MATHEMATICAL MODEL FOR GEOSTATIONARY SPACECRAFT DISTURBING TORQUES DETERMINATION

Толық мәтін

Introduction. Since 2011 in Russia on the basis of modern platforms of medium (”Express 1000H”) and heavy (”Express-2000”) classes several geostationary spacecrafts (SC) of communication have been launched: ”Luch 5A”, ”Luch 5B”, ”Luch 5B”, ”Express AM5”, ”Express AM6”, ”Express AT1”, ”Express AT2”, ”Yamal-300K”, ”Yamal-401”, etc. In the course of a SC normal operation it experiences different disturbance torques which are compensated by orientation and stabili- zation active control system with flywheels application. Such system is very widespread among relay satellites [1-8]. The main advantage of flywheels systems is an opportunity to create the operating moments in external force fields absence or when this field is the disturbing factor. Also, flywheels do not expend a working body as jet engines do. But at disturbance torques long influence, flywheels rotation speed increases. Flywheels rotation maximal speed restriction leads to unloading due to jet engines inclusion [2]. Therefore it is necessary to have a fuel reserve for flywheels unloading during the whole term of active existence on modern SCs. The authors of works [9-13] presented the methods of geostationary SC flywheels unloading by solar pressure force use that allows to save the fuel. At the same time one of the speci- fied methods first problems is the definition of a system kinetic moment which consists of a SC kinetic moment and a flywheels kinetic moment. The SC kinetic moment is calculated by multiplication of a SC angular velocity of rotation by SC inertia moments, the flywheels kinetic moment is calculated by multiplication of flywheels rota- tion speed by a flywheel inertia moment. Using this method of SC kinetic moment determination an internal kinetic moment (for example, from the thermal regulation system [14]) and an error of SC inertia moments knowl- edge are not considered. As shown in [15], one more error source is the influence of unknown external disturbing torques. All these errors, at accurate pointing and stabili- zation of a SC, are compensated by a flywheel control system. And then it turns out that a flywheel kinetic moment consists of the cumulated kinetic moment from external disturbing forces and errors of SC total kinetic moment knowledge. In a control system which does not consider these errors unloading is performed less effectively. The purpose of this work is to improve unloading ef- fectiveness by solving the following tasks: 1) the development of a mathematical model which will allow to define authentically a cumulated kinetic moment from the external disturbing forces on flywheels; 2) to confirm mathematical model adequacy accord- ing to telemetric data from the SCs in flight. A system model. Let us introduce the following right orthogonal systems of coordinates: 1) inertial coordinate systems (ICS), which corre- sponds to the international celestial coordinate system [16] and is indicated by OIXIYIZI (fig. 1); 2) coupled coordinate system (CCS) of OXYZ, its origin is in the spacecraft cog, the axes are its principal central axes of inertia (fig. 1). Fig. 1. Inertial and coupled coordinate systems Рис. 1. Инерциальная и связанная системы координат Fig. 2. The vector of the kinetic and disturbing torques in the inertial and coupled coordinate systems Рис. 2. Векторы кинетического и возмущающих моментов в инерциальной и связанной системах координат For a SC in the coupled coordinate system the law of a kinetic moment change is valid [1; 15]: Hɺ + ω ´ H = Μdist + Μctrl , (1) where H = J·ω - SC total kinetic moment; a derivative is indicated by a point above H, N·m/s; J - matrix of SC inertia tensor, kg · m2; ω - SC angular velocity in an iner- tial coordinate system, rad; Mdist and Mctrl - external forces moment and operating moment respectively, Nm. Let us consider the geostationary SC of communica- tion operating in a normal mode. Such SC characteristics are: 1) SC uniform circular orbiting with a day period; 2) SC stabilization and orientation to the given point of the Earth surface; 3) total absence of the atmosphere impact and con- siderably reduced values of gravitational and magnet moments in comparison with low-flying SCs [17]. The angular velocity of SC rotation around an OZ axis presence follows from the first and second characteristics, which is