of the Foton M-2 satellite using measurements with the accelerometer TAS-3 are presented. The attitude motion of this satellite has been previously determined by the measurement data of the Earth’s magnetic field and the angular velocity. The TAS-3 data for this purpose are used for the first time. These data contain a well-pronounced additional component which made impossible their direct employment for the reconstruction of the attitude motion and whose origin was unknown several years ago. Later it has become known that the additional component is caused by the influence of the Earth’s magnetic field. The disclosure of this fact allowed us to take into account a necessary correction in processing of TAS-3 data and to use them for the reconstruction of the attitude motion of Foton M-2. Here, a modified method of processing TAS-3 data is described, as well as results of its testing and employing. The testing consisted in the direct comparison of the motion reconstructed by the new method with the motion constructed by the magnetic measurements. The new method allowed us to find the actual motion of Foton M-2 in the period June 9, 2005–June 14, 2005, when no magnetic measurements were carried out.     

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.     

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.     

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 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.     

Rotational motion of <Emphasis Type="Italic">Foton M-4</Emphasis>

The actual controlled rotational motion of the Foton M-4 satellite is reconstructed for the mode of single-axis solar orientation. The reconstruction was carried out using data of onboard measurements of vectors of angular velocity and the strength of the Earth’s magnetic field. The reconstruction method is based on the reconstruction of the kinematic equations of the rotational motion of a solid body. According to the method, measurement data of both types collected at a certain time interval are processed together. Measurements of the angular velocity are interpolated by piecewise-linear functions, which are substituted in kinematic differential equations for a quaternion that defines the transition from the satellite instrument coordinate system to the inertial coordinate system. The obtained equations represent the kinematic model of the satellite rotational motion. A solution of these equations that approximates the actual motion is derived from the condition of the best (in the sense of the least squares method) match between the measurement data of the strength vector of the Earth’s magnetic field and its calculated values. The described method makes it possible to reconstruct the actual rotational satellite motion using one solution of kinematic equations over time intervals longer than 10 h. The found reconstructions have been used to calculate the residual microaccelerations.     

Study of the mode of angular velocity damping for a spacecraft at non-standard situation

Non-standard situation on a spacecraft (Earth’s satellite) is considered, when there are no measurements of the spacecraft’s angular velocity component relative to one of its body axes. Angular velocity measurements are used in controlling spacecraft’s attitude motion by means of flywheels. The arising problem is to study the operation of standard control algorithms in the absence of some necessary measurements. In this work this problem is solved for the algorithm ensuring the damping of spacecraft’s angular velocity. Such a damping is shown to be possible not for all initial conditions of motion. In the general case one of two possible final modes is realized, each described by stable steady-state solutions of the equations of motion. In one of them, the spacecraft’s angular velocity component relative to the axis, for which the measurements are absent, is nonzero. The estimates of the regions of attraction are obtained for these steady-state solutions by numerical calculations. A simple technique is suggested that allows one to eliminate the initial conditions of the angular velocity damping mode from the attraction region of an undesirable solution. Several realizations of this mode that have taken place are reconstructed. This reconstruction was carried out using approximations of telemetry values of the angular velocity components and the total angular momentum of flywheels, obtained at the non-standard situation, by solutions of the equations of spacecraft’s rotational motion.     

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.     

Uncontrolled Attitude Motion of the Foton-12 Satellite and Quasi-Steady Microaccelerations onboard It

The results of determination of the uncontrolled attitude motion of the Foton-12 satellite (placed in orbit on September 9, 1999, terminated its flight on September 24, 1999) are presented. The determination was carried out by the onboard measurement data of the Earth's magnetic field strength vector. Intervals with a duration of several hours were selected from data covering almost the entire flight. On each such interval the data were processed simultaneously using the least squares method by integrating the satellite's equations of motion with respect to the center of mass. The initial conditions of motion and the parameters of the mathematical model employed were estimated in processing. The results obtained provided for a complete representation of the satellite's motion during the flight. This motion, beginning with a small angular velocity, gradually sped up. The growth of the component of the angular velocity with respect to the longitudinal axis of the satellite was particularly strong. During the first several days of the flight this component increased virtually after every passage through the orbit's perigee. As the satellite's angular velocity increased, its motion became more and more similar to the regular Euler precession of an axisymmetric rigid body. In the last several days of flight the satellite's angular velocity with respect to its longitudinal axis was about 1 deg/s and the projection of the angular velocity onto the plane perpendicular to this axis had a magnitude of approximately 0.15 deg/s. The deviation of the longitudinal axis from the normal to the orbit plane did not exceed 60°. The knowledge of the attitude motion of the satellite allowed us to determine the quasi-steady microacceleration component onboard it at the locations of the technological and scientific equipment.     

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.     

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.     

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.     

Estimating the accuracy of the technique of reconstructing the rotational motion of a satellite based on the measurements of its angular velocity and the magnetic field of the Earth

The paper has studied the accuracy of the technique that allows the rotational motion of the Earth artificial satellites (AES) to be reconstructed based on the data of onboard measurements of angular velocity vectors and the strength of the Earth magnetic field (EMF). The technique is based on kinematic equations of the rotational motion of a rigid body. Both types of measurement data collected over some time interval have been processed jointly. The angular velocity measurements have been approximated using convenient formulas, which are substituted into the kinematic differential equations for the quaternion that specifies the transition from the body-fixed coordinate system of a satellite to the inertial coordinate system. Thus obtained equations represent a kinematic model of the rotational motion of a satellite. The solution of these equations, which approximate real motion, has been found by the least-square method from the condition of best fitting between the data of measurements of the EMF strength vector and its calculated values. The accuracy of the technique has been estimated by processing the data obtained from the board of the service module of the International Space Station (ISS). The reconstruction of station motion using the aforementioned technique has been compared with the telemetry data on the actual motion of the station. The technique has allowed us to reconstruct the station motion in the orbital orientation mode with a maximum error less than 0.6° and the turns with a maximal error of less than 1.2°.     

