Annotation of the results obtained in 2021
As part of the 2021 project’s stage, the team conducted research on the following topics: 1. Investigation of the dynamics of nanosatellites of various types with internal moving elements during the implementation of new and modified control schemes for its angular motion. 2. Simulation of the dynamics of the motion of coaxial bodies and spacecraft of variable composition, taking into account the complex laws of variation of inertial mass parameters. 3. Study of the dynamics of multi-rotor spacecraft with a variety of laws of control by internal rotors. 4. Dynamics of tethered towing of a passive satellite by an active spacecraft, taking into account the angular motion of the satellite. 5. Movement of satellites with large elastic panels in a stationary orbit through electrostatic interaction with an active spacecraft. 6. Investigation of the chaotic motions of a nanosatellite under the influence of an ion flow created by an active spacecraft. In accordance with the above research topics, the following description of the work performed in 2021 can be given. 1. The motion of a nanosatellite of the CubeSats format with elastic movable panels, taking into account aerodynamic damping, was investigated. A mathematical model describing the plane motion of a nanosatellite with deflectable elastic panels and internal movable elements in low Earth orbit was developed. An analysis of its attitude motion was carried out. The existence of chaotic motion modes was demonstrated using Poincare sections and Lyapunov exponents. The influence of the orbital altitude and geometric layout of the nanosatellite on the equilibrium positions and its type, as well as on the possibility of chaos, was investigated. Also, a study of the dynamics of the angular motion of a nanosatellite with a mobile module on extendable torsion bars was carried out using the canonical variables of Andoyer-Deprit. The appearance of chaotic regimes in the presence of small oscillations in the angle of the relative position of the mobile module and the possible elimination of chaos due to dissipative processes were shown. The type and layout of the internal gravitational damper of spatial motion energy based on the interaction with the external gravitational field of a small internal body floating in a spherical cavity with a viscous filling inside one of the modules of the nanosatellite body were proposed. Energy dissipation by means of a gravitational damper can be a mechanism for eliminating chaos in the dynamics of the spatial motion of nanosatellites. 2. The simulation and the qualitative study of the dynamics of motion of coaxial bundles of spacecraft and upper stages of variable composition, taking into account the complex laws of change in inertial-mass parameters, have been carried out. Within the framework of modeling and research, a mathematical model was built, and a number of cases of dynamics with optimal inertial-mass layouts were analyzed, when charges placed inside the spacecraft during burnout and change in their masses and volumes provide a natural approximation of the thrust vector, issued along the longitudinal axis of the spacecraft, to the precession axis, which is given direction of impulse output. To carry out this study, a generalization and development-modification of the method of qualitative study of the dynamics of angular motion of bodies of variable composition was carried out. This method allows to analyze and synthesize the nutation-precession motion of a spacecraft of variable composition with asymmetric dynamic layouts. The generalized qualitative method uses an analytical estimate of the averaged value of the curvature of the phase trajectory of the apex of the longitudinal axis of the spacecraft. It allows to formulate the conditions for the optimality of the inertial-mass configurations of the spacecraft that provide the dynamics of the longitudinal axis along the "twisting cone of nutation" and bring the longitudinal axis closer to the axis of precession, which is the target direction of delivery of the interorbital transient pulse. This approximation of the axis, in turn, provides an increase in the accuracy of interorbital transitions due to the natural properties of the dynamics of angular motion of a spacecraft of variable mass. 3. The modeling and the study of the dynamics of multi-rotor spacecraft during the initiation of nonholonomic internal connections arising from the implementation of direct closings of the internal rotors to each other were carried out. A nonholonomic model of the dynamics of the spatial motion of a multi-rotor system, which is a multi-rotor core of gyroscopic control of the angular motion of a multi-rotor spacecraft, was constructed. The modeling and study of the dynamics of the spatial motion of a multi-rotor spacecraft with "freezes", releases and / or direct closures of the internal rotors in their relative rotation with respect to the spacecraft body-frame with the corresponding imposition of nonholonomic kinematic connections was carried out. The modeling and study of the possibility of the spacecraft exit from the chaotic mode of its initial motion by means of initiation of direct rigid connections-connections of internal rotors with an instant exchange of the relative values of the angular momentum distributed on the rotors and the main body of the spacecraft was carried out. The modeling showed that such direct closures can provide a way out of the chaotic regime, which formulates a new mechanism for the instant elimination of chaos that has arisen in the dynamics of the angular motion of the spacecraft. 