calculated as - ω0 = 2π/86164 с = 7,27·10-5 rad/s. It allows to rewrite the equation (1) as a set of equations: From the listed characteristics and system (2) it fairly follows: 1) angular velocity ω0 on an OZ axis presence leads to the fact that the vector of a SC kinetic moment of H, remaining invariable in an inertial coordinate system, makes a circle in the coupled coordinate system (fig. 2); 2) SC rotation in relation to the external disturbing forces in the coupled coordinate system leads to the divi- sion of the external disturbing torque in the coupled coor- dinate system into a variable and a constant component (fig. 2). In fig. 2 two points of a SC orbit are presented. For each orbit point the mutual positioning of inertial (OIXIYIZI) and coupled (OXYZ) systems of coordinates is given. Projection of a vector of a SC kinetic moment H on an inertial coordinate system axis in which it re- mains invariable is done. Also this vector is projected on the coupled coordinate system axis from which the change of a vector H projection due to the angular SC turn, and respectively the coupled coordinate system, in an inertial coordinate system is visible. The similar situation is with the variable external disturbing torque, and is indicated as Mvar. The constant external disturb- ìHɺ X í Y ïHɺ + w0 H - 0 X Y Y w H = M dist + M ctrl , Y X X = M dist + M ctrl , (2) ing torque, on the contrary, rotates together with the SC coupled system, and is indicated as Mconst. For the con- venience the constant external disturbing torque is lo- î Z Z Z Hɺ = M dist + M ctrl , cated on the OX axis. The graphs of the projection of the kinetic moments H Подпись: External forceswhere ω0 - angular velocity of SC rotation, which is 7,27·10-5, rad/s. The third SC characteristic allows to draw a conclu- sion that the external disturbing torque will mainly occur due to the solar pressure. The solar pressure impact on a SC design is well described in [17; 18]. vector on the axis OX and OY are presented in fig. 3. The graphs of the projection of the kinetic moment vector on the axis OX and OY are well combined with formula (2). If we draw the graph with HX on the abscissa axis and HY on the ordinates axis, the graph will resemble the spinning spiral that is well shown in work [9]. Fig. 3. Graphs of the projection of the kinetic moment H vector on the axis OX and OY Рис. 3. Графики проекции вектора кинетического момента H на оси OX и OY Considering that flywheels form the operating mo- ment in the coupled frame [19-22], using (1) and the fact that the SC kinetic moment without flywheels remains Then from formulas (4) and (5) taking into considera- tion replacement t1 = t0 + ∆t follows: constant ( Hɺ + ω×H = const), the flywheels kinetic mo- ìH (t ) = (H (t ) + M var ×Dt )´ ment is determined as: t1 ï X XY 0 ï XY M const H (t ) = H (t ) + (Hɺ + ω ´ H - Mdist )dt = ï´ cos(w0 (t0 + Dt )) + Y , 0 ò ï t0 ïH (t ) = (-H w0 (t ) + M var ×Dt )´ (6) 0 = H (t ) - Mdist × Dt , (3) í Y XY 0 ï XY const where H(t) - flywheel kinetic moment, N·m/s; t0, t1 - initial and terminal time instant, s; ∆t = t1 - t0. To receive a set of equations of flywheels kinetic mo- ment in axes of the coupled frame we will do the same ï´ sin (w0 (t0 ï ï ïH (t ) = H + Dt )) + M X , w0 (t ) - (M var + M const )×Dt, [23] taking into consideration unperturbed motion: î Z Z 0 Z Z ìHX (t ) = HX (t0 )cos(w0t1 ) + ï ï+ H (t )sin (w t ) - M dist ×Dt, where HXY(t0) - kinetic moment vector projection in the XOY plane, N·m/s. As it was already emphasized in practice the external ï Y 0 0 1 X H í Y (t ) = -HX (t0 )sin (w0t1 ) + ï+ H (t ) cos(w t ) - M dist ×Dt, (4) disturbing torques knowledge is unknown, but flywheels kinetic moments are known. Therefore the inverse task - determination of the external disturbing torques on fly- ï Y 0 0 1 Y î Z Z 0 Z ïH (t ) = H (t ) - M dist ×Dt. As it was shown above the external disturbing torque involves a variable and a constant component, and then proceeding from fig. 2 the external disturbing torque is calculated: ì dist var M const ïM (t ) = M × cos(w t ) + Y , Подпись: 0 1 wheels kinetic moments on mathematical model (6) is considered further. External disturbing torques determination. The flywheels kinetic moment on an OZ axis does not depend on a kinetic moment of two other axes that allows to di- vide a problem of external disturbing torques determina- tion into two. At the same