THE USE OF MOBILE CONTROL METHODS FOR STABILIZATION OF A SPACECRAFT WITH AEROMAGNETIC DEORBITING SYSTEM

Annatation. The search for optimal control algorithms for spacecrafts is one of the key areas in rocket and space technology. Taking into account certain restrictions and requirements in a specific space mission, the selection of certain executive devices of the spacecraft is carried out and the corresponding control law is synthesized. One of such space missions is the providing of angular motion stabilization of a utilized spacecraft with aeromagnetic deorbiting system. The stabilization of spacecraft angular motion is needed for the orientation of aerodynamic element perpendicular to the vector of atmosphere dynamic flux with the aim of increasing of aerodynamic braking force. In this mission, the main optimization criterion is the minimization of the on-board electrical energy consumption which is needed for the control of angular motion. The original construction of the aeromagnetic deorbiting system consists of aerodynamic flat sails element and executive control devices with permanent magnets. However, not all spacecraft can be equipped with additional executive control devices with permanent magnets. That’s why with the aim of expansion of aeromagnetic deorbiting system application, using extra source of electromagnetic control executive devices is proposed in this research. The purpose of the article is the search of the control law which provides minimal consumption of electrical on-board energy by electromagnetic control executive devices during long-term deorbiting mission. For satisfying this criterion of optimization using of mobile control methods to orientate the spacecraft with aeromagnetic deorbiting system are proposed in this investigation. Computer modeling of orbital motion of spacecraft with aeromagnetic deorbiting system show the efficiency of using proposed mobile methods for angular motion control which realized by electromagnetic devices – magnetorquers. It has been showed that because of using mobile control method consumption of on-board electrical energy significantly less than with classical approach. The advantages and disadvantages have been determined.

are the main source of space debris [1]. So, on November, 2019, about 14598 space debris objects (SDO) were catalogued by NASA [2]. In turn, the highest concentration of SDO is observed in the near circular Low Earths Orbits (LEO) with altitude under 2000 km and Geosynchronous High elliptical communication orbit up to an altitude of 20000 km at its apogee [3]. So, many methods and means were developed for decreasing the tendency of SDOs number growth.
Today, there are two main approaches of development methods and means for deorbiting of SDO. The first approach is based on creation of active deorbiting systems (ADS). ADS include space servicing vehicles, propulsion deorbiting devices, robotic manipulators for capturing of SDO, electromagnetic devices, system "LEOSweep" with ion beam etc. [4]. These systems are needed control during deorbiting operation and consume on-board electrical energy or fuel. Considering the fact of the opportunity of some spacecrafts control systems failure, propulsion and energy systems failure at the end of lifetime, the extra propulsion or electromagnetic ADS can't be used in all deorbiting space missions. As for space servicing vehicles equipped by robotic manipulators, system "LEOSweep", they are required extra launch to SDOs orbits. In turn, extra launches are required a lot of costs and risks, which make this technology of deorbiting expensive nowadays. Paid attention to these disadvantages of ADS the second approach of deorbiting systems creation had been developed. This approach is based on using passive deorbiting systems (PDS) which practically don't require consumption fuel and on-board energy of spacecraft. PDS are aerodynamic deorbiting systems, electrodynamic tether systems, plasmodynamic deorbiting systems and solar sails. However, when using PDS the process of deorbiting takes a long time. That's why, there are some difficulties which are connected with PDSs reliability assurance during longtime missions. Moreover, some PDS require significant mass and volume for providing the efficiency of their usage. For example, aerodynamic inflatable deorbiting systems requires extra boost system for inflating and deployment, which takes extra useful volume of spacecrafts. Furthermore, during longtime exploitation casing of inflatable element is influenced by space environment factors (atomic oxygen, atmospheric electricity, etc.) which can damage the casing. From the other hand, there is aerodynamic deorbiting deployable sailing systems which don't need extra boost system and less affected by the space environment. But, unlike inflatable deorbiting systems, the high effectiveness of deployable sailing systems is achieved when aerodynamic sailing element is orientated perpendicular to the vector of atmosphere dynamic flux, which increases the aerodynamic braking force.