4. The dynamics of a space tether system consisting of an active spacecraft, an extended elastic tether, and a passive satellite, which is a rigid body, was investigated. An analysis of the forces and torques acting on the space tether system was carried out. A mathematical model of the system was built, and the influence of the active spacecraft’s engine thrust and the tether system parameters on the deformation of phase portraits and oscillations of the tether system was investigated. The analysis of chaotic motion of the tether system was carried out using Poiecare sections. Recommendations on the choice of the active spacecraft’s engine thrust excluding the possibility of the appearance of chaos were formulated. The problem of a tether deployment from a satellite located at the libration point L1 is considered. The equations of the space tether system were written under the conditions of the restricted circular three-body problem. The analysis of the tether system oscillations in the vicinity of the libration point was carried out. The energy integral and the period of the tether oscillations were found. A three-phase scheme for the tether deployment from the satellite in libration point was proposed. 5. A new restricted circular three-body problem for an electrically charged spacecraft moving near an oppositely charged satellite placed at the libration point was formulated and studied. It was shown that the presence of an electrostatic field leads to a splitting of the libration point for the charged spacecraft. New libration points were found, and their stability was investigated. The Jacobi integral was found in analytical form. Its correctness was confirmed by the results of numerical simulations. 6. A mathematical model describing the plane motion of a nanosatellite with a movable internal mass with a damper during ion beam assisted transportation by an active spacecraft was developed. The motion of the nanosatellite relative to its center of mass in the Kepler orbit was studied. The existence of chaos was demonstrated using Poincare sections. The ion beam control law, which transfers the nanosatellite to the required angular motion mode, was proposed. Three control strategies for the nanosatellite attitude motion during its ion beam assisted transportation by the active spacecraft were proposed. Comparison of fuel consumption in the absence of angular motion control and when using the described control strategies was carried out. A mathematical model describing the spatial motion of a passive space object under the influence of the torque generated by the ion beam of the active spacecraft was developed. Simplified equations describing the attitude motion of an axisymmetric object were obtained. Possible stationary modes of motion were found. The control law for the active spacecraft’s thrust, which transfers a passive object to a stationary mode, was proposed. For a passive object with given mass-geometric characteristics, the boundaries of applicability of the proposed control law were obtained numerically. The influence of the eccentricity of the orbit on the oscillations of the object relative to its center of mass was also investigated. The possibility of chaos in an elliptical orbit with a small eccentricity was shown. In the framework of these studies, original and modified existing mathematical models were developed, new control laws were proposed, and schemes of transport operations were developed. Based on the results of the work in 2021, 8 publications were completed in publications indexed in the Web of Science and Scopus databases (of which 6 are in the Q1 journals), and two more publications were accepted for publication in 2022, which will be indexed in Scopus / RSCI. A number of oral and online reports were made, including reports at the international conference "Second International Nonlinear Dynamics Conference NODYCON-2021" (Sapienza University, Rome, February 2021), at the All-Russian conference "XLV Academic Readings in Cosmonautics" (Moscow, March- April 2021), reports "The International MultiConference of Engineers and Computer Scientists - IMECS 2021" (Hong Kong, October 2021), as well as at the XV International IEEE scientific and technical conference "Dynamics of systems, mechanisms and machines" (Omsk, November 2021). Based on the results of the project, a monograph was prepared and accepted for publication in Elsevier in 2022: Aslanov V.S., Ledkov A.S. "Attitude dynamics and control of space debris during ion beam transportation."