time external disturbing torques determination on an OZ axis comes down to the problem X XY ï ï dist var w0 × Dt M const of approximation which solution can be found in [24; 25] and does not represent complexity and interest. íMY ï (t ) = -M XY ×sin (w0t1 ) + X , w0 ×Dt (5) The detailed consideration of the mathematical model (6) allows to reveal the known kinetic moment depend- ïM dist (t ) = M var + M const . ence on unknown parameters: M const , M const , M var , ï Z Z Z ï HXY and t0. X Y XY Let us write down unknown parameters in the matrix: presented by diagonal matrixes, with the main diagonal values according to tab.1. a = éëM const M const M var H t ùû . (7) Numerical modeling results. Filters model operation k X Y XY XY 0 There is no analytical solution of parameters (7) de- termination from the mathematical model (6), therefore the solution of a task comes down to numerical methods application and the problem of parameters (7) identifica- tion is considered with the use of mean-square indications processing by a filtration method. The Gaussian and the Kalman filters are widespread methods of filtration. The Gaussian filter is a recursive (iterative) filter for the linear estimation of system constant parameters. For the prob- lem with variable defined parameters solving the most known method is the Kalman filter. It is necessary for filters operation: 1. To define interrelation of a required parameters vec- tor and parameters measured on k iteration. In a general view this interrelation is defined from the ratio: yk = h (ak , t ) + Rk , (8) where h(αk, t) - function defining the model of measure- ments. Then from formula (6) formula (8) is written as: é(H (t ) + M var ×Dt )´ ù was performed in mathematical computing environment GNU Octave (ver. 4.2.1, John W.Eaton research, etc.). The modeling method includes: 1) initial parameters setting; 2) SC telemetric information from the database proc- essing; 3) averaging of kinetic moments from flywheels at the period of 1 minute; 4) calculation of filter parameters according to sense telemetric information; 5) filter operation results conclusion. Modeling was performed according to the data of telemetric information from the flying SC of average (”Express-1000H” platform) and heavy (”Express 2000” platform) classes. For the middle class SC modeling was performed for the interval from 2 o’clock 23.04.2017 till 10 o’clock 28.04.2017, for the heavy SC - from 0 o'clock 9.10.2017 for 6 o’clock 11.10.2017. In fig. 4 the graphs of calculation of the difference be- tween the kinetic moment from flywheels and estimation of the kinetic moment for the middle class spacecraft on mathematical model are given. In fig. 5 similar graphs for heavy class SC are given. Mean-square deviation ê XY 0 ê XY ú M const ú (MSD) - σ designation is introduced in all drawings. ê´ cos(w0 (t0 + Dt )) + Y ú The graphs analysis showed that with the increase in X = éH (t )ù ê ê H (t )ú ê w0 ú + R . (9) ú k var measurements the difference between flywheels kinetic moments and its estimation decreases and does not exceed ë Y ûk ê(-HXY (t0 ) + M XY ×Dt )´ ú ê M const ú 0.2 N·m/s for the Gaussian filter (fig. 4, a and 5, a) and ê´ sin (w0 (t0 + Dt )) + X ëê w0 ú úûk 0.06 N·m/s for the Kalman filter (fig. 4, b) and 5, b). In fig. 6 and 7 the graphs of calculation of the external disturbing torques respectively by the Gaussian and the The matrix of the previous estimation transfer to a new point will be single: Фk = E2x5. 2. To linearize a set of equations (6). After lineariza- tion the system is as follows: U = ¶h (ak , t ) = k ¶a = ê é1 w 0 t ×cos(u) cos(u) -(H + M const ×Dt )×w ×sin(u)ù Подпись: 0 Kalman filters for the middle class SC are presented. In fig. 8 and 9 similar graphs for the heavy class SC are pre- sented. The errors convergence confirms the accepted mathe- matical model adequacy (6). Convergence time equal to one turn should be noted, that corresponds to one day of a SC on the GSO. Disturbing torques computing results are presented in XY XY 0 ú Подпись: 0 ê 0 1 w t ×sin(u) -sin(u) (-H + M const ×Dt )×w ×cos(u)ú tab. 2. ë 0 XY XY û The disturbing torques error by the Gaussian filter is where u = ω0 (t0 + ∆t). 