RU2332334C1 - Method of semi-passive three-axis stabilisation of dynamically symmetric artificial earth satellite - Google Patents

Abstract

FIELD: transport.

SUBSTANCE: invention relates to space engineering and can be used for stabilisation of artificial Earth satellites (AES) using a geomagnetic field. In compliance with the proposed method, a double electrically static layer is distributed on over a part of the satellite surface. Because of the AES orbital flight the said layer interacts with the Earth magnetic field to generate a control moment of Lorentz forces that can, in fact, exceeds the moment of gravitation acting on larger AES with a dynamic symmetry approximating to a complete symmetry, or acting on a gravitation-oriented AES. To control orientation of AES with an axial dynamic symmetry, the magnitude and direction of the double layer dipole moment vector are changed to comply with projections of the AES centre of mass velocities and geomagnetic induction vector onto the axes of the orbital coordinate system. The derive law of varying the said vector allows stabilising the AES position nearby the point of equilibrium whereat the AES dynamic symmetry axis is aligned with the local vertical. Given the damping, this AES is stable under effects of gravitational and the like minor disturbances.

EFFECT: expanded range of application of the said AES orientation control method.

5 dwg

Description

The invention relates to the field of space technology and can be used to stabilize artificial Earth satellites (AES) by creating a controlled moment of Lorentz forces.

Known methods for controlling the orientation of a satellite by using forces of a different nature - reactive or external forces with respect to a satellite that create control moments. Factors external to the satellite that create control moments without the consumption of the working fluid are used in passive stabilization methods / 1 /. The use of passive methods of controlling the orientation of the satellite is preferable in cases when during the active phase of the flight the deviation of the satellite from a given position in space should not exceed several degrees, and also when complex program turns and counteraction to large disturbing moments are not required / 2 /.

Known methods and devices for passive orientation (stabilization) of satellites based on the use of natural geophysical factors, for example, gravitational forces, forces of interaction of a satellite’s magnetic field with the Earth’s external magnetic field (MPZ), solar radiation pressure, aerodynamic forces. However, the known methods are characterized by low efficiency either due to the fact that the creation of a control (stabilizing) moment requires significant structural complications, or due to the fact that the field of their practical application is quite limited. So, magnetic control systems (MCS), having significant dimensions, create small control moments and are a powerful source of magnetic fields on board an artificial satellite, which affects the performance of the ISU and reduces the efficiency and noise immunity of the ISU / 1 /. Methods using the pressure of solar radiation / 3 / can only be effective for satellites in high orbits / 1 /, and the corresponding devices / 4 / require large working surfaces. The use of aerodynamic systems based on the stabilizing effect of a stream of rarefied atmospheric layers incident on a satellite is limited by small orbital altitudes and a significant influence of the atmosphere on the orbit parameters, which reduces the effectiveness of the corresponding satellite control devices / 5 /. A common disadvantage of known methods for passive stabilization of satellites is the difficulty of changing the control moment and the limited use due to the inability to compensate for eccentric vibrations arising in elliptical orbits.

двойного слоя. При этом получение управляемого крутящего момента производят путем согласованного изменения величины и направления вектора

.A known method of controlling the orientation of the satellite / 6 /, including obtaining a controlled torque of Lorentz forces by distributing on the surface part of the satellite a double electrostatic layer with a given total dipole moment

double layer. At the same time, controlled torque is produced by a coordinated change in the magnitude and direction of the vector

.

The disadvantages of this method are as follows. The method is applicable for stabilization of a satellite only in a direct equilibrium position in an orbital coordinate system. The method does not work for satellites in orbits with inclinations greater than 60 °.

The known method is selected as the closest analogue to the claimed invention.

The objective of the invention is to expand the field of applicability of the method for controlling the orientation of satellites.