 

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Annotation of the results obtained in 2019
As part of the 2019 project’s stage, the team conducted research on the following topics: 1. Simulation of the dynamics of movement of a 3U nanosatellite with a movable module and the development of control schemes for its angular motion. 2. Analysis of the angular motion of a small spacecraft, taking into account small perturbations. 3. Study of the motion dynamics of multi-rotor spacecraft during the implementation of complex control schemes for the gyrostatic rotor moments. 4. Study the chaotic motion of space tether system (an orbital space elevator) with the moving climber along the tether. 5. The study of the evolution of the motion of the space tether system in low Earth orbit. 6. Study of the dynamics and synthesis of the control laws of an active spacecraft with electrodynamic engines, during the transport of a passive space object by means of an ion flow. 7. Study of the dynamics and synthesis of the control laws of an active spacecraft during the transport of a passive space object using electrostatic interaction. 8. A new way to transport passive space objects in geostationary orbit due to gravitational interaction with a heavy orbital collector. In accordance with the above research topics, the following description of the work performed in 2019 can be given. 1. Simulation of the dynamics of movement of a 3U nanosatellite with a movable module and the development of control schemes for its angular motion: A complete mathematical model of the motion of the nanosatellite with a moving module, and a simplified linearized model were obtained. A design scheme for controlling the angular displacements of the movable module, which is based on the use of retractable flexible rods connecting the movable module with the main body in which the fixed-thrust jet engine is fixed, was proposed. This scheme allows the use of a functional element as a control body for the dynamics of spatial motion. Based on the above models and schemes, laws for controlling the dynamic parameters of the nanosatellite with the movable module were developed. Analytical estimates for the reorientation angles of the body in the inertial space during the implementation of the pulse control circuit were made. Preliminary modeling of the nanosatellite dynamics with complex laws of controlling the angles of the movable module relative position, taking into account feedback, was carried out. 2. Analysis of the angular motion of a small spacecraft, taking into account small perturbations: A mathematical model describing the motion of a small spacecraft taking into account its attitude motion relative and the influence of small perturbations was developed. A simplified mathematical model describing the angular motion of a small spacecraft was obtained using the linearization method. A comparison of the results of numerical integration of the full and linearized models was made. 3. Study of the motion dynamics of multi-rotor spacecraft during the implementation of complex control schemes for the gyrostatic rotor moments: Mathematical models for advanced modeling of special dynamics modes of multi-rotor spacecraft taking into account the presence of intrinsic gyrostatic moments of rotors and their changes were developed. The control laws for the relative rotation of the rotors to provide special modes of angular motion of the main carrier body, including the initiation of strange chaotic attractors in the phase space of a multi-rotor spacecraft, were found. Analytical expressions for determining the necessary initial conditions for the implementation of strange chaotic attractors in the system’s dynamics were obtained. 4. Study the chaotic motion of space tether system (an orbital space elevator) with the moving climber along the tether: The plane motion of an orbiting space elevator consisting of two space stations connected by a tether along which the climber moves was considered. A system of differential equations describing the motion of the mechanical system was obtained. The study of the climber motion influence on the oscillations of the heavy space station was carried out for a special case when the mass of one station significantly exceeds the mass of the second station and the climber, and the center of mass of the heavy station moves in a circular orbit. Using the Poincare sections, it was shown that the climber motion along the tether leads to the appearance of chaotic motions of the heavy station, which can lead to its rotation. The results of numerical modeling showed that uncontrolled sliding of the climber along the tether leads to its relative velocity when it reaches the space station can take large values, which can lead to an accident. For the orbital elevator with the given parameters, the moment of turning on the brake mechanism, which ensures the stop of the climber upon reaching the space station, was numerically determined. 5. The study of the evolution of the motion of the space tether system in low Earth orbit: The evolution of the orbit parameters of the space tether system under the influence of the aerodynamic drag of the atmosphere was investigated. The space tether system was considered as two material points connected by a massless inextensible rod. An approximate system of equations averaged over the period of oscillations of the tether describing the motion of the center of mass of the system was obtained. For an approximate system of equations, fast and slow motions were identified and averaging was performed over the fast variable. The system of approximate equations obtained in this way allows studying the evolution of the orbit of the center of mass of the space tether system experiencing atmospheric resistance over long time intervals. 