Electrodynamic Attitude Stabilization of a Spacecraft in an Elliptical Orbit

Abstract

One of the fundamental problems of spacecraft dynamics related to ensuring its angular orientation in the basic coordinate system is considered. The problem of electrodynamic attitude control for a spacecraft in an elliptical near-Earth Keplerian orbit is studied. A mathematical model describing the attitude dynamics of the spacecraft under the action of the Lorentz torque, the magnetic interaction torque, and the gravitational torque is constructed. The multipole model of the Earth’s magnetic field is used. The possibility of electrodynamic attitude control for the spacecraft’s angular stabilization in the orbital frame is analyzed based on the Euler–Poisson differential equations. The problem of electrodynamic compensation of disturbing torque due to the orbit eccentricity is solved. The control strategy for spacecraft electrodynamic attitude stabilization is presented. Electromagnetic parameters that allow stabilizing the spacecraft’s attitude position in the orbital frame are proposed. The disturbing gravity gradient torque is taken into account. The convergence of the control process is verified by computer modeling. Thus, the possibility and advisability of using the electrodynamic method for the spacecraft attitude control and its angular stabilization in the orbital coordinate system in an elliptical orbit is shown.

1. Introduction

The electrodynamic interaction of spacecraft with the Earth’s magnetic field has a significant effect on the attitude dynamics of the spacecraft and can be used in the construction of spacecraft attitude control systems. Magnetic attitude control systems (MACS) based on the interaction of the intrinsic spacecraft’s magnetic moment with the geomagnetic field, their advantages, disadvantages and various application options, are described in [1,2,3,4,5,6,7]. The method of a spacecraft attitude stabilization based on the use of the moment of Lorentz forces (LACS), proposed in [8], has been developed in [9,10,11,12,13]. The interest in electrodynamic control systems is based, in particular, on the fact that the creation of a controlling Lorentz torque, significantly exceeding the gravitational and other disturbing torques in magnitude, does not cause technical difficulties [8]. The electrodynamic system of a spacecraft attitude stabilization [14] uses both the torque of magnetic interaction and the Lorentz torque at the same time and thereby removes the underactuation problem inherent in both MACS and LACS separately. In [14,15], a mechanism for the synthesis of restoring and dissipative components of control torques is formulated. Subsequently, the electrodynamic attitude control system for spacecraft was developed in [16,17,18,19] and other investigations by the authors of this paper aimed at solving a number of problems of stabilization of various modes of spacecraft attitude motion. All these issues discussed in this study belong to a wide range of problems devoted to magneto-electro-mechanical coupling and are relevant in connection with numerous applications not only in cosmodynamics, but also in other areas of modern mechanics [20,21].

In all the above-mentioned publications using the construction of electrodynamic controls to support certain space missions, the research is based on the assumption that the spacecraft’s orbit is circular. This assumption significantly simplifies the mathematical model of the problem and allows the use of well-known research methods based on both the consideration of linear approximation equations and nonlinear differential equations.

However, in some problems of the attitude dynamics of spacecraft, it is fundamentally important to take into account the ellipticity of the spacecraft’s orbit. For example, see papers [22,23,24,25,26]. The mathematical model of the spacecraft attitude motion in an elliptical orbit is fundamentally more complex than in the case of a circular orbit, and the effects of physical factors caused by the ellipticity of the orbit (the so-called eccentric oscillations of spacecraft) complicate the process of angular stabilization of spacecraft, regardless of which attitude stabilization system is used, gravitational, magnetic, Lorentz or electrodynamic. In this paper, for the first time, the application of an electrodynamic attitude control system for the angular stabilization of a spacecraft moving in an elliptical orbit is considered. The basic coordinate system is the orbital one. The possibility of electrodynamic compensation of the disturbing torque caused by the ellipticity of the orbit and the implementation of the program mode of spacecraft attitude motion is shown. In this case, the quadrupole approximation of the Earth’s magnetic field is used as a well-proven model, which is not bulky and at the same time quite accurate and devoid of the disadvantages inherent in the magnetic dipole model [27]. Computer modeling confirms the feasibility of the proposed control and indicates a fairly rapid convergence of transients.