двойного слоя, а получение управляемого крутящего момента производят путем согласованного изменения величины и направления вектора

, удовлетворяющего условиямThe problem is solved in that in the known method for controlling the orientation of the satellite, including obtaining controlled torque of the Lorentz forces, in accordance with the invention, a double electrostatic layer with a given total dipole moment is distributed on a part of the surface of the satellite

double layer, and obtaining controlled torque is produced by a coordinated change in the magnitude and direction of the vector

satisfying the conditions

на главные центральные оси инерции ИСЗ, υ_Cξ, υ_Cη - проекции вектора

скорости центра масс ИСЗ относительно геомагнитного поля на оси орбитальной системы координат, B_Cξ, B_Cη, B_Cζ - проекции вектора

индукции геомагнитного поля в центре масс ИСЗ на оси орбитальной системы координат, ψ₀ - угол поворота ИСЗ относительно оси, совпадающей в режиме ориентированного движения с местной вертикалью, k - постоянный коэффициент.where P _x , P _y , P _z are projections of the vector

on the main central axis of inertia of the satellite, υ _Cξ , υ _Cη are projections of the vector

the velocity of the center of mass of the satellite relative to the geomagnetic field on the axis of the orbital coordinate system, B _Cξ , B _Cη , B _Cζ are projections of the vector

induction of the geomagnetic field in the center of mass of the satellite on the axis of the orbital coordinate system, ψ ₀ is the rotation angle of the satellite relative to the axis coinciding in the mode of oriented movement with the local vertical, k is a constant coefficient.

The technical result achieved by the invention is that the creation of a double electrostatic layer on the satellite surface leads to the excitation of the Lorentz forces acting on the satellite and exerts an orienting effect, and the fulfillment of conditions for a coordinated change in the polarization distribution ensures the existence and stability of the equilibrium position even in the presence of damping (provided by any of the known methods, for example, by using hysteresis rods) solves the satellite stabilization problem in orbit coordinate system.

The invention is illustrated by Figure 1, which shows the orbital coordinate system, which is the basic system for solving the satellite stabilization problem, Figure 2, which shows the satellite orientation angles in the orbital coordinate system, Figure 3, which shows the maximum possible values of the vector module the dipole moment of the double layer, Fig.4, 5, which presents the results of a numerical calculation of the stabilization process of a satellite in accordance with the invention and without it.

относительно МПЗ, характеризуемого магнитной индукцией

распределение на части поверхности ИСЗ с площадью S (далее - оболочка) двойного электростатического слоя с плотностью поляризации

приводит к возникновению момента

лоренцевых сил, определяемого по формулеThe invention consists in the following. For a satellite whose center of mass (point C) moves at a speed

with respect to magnetic induction characterized by magnetic induction

distribution of a double electrostatic layer with a polarization density on a part of the satellite surface with an area S (hereinafter referred to as the shell)

leads to a moment

Lorentz forces, determined by the formula

- суммарный дипольный момент оболочки, оказывающего при определенных условиях ориентирующее воздействие на ИСЗ и используемого в качестве восстанавливающего момента.Where

- the total dipole moment of the shell, which under certain conditions has an orienting effect on the satellite and is used as a restoring moment.

To obtain a reducing moment that exceeds the existing disturbing moments, it is possible to vary the parameters of the double electrostatic layer. Estimates show that for characteristic values of the satellite’s orbit parameters R≈7 · 10 ⁶ m, the area of the double electrostatic layer S ≈ 300 m ² , the strength E of the electrostatic field in the system of electrodes creating this double layer is E≈10 ⁷ V / m, and the thickness double electrostatic layer d≈0.05 m the moment of Lorentz forces created by the electrostatic layer will be of the order of 10 ^-4 N · m, which exceeds the moment of gravitational forces acting on a large satellite with a balanced distribution of masses, or on a satellite previously set to b Low to the position of the gravitational orientation.