6. Study of the dynamics and synthesis of the control laws of an active spacecraft with electrodynamic engines, during the transport of a passive space object by means of an ion flow: A mathematical model describing the plane motion of a mechanical system consisting of an active spacecraft equipped with two oppositely directed electrodynamic low-thrust engines and a passive space object was developed. For the case of movement of a cylindrical passive object in a circular orbit, control laws that provide stabilization of the passive object in an upright position were proposed: the control law of the thrust of an ion engine; the law of the axis of the ion flow. Comparison of the developed control laws showed that the control of the flow axis is more effective in terms of minimizing the stabilization time of the passive object, in addition, this law allows you to stop the rotation of the passive object. The chaotic motion of a cylindrical passive object in an elliptical orbit under the action of an ion flux was investigated for the case when its center of mass lies at the intersection of the axis and plane of symmetry. The presence of chaos in the system was proved by calculating the spectrum of Lyapunov exponents and constructing Poincare sections. The movement of CubSat3U nanosatellite in a low Earth orbit under the influence of an ion beam from an active spacecraft was considered. Spacecraft engine thrust control laws were proposed for the stages of approaching and transporting a nanosatellite to the boundary of the atmosphere. 7. Study of the dynamics and synthesis of the control laws of an active spacecraft during the transport of a passive space object using electrostatic interaction: The problem of transporting and stabilizing a passive space object in three-dimensional space due to the forces of electrostatic interaction with an active spacecraft was considered. The spacecraft was considered as a spherical body equipped with three engines oriented mutually perpendicularly, and the passive object was considered as an elongated cylinder. The equations of motion of the system were obtained, and the control law of the active spacecraft engines thrust was found, providing a constant distance between the centers of mass of the spacecraft and the passive object. The control law of the active spacecraft charge, which ensures stabilization of the angular motion of the passive object in equilibrium, was proposed. 8. A new way to transport passive space objects in geostationary orbit due to gravitational interaction with a heavy orbital collector: A fundamentally new scheme for transporting passive objects in geostationary orbit was proposed. It is based on the gravitational interaction of an object with an artificially created composite heavy orbital collector. The heavy collector is a massive body that acts as the Moon. The heavy collector flies up to a passive object. If a passive object falls into Hill sphere of the collector, it will either begin to move around the collector or will be attracted to its surface. After capturing the object, the engines of the collector are turned on, and it, together with the transported object, flies to the target orbit. There the object is discarded, and the collector flies to the next target. For the described mechanical system, a mathematical model that describes the plane relative motion of a passive object in the Cartesian Earth-Collector coordinate system was developed. The results of the numerical simulation confirm the feasibility of the proposed transportation scheme. Estimates of fuel consumption and the time required to move a passive object from a geostationary orbit to a circular disposal orbit were given. The permissible thrust force of the heavy collector's propulsion system was determined and recommendations on the formation of the appearance of a gravitational collector were formulated. During these studies, original and modified mathematical models were developed, new control laws were proposed, transport operations schemes were developed. Based on the results of the work, 5 articles were prepared for journals included in the Web of Science and Scopus databases, including 3 of them in Q1 journals (thus, taking into account doubling coefficients, the number of articles WoS/Scopus is 8), One article was prepared for a Russian-language journal indexed by RSCI. Four oral reports were made at international conferences and one report at an all-Russian seminar. Also in 2019, the thesis for the degree of Doctor of Physics and Mathematics was defended by the head of the project: A.V. Doroshin "Regular and chaotic dynamics of gyrostat satellites under the action of small perturbations." The dissertation was defended on October 17, 2019 at the Dissertation Council D 002.240.01 at the Institute of Problems of Mechanics named after A.Yu. Ishlinsky RAS.

 

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Annotation of the results obtained in 2020
As part of the 2020 project’s stage, the team conducted research on the following topics: 1. Study of the dynamics and possible chaotisation of the motion of a nanosatellite with a moving module and the development of control schemes for its angular motion. 2. Study of the chaotic motion of a nanosatellite with elastic aerodynamic surfaces at the motion in a rarefied atmosphere. 3. Search and analysis of strange attractors in the dynamics of the attitude motion of multi-rotor spacecraft. 4. Study of the dynamics of the orbital space elevator with moving climber in an elliptical orbit. 5. Deorbiting a rocket stage from low Earth orbit using an active spacecraft-tug 6. Movement of satellites with large elastic panels in a stationary orbit through electrostatic interaction with an active spacecraft. In accordance with the above research topics, the following description of the work performed in 2020 can be given. 1. Study of the dynamics and possible chaotisation of the motion of a nanosatellite with a moving module and the development of control schemes for its angular motion: A mathematical model describing the attitude motion of a nanosatellite with a mobile module in Andoyer-Deprit variables under the action of internal disturbances was constructed. It was used to analyze the presence of regular and chaotic attractors in the phase space of the system. On the basis of this analysis, the nanosatellite attitude motion control schemes were proposed. The analytical confirmation of the possibility of chaotic regimes was fulfilled on the base of the Melnikov’s method. A series of numerical simulations of the disturbed dynamics of a nanosatellite under the action of internal disturbances was carried out, which confirmed the possibility of realizing chaotic regimes. 