The results of repeated processing of magnetic measurements carried out on the Photon-12 satellite (which was in orbit from September 9 to 24, 1999) are described. The processing was performed in order to reconstruct the uncontrolled rotational motion of this satellite. In re-processing, the simplified mathematical model of rotational motion was used. The actual orbit of a satellite (the apogee height is 380 km, the perigee height is 220 km) is replaced by a circular orbit; the expression for the aerodynamic moment acting on a satellite is simplified. The system of differential equations underlying the new model is autonomous and proved to be sufficiently accurate to reconstruct the satellite motion based on magnetic measurements in the case in which the satellite angular velocity was not very low and grew gradually. In the last third of the flight, when the satellite motion became virtually stable and had a sufficiently high angular velocity, this system could be reduced to a generalized-conservative system. Such a reduction makes more definite the set of its solutions suitable for approximate description of the actual motion of a satellite. In some segments of motion, the combining of which covers about 3 days, it was possible to use, for this purpose, the periodic solutions continued from the Lyapunov periodic solutions.

The results of reprocessing of magnetic measurements performed on the Foton-12 satellite (in orbit on September 9–24, 1999) are described. The processing was carried out in order to reconstruct the uncontrolled rotational motion of this satellite. The reprocessing used a simpler mathematical model of the rotational motion than the processing performed shortly after the flight. The simplifications were made in such a way that the new model was accurate enough to match the models used by V.V. Beletskiy when studying the evolution of the uncontrolled rotational motion of a satellite in the case in which this motion is close to the regular Euler precession of an axisymmetric rigid body. The results of processing measurements in 25 short (3.5 h) time intervals were directly compared with the averaged equations arising in Beletskiy’s theory. The evolution of the Foton-12 motion is well described by these equations in the second half of the flight, when the angular velocity of the satellite was within 0.8°–1.0°/s.

The rotational motion of an axisymmetric artificial satellite with a constant magnet under an effect of a torque produced by the Earth’s magnetic field (EMF) influence on the magnet is studied. The orbital motion of a satellite is calculated taking into account the noncentral nature of the Earth’s gravitational field and the atmospheric drag; the proper magnetic moment of a satellite is parallel to the axis of symmetry. The steady-state motions of a satellite are constructed, in which the axis of symmetry makes a small angle with the EMF intensity vector. The IGRF model is used as the EMF model. It is shown that such motions can be approximated by a sequence of periodic solutions of modified equations of motion. Steady-state motions contain two basis frequencies—the orbital and angular velocity of the Earth’s rotation. Periodic solutions have an orbital period, but the spectrum of the approximating sequence composed of them virtually coincides with the spectrum of the original steady-state mode.

The results of the Sreda–MKS space experiment showed that visual data on the vibrations of the ISS structural elements allow one to acquire quantitative characteristics of these vibrations. Such characteristics were found as a result of the analysis of time series obtained by tracing some objects in the ISS structure on video sequences. The numerical data are the vertical and horizontal coordinates of the selected point of the structural element in the frame expressed in pixels. Analysis of the time series allows us to reconstruct the actual dependence of these coordinates on time. In some cases, this dependence is oscillatory in nature and can be represented as the sum of a finite number of harmonics, the amplitudes and frequencies of which are determined by means of spectral analysis.

The uncontrolled rotational motion of the Progress M-24M and M-25M cargo spacecraft in the mode of one-axis solar orientation (the so-called “spin on the Sun”) has been reconstructed. The initial conditions of motion in this mode are the spinning of the spacecraft with an angular velocity of 2.2–2.4°/s around the normal to the sunward plane of the solar arrays. The duration of the mode is several orbits. The reconstruction was carried out using an integral statistical method based on telemetric values of the current taken from its solar arrays. As a result, the rotational motion of the spacecraft relative to the Earth–Sun direction is determined. A study of the spectrum of current oscillations during spinning was carried out, which explains the possibility of reconstruction based on such, at first sight, little informative data.

The results of the reprocessing of magnetic measurements described below were performed with onboard sensors on the Foton M-2 satellite (orbiting May 31 June 16, 2005). Processing was carried out in order to reconstruct the uncontrolled rotation. The reprocessing used a simpler mathematical model of rotational motion in comparison with the processing carried out immediately after the flight. Simplifications were made so that the new model matched the models used by V.V. Beletsky in his theoretical analysis of the evolution of the uncontrolled rotational motion of a satellite in the case when this motion was close to the regular Euler precession of an axisymmetric rigid body. As a result, due to some coarsening of the reconstruction, it was possible to directly compare the obtained experimental results with the theory and analyze them.