Uncontrolled rotational motion of the <Emphasis Type="Italic">Aist</Emphasis> small spacecraft prototype

The results of reconstructing the uncontrolled rotational motion of the Aist small spacecraft prototype during its flight in early 2014 have been presented. The reconstruction was carried out by processing data from onboard measurements of the Earth’s magnetic field. The processing procedure used portions of data covering intervals of time with durations ranging from a few dozen minutes to three hours. Data obtained in each such interval were processed jointly by the least-squares method by integrating the equations of the satellite motion relative to the center of mass. The initial conditions of the motion and the parameters of the used mathematical model during processing have been estimated. The results of processing for several data intervals have provided a fairly complete picture of the satellite motion. This was the weakly disturbed Euler–Poinsot motion.     

Determination of parameters of rotational motion of the <Emphasis Type="Italic">Monitor-E</Emphasis> spacecraft from the telemetry data on solar batteries’ current

The Monitor-E spacecraft executed uncontrolled flight due to emergency situation, no telemetry information on parameters of the spacecraft’s attitude motion being available. So, the problem arose to determine the spacecraft’s rotational motion from the accessible indirect information—the electric current provided by solar batteries. In this paper the integrated statistical technique is described, that allows one to solve this problem. The values of current, obtained over the time interval some tens of minutes long, have been processed simultaneously by the least squares method using the integration of the equations of spacecraft’s rotational motion. As a result of processing, the initial conditions of motion were estimated, and the spacecraft’s moments of inertia were updated, as well as the angles, specifying solar batteries position in the spacecraft-fixed coordinate system. The results of processing of 12 data sets are presented, which allowed us to reconstruct the actual rotational motion of the spacecraft.     

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.     

Reconstruction of spacecraft rotational motion using a Kalman filter

Quasi-static microaccelerations of four satellites of the Foton series (nos. 11, 12, M-2, M-3) were monitored as follows. First, according to measurements of onboard sensors obtained in a certain time interval, spacecraft rotational motion was reconstructed in this interval. Then, along the found motion, microacceleration at a given onboard point was calculated according to the known formula as a function of time. The motion was reconstructed by the least squares method using the solutions to the equations of satellite rotational motion. The time intervals in which these equations make reconstruction possible were from one to five orbital revolutions. This length is increased with the modulus of the satellite angular velocity. To get an idea on microaccelerations and satellite motion during an entire flight, the motion was reconstructed in several tens of such intervals. This paper proposes a method for motion reconstruction suitable for an interval of arbitrary length. The method is based on the Kalman filter. We preliminary describe a new version of the method for reconstructing uncontrolled satellite rotational motion from magnetic measurements using the least squares method, which is essentially used to construct the Kalman filter. The results of comparison of both methods are presented using the data obtained on a flight of the Foton M-3.     

An analysis of low-frequency component in microacceleration measurements made onboard the <Emphasis Type="Italic">Foton M-2</Emphasis> satellite

The low-frequency component is investigated in the data of measurements performed onboard the Foton M-2 satellite with the three-component accelerometer TAS-3. Investigations consisted in comparison of this component with its calculated analog found from a reconstruction of the satellite’s attitude motion. The influence of the Earth’s magnetic field on the accelerometer readings is discovered by way of spectral analysis of the functions representing the results of determining the low-frequency microacceleration by two methods. After making correction for this influence, the results obtained by these two methods coincided within a root-mean-square error of less than 10^?6 m/s².     

On the belt of energetic electrons at <Emphasis Type="Italic">L</Emphasis> = 2.75 in the Earth’s magnetosphere

In 1964, during flights of the ELECTRON satellites the narrow belts of energetic electrons (E _e ≈ 6MeV) have been discovered in the Earth’s magnetosphere at L ≈ 2.75. The same structures approximately at the same magnetic shells were found in 2004 by the CORONAS-F and SERVIS-1 satellites. A comparison of the results of these experiments is presented. It is shown that the additional narrow belts of energetic electrons occur after intense magnetic storms (D _st > 100 nT), in our cases, having a double-triple structure. The lifetime of these belts is a few months and their disappearance had a gradual character. The obtained results separated in time by 40 years suggest the constancy of the sources of particles of the Earth’s radiation belts and processes occurring in the magnetosphere, which ensures not only existence of the radiation belts, but also the recurrence of various exotic phenomena in the belts similar to the belt of energetic electrons at the inner magnetic shells.     

A simplified technique for determining the rotational motion of a satellite based on the onboard measurements of the angular velocity and magnetic field of the Earth

The mathematical model, which allowed us to reconstruct the rotational motion of the Bion M-1 and Foton M-4 satellites by processing the measurements of onboard magnetometers and the angular velocity sensor, is sufficiently detailed and accurate. If we slightly lower the requirements for accuracy and transfer to a rougher model, i.e., we will not update the biases in measurements of the angular velocity component, then the measurement processing technique can be significantly simplified. The volume of calculations in minimizing the functional of the least-square technique is reduced; the most complicated part of calculations is performed using the standard procedure of computational linear algebra. This simplified technique is described below, and the examples of its application for reconstructing the rotational motion of the Foton M-4 satellite are presented. A noticeable distinction in the reconstructions of motion, constructed by simplified and more exact techniques, is revealed in processing the measurements over time intervals longer than 4 hours.