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.     

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.     

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.     

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.     

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°.     

Experiments with the DAKON-M convection sensor

We have described an express technique for processing the results of experiments with a DAKON-M convection sensor on board the Service Module of the International Space Station (ISS) in 2011. The technique uses a certain rule to compare the sensor measurements with the calculated data on the quasistatic component of microacceleration at the point of installation. The sensor experiments have been conducted during shuttle docking and undocking, when low-frequency microaccelerations on the ISS were significant. The microaccelerations have been calculated using measurement data of the MAMS low-frequency accelerometer installed in the Lab module and the telemetry data on the ISS rotational motion. This has made it possible to convert the MAMS measurement data to the DAKON-M convection sensor installation point. A comparison of sensor readings with calculated microaccelerations has revealed fairly good agreement between them.     

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.     

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.     

Forecasting the velocity of quasi-stationary solar wind and the intensity of geomagnetic disturbances produced by it

A brief review is given of contemporary approaches to solving the problem of medium-term forecast of the velocity of quasi-stationary solar wind (SW) and of the intensity of geomagnetic disturbances caused by it. At the present time, two promising models of calculating the velocity of quasi-stationary SW at the Earth’s orbit are realized. One model is the semi-empirical model of Wang-Sheeley-Arge (WSA) which allows one to calculate the dependence V(t) of SW velocity at the Earth’s orbit using measured values of the photospheric magnetic field. This model is based on calculation of the local divergence f _S of magnetic field lines. The second model is semi-empirical model by Eselevich-Fainshtein-Rudenko (EFR). It is based on calculation in a potential approximation of the area of foot points on the solar surface of open magnetic tubes (sources of fast quasistationary SW). The new Bd-technology is used in these calculations, allowing one to calculate instantaneous distributions of the magnetic field above the entire visible surface of the Sun. Using predicted V(t) profiles, one can in EFR model calculate also the intensity of geomagnetic disturbances caused by quasi-stationary SW. This intensity is expressed through the K _p index. In this paper the EFR model is discussed in detail. Some examples of epignosis and real forecast of V(t) and K _p (t) are discussed. A comparison of the results of applying these two models for the SW velocity forecasting is presented.     

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.     

Determination of two Satellites’ relative motion using tracking measurements

A technique to determine the initial relative velocity of two satellites at the initial time instant, based on the results of NORAD system’s measurements, is presented. The determination of a relative initial position is shown to result in the unsatisfactory accuracy. This technique is applied to determining the initial velocity of separation of the first Russian nanosatellite TNS-0 from the International Space Station.     

A comparison of polarization jet characteristics at the greatly separated stations Yakutsk and Podkamennaya Tunguska

Characteristics of the polarization jet (PJ) are considered on the basis of measurements made in 1989–1992 at the ionospheric stations Yakutsk (L = 3.0, λ = 129.6°) and Podkamennaya Tunguska (L = 3.0, λ = 90.0°) separated in longitude. Using the data of these stations, the result obtained earlier (that the formation of PJ during disturbances in the near-midnight sector occurs at the expansion phase of a substorm) is confirmed. At isolated magnetic disturbances with AE > 500 nT in the 11:00–16:00 UT interval, the PJ band covers an MLT sector of 3 h between the Yakutsk and Podkamennaya Tunguska stations. The time of the PJ beginning at the Podkamennaya Tunguska station for the majority of events is 45–60 min behind the PJ beginning at the Yakutsk station. This corresponds to the westward motion of a PJ source with a velocity of ～3 MLT h per hour.     

Families of periodic solutions in three-dimensional restricted three-body problem

Planar orbits of three-dimensional restricted circular three-body problem are considered as a special case of three-dimensional orbits, and the second-order monodromy matrices M (in coordinate z and velocity v _z ) are calculated for them. Semi-trace s of matrix M determines vertical stability of an orbit. If |s| ≤ 1, then transformation of the subspace (z, v _z ) in the neighborhood of solution for the period is reduced to deformation and a rotation through angle φ, cosφ = s. If the angle ? can be rationally expressed through 2π,φ = 2π·p/q, where p and q are integer, then a planar orbit generates the families of three-dimensional periodic solutions that have a period larger by a factor of q (second kind Poincareé periodic solutions). Directions of continuation in the subspace (z, v _z ) are determined by matrix M. If |s| < 1, we have two new families, while only one exists at resonances 1: 1 (s = 1) and 2: 1 (s = ?1). In the course of motion along the family of three-dimensional periodic solutions, a transition is possible from one family of planar solutions to another one, sometimes previously unknown family of planar solutions.     

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.     

Study and comparison of the parameters of five hot flow anomalies at a bow shock front

Five hot flow anomalies (HFA) recorded by the Tail Probe of the INTERBALL satellite in 1996 are analyzed in present work. For the five chosen events the authors determined the characteristics of current sheets whose interaction with the bow shock front led to formation of an HFA, as well as the directions of external electric fields and the directions of motion of these HFAs over a shock front. The analysis of plasma convection in an HFA body is carried out; the average velocities of plasma motion in the HFA are determined in a coordinate system linked with the normal to a current layer and with the normal to the bow shock. According to the character of plasma convection in an HFA body, these five events may be divided into two types, which also differ in the direction of the motion over the front of the bow shock. In the first-type HFAs, the convection of plasma has a component directed from the intermediate region confirming its identification as a source of energy for the formation of an HFA. In the second-type HFAs, plasma motion from the intermediate region in leading and trailing parts is less expressed. This fact, as well as the great variation of peculiar velocities in the body of anomalies, allowed the assumption that second-type anomalies are nonstationary. Evidence is presented that the anomalies considered in the paper are bordered with shocks formed in solar wind passing a large-scale, decelerated body of heated plasma.     