Analysis of previous researches. Considering these advantages and disadvantages of ADS and PDS the new approach of development hybrid deorbiting systems was proposed [5][6][7][8]. One of these systems is aeromagnetic deorbiting system (AMDS) which consists of two modules: aerodynamic flat sailing element and electromagnetic control system [7]. In the first modification electromagnetic control system AMDS consists of rotational permanent magnets and step electrical engines which rotate these magnets for change the polarity. The study shows that the consumption of electrical energy when using this control systems significant less than using electromagnets [7]. But  The cinematic equations can be presented in the following form: cos sin sin sin cos cos cos sin cos , cos sin cos where , ,    -Krylov angles (yaw, roll, pitch).
Synthesis of mobile control law for spacecraft with aeromagnetic deorbiting system. For the synthesis of the controller, using the feedback linearization method, it is convenient to represent a nonlinear mathematical model of the relative motion of the spacecraft (1) in the form of a state space in the following discrete form: For the synthesis of control law in each channel the pole placement method is proposed in this problem [9][10][11]. That's why the matrix and the perturbation vector are not taken into account in discrete equation (3). At the same time, these perturbations are taken into account in the model (1). In turn, the efficiency of the controller is determined by the bandwidth and the ability to compensate for the disturbance.
So, the linear control vector can be represented as follows: where K -matrix of gain coefficients.
In turn, this matrix of gain coefficients with using binomial distribution can be written as follows: , The converting from linear model (6) to real nonlinear model is realized with the transformation: where J -diagonal matrix of inertia with of the spacecraft with aerodynamic sailing flat element; , , tan sin tan cos cos sin sec sin sec cos where control -I, II, III -3 main algorithms which will be used in mobile control law; . , , x y z    -the functions of the sign, which provide necessary polarity of magnetic dipole moments of magnetorquers coils.
In turn, the switch function which change the control -I, II or III at the certain point in time depends on initial maximum error can be written as:  Orbital translational motion of spacecraft center of mass can be described with the equations in [14] or [15]. So, the orbit with next parameters is chosen for the simulation: eccentricity is 0.0001; altitude is 600 km; inclination is 60 degree.
Considering aerodynamic and gravitational perturbations, model [14], formulas (1)-(10) with using atmosphere model [16], the modeling of orbital motion during three axial rough stabilization of spacecraft with aeromagnetic deorbiting system by magnetorquers was carried out. The values of the regulator performance  0 are 0.012 in each channel. With the maximum value of yaw, roll, pitch errors of 0.2 radian it has been obtained following values of stabilization by each angle for period of 10800 s ( fig. 2-4).
Based on the analysis of the simulation results ( fig. 2-4), it can be concluded that the system remains stable in all three angles at a given control period. Despite the fact that the transition time in some channels is significant    Considering, the fact the magnetorquers is the electromagnets, the magnetic dipole moment is calculated by: m p I S N    (11) where I -the value of supply current for electromagnet; S -cross section area of the coil; N -number of turns.
So, the energy consumption of on-board energy can be calculated with using classical formula for electrical energy: where U -the value of supply voltage for electromagnet; work t -the time of magnetorquers work.
Thus, the full consumption of on-board electrical energy with using formulas (11) and (12) was calculated during simulation and amounted to 0.00107 kWh per the period of control. In turn, when using approach of classic continuous control, the consumption of on-board electrical energy is about 0.04 kWh for magnetorquers of the same class. Thereby, it can be concluded that the consumption of on-board electrical energy during using of mobile control law more than when using devices with rotating permanent magnets in [7], but significantly less than during using approaches of continuous control.
Based on this, mobile control methods can be used for spacecraft that cannot be equipped with devices with rotating permanent magnets, which extends the boundaries of the effective use of aeromagnetic deorbiting systems.
Conclusions. 1. The synthesis of mobile control law for stabilization of spacecraft with the aeromagnetic deorbiting system has been carried out in this research. With using of linear regulator, nonlinear transformation and algorithms of mobile control, the mobile control law for stabilization by magnetorquers was developed.
2. Computer modeling has shown the feasibility of using the synthesized law of mobile control to stabilize the spacecraft with aeromagnetic deorbiting system in long-term deorbiting missions. The use of this control law has shown sufficient values of stability and quality of control to ensure rough stabilization, which meets the conditions of the task.