которой направлена по касательной к орбите в сторону движения, ось

- по нормали к плоскости орбиты, ось

- вдоль радиуса-вектора

центра масс ИСЗ относительно центра Земли О_З. Здесь и в дальнейшем используются правые декартовы прямоугольные системы координат. Исследование проводится с учетом вращения орбитальной системы координат относительно инерциальной системы с угловой скоростью ω₀. В качестве инерциальной системы координат принимается система O_ЗX_*Y_*Z_*, ось

которой направлена по оси собственного вращения Земли, ось

- в восходящий узел орбиты, а плоскость (Х_*Y_*) совпадает с плоскостью экватора. Используется также жестко связанная с ИСЗ система его главных центральных осей инерции Cxyz (орты

Ориентация орбитальной системы координат относительно системы O_ЗX_*Y_*Z_* определяется на основании равенствTo determine the conditions under which the stabilized motion of a satellite is achieved, we consider a satellite whose center of mass moves in a gravitational field in a circular Kepler orbit. It is assumed that the Earth’s gravitational field is Newton’s central. The MPZ is approximated taking into account the dipole and quadrupole components / 8 /. We study the rotational motion of the satellite relative to its center of mass in the orbital coordinate system Сξηζ (Fig. 1) with the origin in the center of mass of the satellite, axis

which is directed along the tangent to the orbit in the direction of motion, the axis

- normal to the orbit plane, axis

- along the radius vector

center of mass of the satellite relative to the earth center O _W. Hereinafter, the right Cartesian rectangular coordinate systems are used. The study is carried out taking into account the rotation of the orbital coordinate system relative to an inertial system with an angular velocity ω ₀ . As an inertial coordinate system, the system O _З X _* Y _* Z _* , the axis

which is directed along the axis of the Earth’s own rotation, the axis

- to the ascending node of the orbit, and the plane (X _* Y _* ) coincides with the plane of the equator. A system of its main central axes of inertia Cxyz (unit vectors) is also used

The orientation of the orbital coordinate system relative to the system O _З X _* Y _* Z _* is determined on the basis of the equalities

- наклонение орбиты;

- аргумент широты, u=ω₀t.Where

- inclination of the orbit;

is the argument of latitude, u = ω ₀ t.

The orientation of the xyz axes relative to the ξηζ axes is determined by the matrix of guiding cosines α _i , β _i , γ _i (i = 1, 2, 3) so that the equalities

if you determine the orientation of the satellite in the orbital coordinate system using the "plane" angles φ, θ, ψ (Figure 2), then the elements of matrix A will take the form

During the motion of the satellite relative to the geomagnetic field, the interaction of the double electrostatic layer with the geomagnetic field leads to the excitation of Lorentz forces. The main moment of these forces relative to the center of mass of the satellite is determined by the formula

можно аппроксимировать выражением (2), гдеIt was shown in / 6, 9 / that for any real satellite and, moreover, for a satellite operating in modes close to oriented motion, the moment

can be approximated by expression (2), where

- скорость центра масс ИСЗ относительно геомагнитного поля,

- угловая скорость суточного вращения Земли,

- вектор магнитной индукции геомагнитного поля в центре масс ИСЗ, заданный своими компонентами в орбитальной системе координат.

- the speed of the center of mass of the satellite relative to the geomagnetic field,

- the angular velocity of the daily rotation of the Earth,

is the vector of the magnetic induction of the geomagnetic field in the center of mass of the satellite specified by its components in the orbital coordinate system.

определяется по формулеIn terms of the quadrupole model of the MPZ vector

determined by the formula

- часовой угол восходящего узла,Where

- the hourly angle of the ascending node,

- мультипольные тензоры 1-го и 2-го рангов, представляющие собой соответственно дипольный и квадрупольный магнитные моменты, выраженные через квазинормированные по Шмидту гауссовы коэффициенты

и

На эпоху 2000.0 гауссовы коэффициенты имеют следующие значения /10/ в нТл:

- multipole tensors of the 1st and 2nd ranks, representing, respectively, dipole and quadrupole magnetic moments, expressed through quasi-normalized Schmidt Gaussian coefficients

and

For the 2000.0 era, the Gaussian coefficients have the following values / 10 / in nT:

на оси x, y, z представим выражение (2) в видеTo find moment projections

on the x, y, z axis, we represent expression (2) in the form

Here υ _Сξ = R (ω ₀ -ω _З cosi), υ _Сη = Rω _З sin i cos u, υ _Cζ = 0.