The rest of the paper is organized in the following way. The problem statement is presented in Section 2. The control torques are derived in Section 3. The Earth’s magnetic field induction in quadrupole approximation is presented in Section 4. The disturbing torque due to the orbital ellipticity is derived in Section 5. The compensation of the disturbing torque is discussed in Section 6. Then, the simulation results are exhibited to verify the effectiveness of the proposed control in Section 7. Section 8 gives the main conclusions.

2. Problem Statement

The paper deals with a spacecraft whose center of mass moves along an elliptical Keplerian orbit of an arbitrary inclination i in the Newtonian central Earth’s gravitational field. The spacecraft is assumed to be equipped with a controlled electrostatic charge $Q=\underset{V}{\int }\sigma dV$ distributed over a certain volume V with a density $\sigma$, and an intrinsic magnetic moment $\overrightarrow{I}$. The Earth’s magnetic field is approximated up to quadrupole components.

First of all, let us introduce the following right-hand Cartesian coordinate systems: $O_{\mathrm{E}}{X}_{*}{Y}_{*}{Z}_{*}$ (Figure 1) is the inertial frame ($O_{\mathrm{E}}{Z}_{*}$ axis is directed along the Earth’s axis of rotation, $O_{\mathrm{E}}{X}_{*}$ is oriented to the vernal equinox point, and the plane $O_{\mathrm{E}}{X}_{*}{Y}_{*}$ coincides with the equatorial plane, the origin coincides with the Earth’s center); the orbital frame $C\xi \eta \zeta$ with unit vectors ${\overrightarrow{\xi }}_0,{\overrightarrow{\eta }}_0,{\overrightarrow{\zeta }}_0$ (Figure 1) with the origin at the spacecraft’s mass center (the $C\eta \mathrm{axis}$ is normal to the plane of the orbit, the $C\zeta \mathrm{axis}$ is directed along the radius vector $\overrightarrow{R}$ of the spacecraft’s center of mass relative to the Earth’s center); $Cxyz$ (Figure 2) is rigidly connected with the spacecraft and the axes are directed along the spacecraft’s principal central axes of inertia (unit vectors $\overrightarrow{i},\overrightarrow{j},\overrightarrow{k}$).

The orientation of the orbital frame $C\xi \eta \zeta$ with respect to the coordinate system $O_{\mathrm{E}}{X}_{*}{Y}_{*}{Z}_{*}$ with unit vectors $\overrightarrow{i_{*}},\overrightarrow{j_{*}},\overrightarrow{k_{*}}$ is determined by the following expressions:

$$\begin{array}{c}\left(\begin{array}{c}\overrightarrow{i_{*}}\\ \overrightarrow{j_{*}}\\ \overrightarrow{k_{*}}\end{array}\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& cosi& -sini\\ 0& sini& cosi\end{array}\right)\left(\begin{array}{ccc}cosu& -sinu& 0\\ sinu& cosu& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}\overrightarrow{{\zeta }_0}\\ \overrightarrow{{\xi }_0}\\ \overrightarrow{{\eta }_0}\end{array}\right)\hfill \\ \hfill =\left(\begin{array}{ccc}cosu& -sinu& 0\\ cosisinu& cosicosu& -sinu\\ sinisinu& sinicosu& cosi\end{array}\right)\left(\begin{array}{c}\overrightarrow{{\zeta }_0}\\ \overrightarrow{{\xi }_0}\\ \overrightarrow{{\eta }_0}\end{array}\right).\end{array}$$

(1)

Here, $i=\left(\widehat{\overrightarrow{k_{*}},\overrightarrow{{\eta }_0}}\right)$ is an orbital inclination angle, $u=\left(\widehat{\overrightarrow{i_{*}},{\overrightarrow{\zeta }}_0}\right)$ is an argument of latitude. In addition, $u={\omega }_{\pi }+\nu$, where ${\omega }_{\pi }$ is a perigee argument, and $\nu$ is the true anomaly.