Study of microaccelerations onboard the <Emphasis Type="Italic">International Space Station</Emphasis> with the DAKON-M convection sensor

The results of experiments with the DAKON-M convection sensor onboard the Russian orbital segment of the International Space Station are described. A comparison of the sensor measurements with the results of calculation of the quasistatic microacceleration component at the point of installation is made. For this comparison we have used three measurement intervals of the experiments in 2009, during which spacecraft were docked with the station, undocked from it, and actuation of jet engines of the attitude control system took place. When calculating microacceleration, we use the measurement data of the low-frequency MAMS accelerometer, installed on the American segment, and the telemetry data on the ISS rotational motion. This information allowed one to convert the MAMS measurements to the point of installation of the DAKON-M convection sensor. A comparison of sensor measurements with calculated microaccelerations showed sufficiently accurate coincidence between the calculated and measured data.     

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.     

Large-scale irregularities of the winter polar topside ionosphere according to data from <Emphasis Type="Italic">Swarm</Emphasis> satellites

An analysis of the electron density measurements (Ne) along the flyby trajectories over the high-latitude region of the Northern Hemisphere under winter conditions in 2014 and 2016 has shown that the main large-scale structure observed by Swarm satellites is the tongue of ionization (TOI). At the maximum of the solar cycle (F_10.7 = 160), the average value of Ne in the TOI region at an altitude of 500 km was 8 × 10⁴ cm^–3. Two years later, at F_10.7 = 100, Ne ~ 5 × 10⁴ cm^–3 and Ne ~2.5 × 10⁴ cm^–3 were observed at altitudes of 470 and 530 km, respectively. During the dominance of the azimuthal component of the interplanetary magnetic field, the TOI has been observed mainly on the dawn or dusk side depending on the sign of B _y . Simultaneous observations of the convective plasma drift velocity in the polar cap show the transpolar flow drift to the dawn (B_y < 0) or dusk side (B _y < 0). Observations and numerical simulation of the Ne distribution have confirmed the significant role of the electric field of the magnetospheric convection in the generation of large-scale irregularities in the polar ionosphere.     

On Saturn’s rotation relative to a center of mass under the action of the gravitational moments of the Sun and Jupiter

Saturn’s rotation relative to a center of mass is considered within an elliptic restricted three-body problem. It is assumed that Saturn is a solid under the action of gravity of the Sun and Jupiter. The motions of Saturn and Jupiter are considered elliptic with small eccentricities e_S and e_J, respectively; the mean motion of Jupiter n_J is also small. We obtain the averaged Hamiltonian function for a small parameter of ε = n_J and integrals of evolution equations. The main effects of the influence of Jupiter on Saturn’s rotation are described: (α) the evolution of the constant parameters of regular precession for the angular momentum vector I₂; (β) the occurrence of new libration zones of oscillations I₂ near the plane of the celestial equator parallel to the plane of the Jupiter’s orbit; (γ) the occurrence of additional unstable equilibria of vector I₂ at the points of the north and south poles of the celestial sphere and, as a result, the existence of homoclinic trajectories; and (δ) the existence of periodic trajectories with arbitrarily large periods near the homoclinic trajectory. It is shown that the effects of (β), (γ), and (δ) are caused by the eccentricity e of the Jupiter’s orbit and are practically independent of Jupiter’s mass (within satellite approximation).     

Turbulent fluctuations of plasma and magnetic field parameters in the magnetosheath and the low-latitude boundary layer formation: Multisatellite observations on March 2, 1996

The results of simultaneous analysis of plasma and magnetic field characteristics measured on the INTERBALL/Tail Probe, WIND and Geotail satellites on March 2, 1996, are presented. During these observations the INTERBALL/Tail Probe crossed the low-latitude boundary layer, and the WIND and Geotail satellites measured the solar wind’s and magnetosheath’s parameters, respectively. The plasma and magnetic field characteristics in these regions have been compared. The data of the Corall, Electron, and MIF instruments on the INTERBALL/Tail Probe satellite are analyzed. Fluctuations of the magnetic field components and plasma velocity in the solar wind and magnetosheath, measured onboard the WIND and Geotail satellites, are compared. The causes resulting in appearance of plasma jet flows in the low-latitude boundary layer are analyzed. The amplitude of magnetic field fluctuations in the magnetosheath for a studied magnetosphere boundary crossing is shown to exceed the magnetic field value below the magnetopause near the cusp. The possibility of local violation of pressure balance on the magnetopause is discussed, as well as penetration of magnetosheath plasma into the magnetosphere, as a result of magnetic field and plasma flux fluctuations in the magnetosheath.