при которых

обращается в ноль в программной ориентации ИСЗ, т.е. является восстанавливающим моментом в окрестности ориентации A₀. Очевидно, что вектор

должен удовлетворять условиюLet the satellite orientation in the orbital coordinate system be given by a certain value of the matrix of guide cosines A = A ₀ = const. In particular, the matrix A ₀ may be unity. This case was considered in / 6 /. Substituting A = A ₀ into expression (2), we find such values of the vector

under which

turns to zero in the program orientation of the satellite, i.e. is a restoring moment in a neighborhood of orientation A ₀ . Obviously, the vector

must satisfy the condition

определяются по формуламwhere k = k (t) is an arbitrary scalar function. Moreover, the components of the vector

determined by the formulas

будет являться восстанавливающим в окрестности заданного положения и может быть использован для поддержания заданной ориентации ИСЗ. Равенства (8) можно рассматривать как закон управления вектором дипольного момента оболочки для осуществления заданной ориентации ИСЗ. В зависимости от вида функции k(t) формулы (8) определяют тот или иной закон полупассивного управления ориентацией ИСЗ.Therefore, if the shell parameters P _x , P _y , P _z will change according to the law (8), then the moment

will be restoring in the vicinity of a given position and can be used to maintain a given satellite orientation. Equalities (8) can be considered as the law of controlling the vector of the dipole moment of the shell for the implementation of a given orientation of the satellite. Depending on the type of function k (t), formulas (8) determine one or another law of semi-passive control of the satellite orientation.

- среднее по времени значение

а P₀ - некоторое фиксированное значение величины

Выберем k исходя из условия

В этом случае на основании (8) получаемConsider the following law of semi-passive control with a constant value of k. Let be

- time average value

and P ₀ is some fixed value

We choose k based on the condition

In this case, on the basis of (8) we obtain

Where

To clarify the possibility of practical implementation of control (8), it is advisable to evaluate the modules of the functions P _x (t), P _y (t) and P _z (t) for the most real values of the parameters of the satellite and its orbit.

для параметров орбиты из диапазона 0≤i≤90°, 6800 км ≤R≤ 13200 км. Результаты расчетов показаны на Фиг.3 для граничных значений радиуса орбиты: кривая 1 соответствует R=6800 км, а кривая 2 - R=13200 км. Промежуточным значениям радиуса соответствуют кривые, заполняющие пространство между кривыми 1 и 2.Using a computer, the calculations of the maximum time values

for orbit parameters from the range 0≤i≤90 °, 6800 km ≤R≤ 13,200 km. The calculation results are shown in Fig. 3 for the boundary values of the orbit radius: curve 1 corresponds to R = 6800 km, and curve 2 corresponds to R = 13200 km. Intermediate values of the radius correspond to curves filling the space between curves 1 and 2.

принимает значения порядка 1 для орбит с произвольными наклонениями и, следовательно, практическая реализация рассматриваемого закона управления не вызовет принципиальных трудностей.Figure 3 shows that

takes values of the order of 1 for orbits with arbitrary inclinations and, therefore, the practical implementation of the control law under consideration will not cause fundamental difficulties.

Let us prove that semi-passive control (8) in the presence of damping in the satellite system solves the problem of stabilizing the satellite in the orbital coordinate system.