A rotation matrix $\mathbf{A}$ transforms the coordinate frame $Cxyz$ to $C\xi \eta \zeta$:

$$\mathbf{A}=\left(\begin{array}{ccc}{\alpha }_1& {\alpha }_2& {\alpha }_3\\ {\beta }_1& {\beta }_2& {\beta }_3\\ {\gamma }_1& {\gamma }_2& {\gamma }_3\end{array}\right),$$

(2)

where

$$\overrightarrow{{\xi }_0}={\alpha }_1\overrightarrow{i}+{\alpha }_2\overrightarrow{j}+{\alpha }_3\overrightarrow{k},\overrightarrow{{\eta }_0}={\beta }_1\overrightarrow{i}+{\beta }_2\overrightarrow{j}+{\beta }_3\overrightarrow{k},\overrightarrow{{\zeta }_0}={\gamma }_1\overrightarrow{i}+{\gamma }_2\overrightarrow{j}+{\gamma }_3\overrightarrow{k}.$$

(3)

The spacecraft’s orientation in the orbital frame is introduced by the aircraft angles $\psi ,\vartheta ,\phi$ (Figure 2). Then, the elements of the matrix $\mathbf{A}$ (2) have the following form:

$$\begin{array}{cc}\hfill & {\alpha }_1=cos\psi cos\vartheta ,{\alpha }_2=-sin\psi cos\phi +cos\psi sin\vartheta sin\phi ,\hfill \\ \hfill & {\alpha }_3=sin\psi sin\phi +cos\psi sin\vartheta cos\phi ,{\beta }_1=sin\psi cos\vartheta ,\hfill \\ \hfill & {\beta }_2=cos\psi cos\phi +sin\psi sin\vartheta sin\phi ,{\beta }_3=-cos\psi sin\phi +sin\psi sin\vartheta cos\phi ,\hfill \\ \hfill & {\gamma }_1=-sin\vartheta ,{\gamma }_2=sin\phi cos\vartheta ,{\gamma }_3=cos\phi cos\vartheta .\hfill \end{array}$$

(4)

Let us denote the spacecraft’s tensor of inertia $\mathbf{J}=diag\left(A,B,C\right)$ relative to the $Cxyz$ coordinate system.

The program orientation of the spacecraft in the orbital frame is defined by a certain value ${\mathbf{A}}_0$ of the rotation matrix $\mathbf{A}$ (2).

3. Control Torques

During the motion of the spacecraft with respect to the geomagnetic field with a magnetic induction $\overrightarrow{B}$, the Lorentz torque ${\overrightarrow{M}}_L$ and the magnetic torque ${\overrightarrow{M}}_M$ are excited. These torques can be written as follows:

$${\overrightarrow{M}}_L=\overrightarrow{P}\times \overrightarrow{T},{\overrightarrow{M}}_M=\overrightarrow{I}\times \overrightarrow{B}.$$

(5)

Here, $\overrightarrow{P}=Q{\overrightarrow{\rho }}_0$, ${\overrightarrow{\rho }}_0=x_0\overrightarrow{i}+y_0\overrightarrow{j}+z_0\overrightarrow{k}=Q^{-1}\underset{V}{\int }\sigma \overrightarrow{\rho }dV$ is the radius vector of the spacecraft’s center of charge relative to its center of mass, $\overrightarrow{\rho }$ is the radius vector of the element $dV$ of the spacecraft with respect to its mass center, $\overrightarrow{T}={\mathbf{A}}^{\top }\left({\overrightarrow{v}}_C\times \overrightarrow{B}\right)$, ${\overrightarrow{v}}_C$ is the velocity of the spacecraft’s center of mass (point C) relative to the geomagnetic field, and magnetic induction $\overrightarrow{B}$ is calculated for the spacecraft’s center of mass. Vector ${\overrightarrow{v}}_C$ has the following form:

$$\begin{array}{c}{\overrightarrow{v}}_C={\displaystyle \frac{p_0}{1+ecos\nu }}\left[\sqrt{\mu p_0^{-3}}{\left(1+ecos\nu \right)}^2-{\omega }_{E}cosi\right]{\overrightarrow{\xi }}_0\hfill \\ \hfill +{\displaystyle \frac{p_0}{1+ecos\nu }}{\omega }_{E}sinicosu{\overrightarrow{\eta }}_0+p_{0}e\sqrt{\mu p_0^{-3}}sin\nu {\overrightarrow{\zeta }}_0,\end{array}$$