3. To set errors matrixes. The Kalman filter accuracy depends on the chosen values of the inverse coefficients matrix, initial values of an error matrix, a motion model error matrix and a measurements noise matrix in a com- plex way. Analytically such dependence can not be ob- tained, therefore such matrixes are manually selected and less (not more than 0,9 %) than by the Kalman filter (not more than 2 %) in some cases, however, the difference between the measured values of the kinetic moment and filters estimation is more. It is explained by smaller sensi- tivity of the Gaussian filter to the deviation of parameters measured from the predicted value unlike the Kalman filter. Table 1 Values of error matrixes main diagonals Error matrix name Value Measurements (Kψ) 0.00012; 0.00012 Required values (Kα) 0.012; 0.012; 0.012; (10-8)2; 12 System models (Qk,)* (10-10)2; (10-10)2; (10-10)2; (10-10)2; 12 * Only for the Kalman filter. а b Fig. 4. The Graphs of the calculation of the difference between the kinetic moment from the flywheels and the estimation of the kinetic moment for the spacecraft of the middle class: a - the Gaussian filter; b - the Kalman filter Рис. 4. Графики расчета разницы между кинетическим моментом с маховиков и оценкой кинетического момента для КА среднего класса: а - фильтр Гаусса; б - фильтр Калмана а b Fig. 5. The Graphs of the calculation of the difference between the kinetic moment from the flywheels and the estimate of the kinetic moment for the spacecraft of the heavy class: a - Gaussian filter; b - the Kalman filter Рис. 5. Графики расчета разницы между кинетическим моментом с маховиков и оценкой кинетического момента по математической модели для КА тяжелого класса: а - фильтр Гаусса; б - фильтр Калмана Fig. 6. The graphs of the disturbing torques calculation by the Gaussian filter for the middle class spacecraft Рис. 6. Графики расчета возмущающих моментов фильтром Гаусса для КА среднего класса Fig. 7. The graphs of the disturbing torques calculation by the Kalman filter for the middle class spacecraft Рис. 7. Графики расчета возмущающих моментов фильтром Калмана для КА среднего класса Fig. 8. The graphs of the disturbing torques calculation by the Gaussian filter for the heavy class spacecraft Рис. 8. Графики расчета возмущающих моментов фильтром Гаусса для КА тяжелого класса Fig. 9. The graphs of the disturbing torques calculation by the Kalman filter for the heavy class spacecraft Рис. 9. Графики расчета возмущающих моментов фильтром Калмана для КА тяжелого класса Table 2 Disturbing torques computing results, Nm Disturbing torque name Gaussian filter Kalman filter Value MSD Value MSD Middle class SC Constant disturbing torque M const X -1.5·10-5 1.6·10-8 (0.1 %) -1.8·10-5 1.2·10-8 (0.1 %) Constant disturbing torque M const Y 7.0·10-7 6.6·10-9 (0.9 %) -1.2·10-7 2.4·10-9 (2.0 %) Variable disturbing torque M var XY 2.1·10-6 1.3·10-9 (0.1 %) 2.1·10-6 1.3·10-8 (0.6 %) Heavy class SC Constant disturbing torque M const X -1.4·10-5 1.3·10-7 (0.9 %) -6.8·10-6 3.4·10-8 (0.5 %) Constant disturbing torque M const Y 5.6·10-5 5.5·10-8 (0.1 %) 5.4·10-5 3.9·10-8 (0.1 %) Variable disturbing torque M var XY 1.4·10-5 5.5·10-8 (0.4 %) 1.3·10-5 6.7·10-8 (0.5 %) Conclusion. As a result of the work done the mathe- matical model considering the spacecraft kinetic moment change, disclosing the interrelation of the flywheel control system kinetic moment and the external disturbing torque was developed. The model differs from the known models for the fact that the constant and variable component of the external disturbing torque is considered separately . A model adequacy check was performed according to tele- metric information of middle and heavy class SC by mean-square indications processing methods - by the Gaussian and the Kallman filters. Mean-square deviation of the disturbing torques as- sessment is less than 0,9 % for the Gaussian filter and less than 2 % for the Kalman filter. The obtained mathematical model of the spacecraft external disturbing torques determination allows to pre- dict the change of a flywheels kinetic moment control system that can be used for flywheels unloading effec- tiveness improvement and respectively to fuel consump- tion decrease.

Авторлар туралы

S. Latyntsev

JSC “Academician M. F. Reshetnev “Information Satellite Systems”

Email: lat.sv@mail.ru
52, Lenin Str., 52, Zheleznogorsk, Krasnoyarsk region, 662972, Russia Federation

A. Murygin

Reshetnev Siberian State University of Science and Technologies

31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation

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