модельного типа, например пропорциональный относительной угловой скорости ИСЗ в орбитальной системе координат:

где

h_i>0 (i = 1, 2, 3),

а

- абсолютная угловая скорость ИСЗ. ТогдаLet there be a damping moment in the satellite system

model type, for example, proportional to the relative angular velocity of the satellite in the orbital coordinate system:

Where

h _i > 0 (i = 1, 2, 3),

but

- absolute angular velocity of the satellite. Then

M _Дx = -h ₁ (ω _x -ω ₀ β ₁ ), M _Дy = -h ₂ (ω _y -ω ₀ β ₂ ), M _Дz = -h ₃ (ω _z -ω ₀ β ₃ ).

We will also take into account the gravitational moment

as the greatest of disturbing moments. Here J = diag (A, A, C) is the satellite inertia tensor in the Cxyz axes, and the Cz axis is the satellite axis of symmetry.

The differential equations of the rotation of the satellite are constructed according to the Euler-Poisson scheme:

позволяет при наличии демпфирования обеспечить стабилизацию ИСЗ в положении равновесия, соответствующем любому наперед заданному значению угла ψ и нулевым значениям углов φ и θ. Для этого подставим в (8) значения φ=θ=0, ψ=ψ₀=const. Закон полупассивного управления ИСЗ в данном случае примет видFrom equations (9), (10) it follows that the satellite has a position of uniaxial gravitational orientation, in which the Cz axis is collinear to the Cζ axis (i.e., φ = θ = 0), and the positions of the Cx and Cy axes are not defined (the angle ψ can take any value). We show that the moment

allows, in the presence of damping, stabilization of the satellite in the equilibrium position corresponding to any previously given value of the angle ψ and zero values of the angles φ and θ. For this, we substitute in (8) the values φ = θ = 0, ψ = ψ ₀ = const. The law of semi-passive control of the satellite in this case will take the form

Let us prove that semi-passive control (11) in the presence of damping in the satellite system solves the problem of triaxial stabilization of the satellite in the orbital coordinate system. First, we consider the case of small values of the inclination i and prove the existence and asymptotic stability of the equilibrium position φ = θ = 0, ψ = ψ ₀ = const, and then we consider the case of arbitrary values of the inclination i and prove the stability of this equilibrium under constantly acting disturbances.

и

могут быть разложены в ряды по степеням этих малых величин. В результате получим их проекции с точностью до членов второго порядка малости в видеWhen considering small satellite oscillations, the assumption about the smallness of the angles φ, θ and ψ ₁ = ψ-ψ ₀ and their derivatives with respect to time is valid. Therefore, in the vicinity of the considered equilibrium position, the moments

and

can be arranged in series in powers of these small quantities. As a result, we obtain their projections up to terms of the second order of smallness in the form

M _Lx = l ₁₁ (t) φ + l ₁₂ (t) θ + l ₁₃ (t) ψ ₁ ,

М _Лy = l ₁₂ (t) φ + l ₂₂ (t) θ + l ₂₃ (t) ψ ₁ ,

M _Лz = l ₁₃ (t) φ + l ₂₃ (t) θ + l ₃₃ (t) ψ ₁ ,

Where

l ₁₁ (t) = - k [(υ _Cη B _Cξ -υ _Cξ B _Cη ) ² + (υ _Cξ cosψ ₀ + υ _Сη sinψ ₀ ) ² B ² _Cζ ],

l ₁₂ (t) = l ₂₁ (t) = - k (υ _Cξ cosψ ₀ + υ _Cη sinψ ₀ ) × (υ _Cη cosψ ₀ -υ _Cξ sinψ ₀ ) B ² _Cζ ,

l ₁₃ (t) = l ₃₁ (t) = k (υ _Cξ B _Cη -υ _Cη B _Cξ ) (υ _Cη cosψ ₀ -υ _Cξ sinψ ₀ ) B _Cζ ,

l ₂₂ (t) = - k [(υ _Cη B _Cξ -υ _Cξ B _Cη ) ² + (υ _Cη cosψ ₀ -υ _Cξ sinψ ₀ ) B ² _Cζ ],

l ₂₃ (t) = l ₃₂ (t) = - k (υ _Cξ B _Cη -υ _Cη B _Cξ ) (υ _Cξ cosψ ₀ + υ _Cη sinψ ₀ ) B _Cζ ,

l ₃₃ (t) = - k (υ ² _Cξ + υ ² _Cη ) B _Cζ .