The possibility of stabilizing the gravitational orientation mode of a massive artificial Earth satellite (AES) by the torque produced by electromagnets interacting with Earth’s magnetic field is shown. As an example, the control of rotational motion of the satellite like Bion M-1 and Foton M-4 is considered. Control is accomplished by changing the currents in the electromagnets. The control law is considered, which provides damping of the disturbed motion of a satellite and its stabilization in the gravitational orientation mode. To form this law, it is sufficient to have the readings of a triaxial magnetometer and of an angular velocity sensor. The control law efficiency is verified by mathematically modeling satellite motion relative to the center of masses, under an effect of gravitational and aerodynamic torques, as well as the torque produced by electromagnets.

We develop a mathematical model of a dynamical testbed for testing accelerometer assemblies created at the Keldysh Institute of Applied Mathematics. It has a movable platform with one degree of freedom. It can rotate about the axis fixed in the prescribed direction. The tested accelerometer assemblies are placed in the platform that rotates arbitrarily. Its real motion is reconstructed a posteriori by measurements of the vision system. The reconstruction is carried out in digital form and allows us to calculate the real acceleration felt by the assemblies. The calculated acceleration is compared with the measured one. The comparison results are used to test and calibrate sensors, electronic units, etc.

Angular motion of an axisymmetrical artificial satellite with a permanent magnet in the real magnetic field of the Earth was investigated. The satellite orbit was calculated taking into account the major perturbing factors, and the satellite dipole moment was parallel to the axis of symmetry. Satellite stationary motions were constructed, where this axis constituted a narrow angle with the Earth's magnetic field vector. Possibility was demonstrated of approximating such motions by a sequence of periodic solutions of modified motion equations. Sequence composed of such solutions also approximated the spectrum of stationary motion.

We consider the motion of an artificial satellite of the Earth of the Progress and Foton-M4 type in different variants of the orbital orientation mode in low Earth orbit. A gyro system (a set of reaction wheels or control moment gyros) is used as the executive part of the satellite control system. The gravitational and restoring aerodynamic torques acting on the satellite are taken into account. Based on the principles of the proportional-differential regulator, we construct the control laws for the gyrostatic momentum, which allow, without its accumulation, maintaining a long and sufficiently accurate orientation of the satellite in the vicinity of gravitationally stable and unstable rest positions that exist in the simplified problem (the satellite moves in a fixed circular orbit under the action of the gravitational torque only; the gyrostatic momentum is zero).

The uncontrolled rotational motion of Progress MS-07 and Progress MS-08 transport cargo spacecraft in the mode of gravitational orientation of a rotating satellite was reconstructed. Modes were implemented in April and August 2018. The reconstruction was conducted using an integral statistical method according to measurements of spacecraft angular velocity. Measurement data obtained at a certain time interval were processed together with the least squares method by integrating the equations of spacecraft motion relative to the center of mass. As a result of processing, the initial conditions of motion and parameters of the used mathematical model were estimated. The correctness of the reconstruction was checked by measuring the current taken from the solar panels. In the mode of gravitational orientation, the spacecraft rotated around its longitudinal axis with an angular velocity of 0.1–0.2 deg/s oscillating relative to the local vertical.

The paper presents the reconstruction results of rotational motion of the AIST-2D small spacecraft by onboard measurements of vectors of angular velocity and the strength of Earth’s magnetic field obtained in summer 2016. The reconstruction method is based on the reconstruction of kinematic equations of the rotational motion of a solid body. According to the method, measurement data of both types collected on a certain time interval are processed together. Measurements of the angular velocity are interpolated by piecewise-linear functions, which are replaced in kinematic differential equations for a quaternion that defines the transformation from the satellite instrument coordinate system to the inertial coordinate system. The obtained equations represent the kinematic model of the rotational motion of a satellite. A solution to 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 Earth’s magnetic field and its calculated values. The initial conditions of the approximating solution, constant bias in angular velocity measurements, and angles specifying the matrices of transformation from magnetometer intrinsic coordinate systems to the instrument coordinate system of the satellite (measurements of the angular velocity are specified in it) are refined. The described method makes it possible to reconstruct the actual rotational motion of a satellite using one solution of kinematic equations over time intervals longer than 10 h.

A second-order differential equation containing a large parameter is considered. Such an equation can be interpreted as an equation of constrained oscillations of a mechanical system with one degree of freedom, provided that the fundamental frequency of the system substantially exceeds the external frequency. We provide a new proof of the existence of a periodic solution of that equation such that it is close to the periodic solution of the corresponding degenerate equation. That proof is obtained by means of the Poincaré method.

We investigate the periodic systematic error found in ground-based measurements of the BOKZ-M60 star sensor. This error can be explained by the periodic pixel structure of the sensor CCD matrix. Our conclusion is based on processing a sufficiently long series of measurement data. The measurements are processed in a few stages. First, we approximate the stellar sky motion on the matrix plane using an appropriate mathematical model. Then, we trace the motion of some particular stars relative to the model sky motion. Using spectral analysis, we then find the periodic components of their motion. The period of those components allow us to make the conclusion given above.