2. Study of the chaotic motion of a nanosatellite with elastic aerodynamic surfaces at the motion in a rarefied atmosphere: A mathematical model describing the motion of a nanosatellite, taking into account its motion around the center of mass, and various external disturbances, was developed. A simplified mathematical model describing the plane oscillations of a nanosatellite with elastic aerodynamic surfaces in a low circular orbit under the influence of a rarefied atmosphere, was developed on the basis of this complete model. The undisturbed motion of the nanosatellite, in which the elastic panels together with the nanosatellite were considered as a single rigid body, was analyzed. An analytical expression for the potential energy of an unperturbed system was written. The equilibrium positions were found, and an analysis of the influence of the orbital altitude on the type of equilibrium positions was carried out. A bifurcation diagram describing the location and type of equilibrium positions of an unperturbed system depending on the orbit height was constructed. The areas where passive aerodynamic stabilization of the nanosatellite angular oscillations is possible due to the deflection of elastic aerodynamic surfaces, and where this stabilization is impossible, but chaotic motions can be realized, were identified. A series of calculations confirming the possibility of aerodynamic stabilization of the 3U CubeSat satellite was carried out. Chaotic oscillations of the nanosatellite were identified using Poincare cross sections. A critical height was found, above which aerodynamic stabilization is impossible due to chaos in the system. It was shown that the use of a pyramidal nose instead of a flat face and a shift of the center of mass closer to the nose make it possible to increase the critical height and lifetime of the nanosatellite in a low near-base orbit. 3. Search of conditions of strange attractors initiation in the dynamics of the attitude motion of multi-rotor spacecraft was realized. Based on the mathematical models developed at the stage of 2019, the search for strange chaotic attractors of a given type in the dynamics of a multi-rotor spacecraft was carried out. The synthesis of new types of strange attractors was performed. An analysis of the properties and characteristics of the emerging deterministic chaos was carried out. The effective software for multiprocessing parallel computer systems was designed for calculation of the Lapunov’s characteristics for phase trajectories along strange chaotic attractors. 4. Study of the dynamics of the orbital space elevator with moving climber in an elliptical orbit: The system of differential equations describing the motion of an orbital space elevator, consisting of two space stations connected by an elastic tether of variable length, along which the climber can move was obtained. It was assumed that the center of mass of the system moves in an elliptical orbit with a small eccentricity. The tether length control law, which ensures the asymptotic stability of the radial position of the orbital elevator, was proposed. Asymptotic stability was proved using the direct Lyapunov method. It was shown by Poincare cross-sections that chaotic motion modes may appear as a result of the presence of eccentricity or periodic oscillations of the climber. However, these modes are not an obstacle to the regular functioning of the system, since they are realized at large angles of the orbital elevator deviation from the radial position. Graphs showing the dependance of the ratio of the area of regular motion region to the area of a chaotic layer on Poincare sections on the eccentricity and the climber velocity were build. A kinematic control law of the climber motion, which transfers payload from one end of the orbital elevator to its other end, was proposed. The control laws of the space station's engine and the tether tension force when solving the problem of the orbital elevator folding were also proposed. 5. Deorbiting a rocket stage from low Earth orbit using an active spacecraft-tug: The transport operation of deorbiting the rocket stage from low Earth orbit using tether towing by an active spacecraft was considered. The transport operation includes three phases: tethered towing of the stage to the atmosphere boundary, uncontrolled descent in a rarefied atmosphere, and uncontrolled descent of stage fragments after its destruction. The equations of motion of the system were written, and the analysis of the influence of the stage motion parameters at the moment of separation from the tether on the moment of its breakup as a result of the action of dynamic and thermal loads was carried out. A numerical simulation of the Ariane 4 rocket stage deorbiting was carried out. Two scenarios were considered: when the stage breakup without an explosion, and when an explosion occurs. Fall areas for stage's fragments were plotted for both scenarios. The study showed that the area of the debris impact footprint is determined by the height of the stage breakup, which in turn depends on the stage motion in the rarefied layers of the atmosphere. It was found that a decrease in the stage angular velocity at the moment of separation from the tether reduces the height of the stage breakup and the area of the debris impact footprint. 6. Movement of satellites with large elastic panels in a stationary orbit through electrostatic interaction with an active spacecraft: A mathematical model describing the motion of a mechanical system consisting of an active spacecraft and a passive satellite with large elastic panels in the presence of electrostatic interaction between them was developed. The control law of the active spacecraft engines, which ensures the transport of a passive satellite, was proposed. Using Poincare sections, it was shown that the presence of elastic panels leads to chaos in angular motion of the passive satellite during its contactless transportation. During these studies, original and modified mathematical models were developed, new control laws were proposed, transport operations schemes were developed. Based on the results of the work, 5 articles were published in 2020 in journals included in the Web of Science and Scopus databases (including 2 in Q1 journals), and two articles were published in Russian journals indexed by RSCI. Three oral reports were made at international conferences.

 

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