Equations (9) in matrix form take the form

Where

нелинейным образом зависящими от φ, θ, ψ₁.X is a vector with components X _j (t, φ, θ, ψ ₁ )

nonlinearly depending on φ, θ, ψ ₁ .

(t), не зависящих от i и членов

содержащих множителем sin i. ТогдаThe components m _ij (t) of the matrix M, depending, generally speaking, on the small parameter sin i, can be represented as the sum of the terms

(t) independent of i and members

containing the factor sin i. Then

а

where M ^{(0) is} obtained from M by replacing l _ij (t) with

but

Consider the linear approximation system of equations (12) for i = 0:

Since for i = 0 the equalities υ _Cξ = R (ω ₀ -ω _З ) _hold , υ _Сη = υ _Сζ = 0, and also on the basis of (6)

то

Where

then

moreover, B _Cξ , B _Cη , B _{Cζ are} determined by formulas (14).

в виде

где

- среднее по t значение функции

ТогдаImagine

as

Where

is the average value over t of the function

Then

We calculate the necessary time averages:

As a result, the matrix M will be represented as

является положительно определенной для любых значений ψ₀. Если, кроме того, потребовать, чтобы выполнялось условие А>С, то матрицаSubstituting the numerical values of the Gaussian coefficients given above, it is easy to verify that, when k> 0, the matrix

is positive definite for any values of ψ ₀ . If, in addition, we require that condition A> C be satisfied, then the matrix

will also be positive definite. If the difference AC can take values of arbitrary sign, then when the inequalities

снова является положительно определенной и, следовательно, нулевое решение системыmatrix

is again positive definite and therefore a zero solution to the system

asymptotically stable.

и оценим ее норму.Next, we consider the matrix

and evaluate its norm.

то и нулевое решение дифференциальной системыSince the constant C ₁ depends only on the matrix

then the zero solution of the differential system

also asymptotically stable / 11 /. Moreover, this asymptotic stability is uniform, since the coefficients of system (18) are almost periodic in t. In addition, the exponential asymptotic stability of the zero solution of system (18) implies the exponentially asymptotic stability of the zero solution of the original nonlinear system (12) for i = 0/12 /. This proves the possibility of semi-passive stabilization of a satellite at i = 0.

In the case of small inclination orbits (i ≠ 0, but sin i is small), it is convenient to write the differential equations of perturbed motion (12) in the form

According to the stability theorem for constantly acting perturbations / 12 /, the uniform asymptotic stability of the zero solution to system (18) is a sufficient condition for the stability of this solution for constantly acting perturbations. As the latter, perturbations small in the norm can be considered

Moreover, from the exponential asymptotic stability of the zero solution to system (18) due to the inequalities

the asymptotic stability of the zero solution of the original nonlinear system (12) follows for sufficiently small values of i / 12 /.

Thus, in the presence of semi-passive control (11) and damping and inequalities (17) under gravitational perturbations, the stability of the stabilized solution φ = θ = 0, ψ = ψ ₀ under constant perturbations and asymptotic stability for sufficiently small values of i are achieved, which justifies the application of the proposed method for orbits of small inclination.

Здесь матрица М_cp - результат покомпонентного усреднения матрицы М по времени, т.е.We turn to the consideration of the problem of stabilizing satellites in medium and large inclination orbits. We represent the matrix M in the form

Here, the matrix M _cp is the result of componentwise averaging of the matrix M over time, i.e.

Where

We calculate the necessary averages:

We first consider the linear approximation system of equations (12):

If we replace the matrix M in it with its average value, we obtain:

Using the replacement τ = ω ₀ t, x ₁ = φ ', x ₂ = θ', x ₃ = ψ ' ₁ , x ₄ = φ, x ₅ = θ, x ₆ = ψ ₁ , system (19) can be reduced to normal dimensionless form

and system (20), to the form x '= Nx, where x = (x ₁ , ... x ₆ ) ^T , the prime denotes the derivative with respect to the dimensionless variable τ, С ₀ = cosψ ₀ , S ₀ = sinψ ₀ ,

The system x '= Nx has bounded solutions on the semiaxis [0, + ∞] if and only if the real parts of the roots of the characteristic equation det (λI-N) = 0 are nonpositive. A numerical analysis carried out using a computer showed that there is a region of satellite parameters and its orbits under which this condition is satisfied and, therefore, in this region, the real parts of the eigenvalues of the matrix N satisfy the inequalities Reλ _j <-Δ, j = 1 ... 6 , Δ = const> 0.

According to / 13 /, we introduce the function

то нулевое решение системы (21) (или (19)) асимптотически устойчиво /13/. Следовательно, нулевое решение исходной нелинейной системы (12) будет устойчивым при постоянно действующих возмущениях /12/.Denote by D the largest value of this function on the semiaxis [0, + ∞]. If

then the zero solution to system (21) (or (19)) is asymptotically stable / 13 /. Consequently, the zero solution to the original nonlinear system (12) will be stable under constant disturbances / 12 /.

A large number of numerical experiments performed shows that if the system parameters satisfy the inequalities

that is, the matrix M _{cp is} positive definite, then all the conditions listed above are satisfied and, therefore, the stabilized motion φ = θ = 0, ψ = ψ ₀ = const is stable under constantly acting perturbations.

Thus, the possibility of using Lorentz forces for triaxial stabilization of a dynamically symmetric satellite in an orbital coordinate system is proved.

The method can be carried out using known technical means, allowing to create a double electrostatic layer on the surface of the satellite. The magnitude and direction of the vector of the dipole moment of the double layer are determined by the above conditions (11).

In order to confirm the effectiveness of the proposed method of semi-passive control of satellite orientation, a series of numerical experiments was carried out using a computer.

For each set of values of the satellite parameters, its orbit, and initial conditions of motion, the calculations were performed according to two mathematical models, respectively, in the presence of semi-passive control according to the law (11) and in its absence. As a result, we plotted the dependences of the “aircraft” satellite orientation angles on the latitude argument — the dimensionless angular quantity u = ω ₀ t in the interval 0≤u≤60, which corresponds to about 10 satellite orbits.

Figure 4 shows the calculation results of the stabilized motion of the satellite φ = θ = 0, ψ = ψ ₀ = 1 for the following parameters of the satellite and its orbit: R = 7 · 10 ⁶ m, i = 1.045 rad, A = 10 ³ kg m ² , C / A = 0.75, h ₁ = h ₂ = h ₃ = h = 0.5. As the initial values, φ (0) = 0.2 rad, ψ (0) = 0.2 rad, θ (0) = 0.2 rad, ω _x (0) = 0.1 rad / s, ω _y (0) = 1.1 rad / s , ω _z (0) = 0.1 rad / s.

были взяты следующие значения: Р_x=Р_y=0, Р_z=5·10^-3 Кл·м.Figure 5 presents the oscillations of the satellite in the absence of control at the same values of the parameters and initial conditions. In this case, for the components of the vector

the following values were taken: P _x = P _y = 0, P _z = 5 · 10 ^-3 C · m.

In Figs. 4 and 5, the solid line shows the dependences φ = φ (u), the dashed curve shows ψ = ψ (u), and the dash-dot curve shows θ = θ (u).

A comparison of the graphs in FIGS. 4 and 5 shows that the introduction of the proposed semi-passive control into the satellite system allows achieving a mode of stable movement in a short time. In this case, there is completely no need to install gyroscopes, flywheels, etc., ensuring stabilization of the satellite, as well as the expenditure of any working substance by the actuator. The obvious simplicity of the control law, which does not require measuring any orientation angles, their derivatives with respect to time, etc. during the motion of the satellite, as well as the reliability and efficiency of the method, testify to the prospects of its use for stabilizing the satellite.

This work was supported by the Russian Foundation for Basic Research (Grant No. 05-01-01073).

Information sources

1. A.P. Kovalenko. Magnetic spacecraft control systems. M .: Engineering, 1975, 248 p.

2. Sarychev V.A., Ovchinnikov M.Yu. Magnetic orientation systems of artificial Earth satellites. "Space exploration, t.23 (Results of science and technology VINITI USSR Academy of Sciences)" M., 1985, 104 S.

3. France, application No. 2550757, MKI B64G 1/36 "Method for regulating the position of satellites."

4. Germany, application No. os 3329955, MKI B64G 1/24, G05D 1/08 "Device for regulating the position of artificial satellites."

5. Japan, application No. 5940679, MKI B64G 1/24 "A device that creates a torque for controlling an artificial satellite and other spacecraft."

6. Petrov K.G., Tikhonov A.A. Patent RU - No. 2191146 - C1 for the invention "A method for the semi-passive stabilization of an artificial Earth satellite and a device for its implementation", IPC 7 В64С 1/32, 1/38 according to the application No. 2001107811, Priority 16.03.01, 34 pp.

7. Trukhanov K.A., Ryabova T.Ya., Morozov D.Kh. Active protection of spaceships. M .: Atomizdat, 1970, 229 p.

8.K.G. Petrov, A.A. Tikhonov. The moment of Lorentz forces acting on a charged satellite in the Earth's magnetic field. Part 1: Earth's magnetic field in the orbital coordinate system // Tomsk State University Journal. St. Petersburg, University. Ser. 1, 1999, Issue 1 (No. 1), pp. 92-100.

9.K.G. Petrov, A.A. Tikhonov. The moment of Lorentz forces acting on a charged satellite in the Earth's magnetic field. Part 2: Calculation of the moment and evaluation of its components // Tomsk State University Journal. St. Petersburg, University. Ser. 1, 1999, Issue 3 (No. 15), pp. 81-91.

10. Mandea M. et al. International geomagnetic reference field - 2000 // Physics of the Earth and planetary interiors. 2000, Vol. 120, pp. 39-42.

11. Bellman R. The theory of stability of solutions of differential equations. Publishing House of Foreign lit., M., 1954, 216 p.

12. Malkin I.G. Theory of motion stability. M .: Nauka, 1966, 530 s.

13. Martynyuk A.A. et al. Stability of motion: the method of integral inequalities. Kiev, Science. Dumka, 1989, 272 p.

Claims (1)

A method of semi-passive triaxial stabilization of a dynamically symmetric artificial Earth satellite, comprising obtaining a control torque of Lorentz forces by distributing a double electrostatic layer with a given total dipole moment on a part of the satellite’s surface

and an agreed change in the magnitude and direction of this moment, characterized in that said dipole moment is changed according to the conditions:

where P _x , P _y , P _z - projection of the vector

to the main central axis of inertia C _x , C _y , C _{z of the} satellite;

- vector projections

the speed of the center of mass (C) of the satellite relative to the geomagnetic field on the axis Cξ, Cη of the orbital coordinate system, and the axis Cξ is directed tangentially to the satellite’s orbit in the direction of its movement; B _Cξ , B _Cη , B _Cζ - projections of the vector

induction of the geomagnetic field at the center of mass of the satellite on the axis Cξ, Cη, Cζ, orbital coordinate system; ψ ₀ is the angle of rotation of the satellite about the Cz axis of its dynamic symmetry, which coincides with the local vertical (-Cζ) in the mode of oriented movement; k is a constant coefficient.

Images