(6)

where $p_0$ is the focal parameter of the orbit, e is an orbital eccentricity, $\mu$ is the Earth’s gravitational parameter, ${\overrightarrow{\omega }}_E$ is the Earth’s rotation rate:

$${\overrightarrow{\omega }}_E={\omega }_E\left(sinicosu{\overrightarrow{\xi }}_0+cosi{\overrightarrow{\eta }}_0+sinisinu{\overrightarrow{\zeta }}_0\right).$$

(7)

The key idea for the stabilization effect of the moment of Lorentz force is the displacement of the center of charge relative to the spacecraft’s center of mass, i.e., a nonzero vector ${\overrightarrow{\rho }}_0$. The Figure 3 and Figure 4 show the restoring effect of the Lorentz torque on a spacecraft whose center of charge O does not coincide with the mass center C, for different positions of the radius vector ${\overrightarrow{\rho }}_0$ relative to the vector $\overrightarrow{R}$.

It is possible to stabilize the attitude motion of the spacecraft by program-controlled electrodynamic parameters $\overrightarrow{P}$ and $\overrightarrow{I}$. In order to create restoring and damping components of control torques ${\overrightarrow{M}}_L$ and ${\overrightarrow{M}}_M$, parameters $\overrightarrow{P}$ and $\overrightarrow{I}$ should consist of two components, respectively:

$$\overrightarrow{P}={\overrightarrow{P}}_{\mathrm{rest}}+{\overrightarrow{P}}_{\mathrm{diss}},\overrightarrow{I}={\overrightarrow{I}}_{\mathrm{rest}}+{\overrightarrow{I}}_{\mathrm{diss}}.$$

(8)

The control parameter components have the following form:

$${\overrightarrow{P}}_{rest}=Q{k}_L{\overrightarrow{T}}_0,{\overrightarrow{P}}_{diss}=Q{h}_L{\overrightarrow{\omega }}^{\prime }\times \overrightarrow{T},$$

(9)

$${\overrightarrow{I}}_{rest}=k_M{\overrightarrow{B}}_0,{\overrightarrow{I}}_{diss}=h_M{\overrightarrow{\omega }}^{\prime }\times \overrightarrow{B}.$$

(10)

Here, ${\overrightarrow{T}}_0=A_0^{\top }\left({\overrightarrow{v}}_C\times \overrightarrow{B}\right)$, ${\overrightarrow{B}}_0=A_0^{\top }\overrightarrow{B}$, ${\overrightarrow{\omega }}^{\prime }=p\overrightarrow{i}+q\overrightarrow{j}+r\overrightarrow{k}$ is the angular velocity of the spacecraft relative to the orbital coordinate system, $k_L,k_M,h_L,h_M$ are scalar coefficients, which can be constant values as well as functions of time.

Thus, the control torques ${\overrightarrow{M}}_L$ and ${\overrightarrow{M}}_M$ take the following form:

$${\overrightarrow{M}}_L=Q{k}_L{\overrightarrow{T}}_0\times \overrightarrow{T}-Q{h}_L\overrightarrow{T}\times \left({\overrightarrow{\omega }}^{\prime }\times \overrightarrow{T}\right),$$

(11)

$${\overrightarrow{M}}_M=k_M{\overrightarrow{B}}_0\times \overrightarrow{B}-h_M\overrightarrow{B}\times \left({\overrightarrow{\omega }}^{\prime }\times \overrightarrow{B}\right).$$

(12)

It should be noted that using only one of the control torques, either the Lorentz or magnetic one, is associated with functional features that limit their capabilities. It follows from the Equation (5) that the torque $\overrightarrow{M_M}$ is orthogonal to the magnetic induction $\overrightarrow{B}$. As a result, it is impossible to create a control magnetic torque directed along the Earth’s magnetic induction vector $\overrightarrow{B}$. A similar shortage occurs in the Lorentz torque $\overrightarrow{M_L}$ (see Equation (5)), since it is impossible to create a control Lorentz torque along the vector $\overrightarrow{T}$. The use of both the Lorentz and magnetic torques eliminates the above functional limitations and allows the control torque to be shifted in any direction. The electrodynamic attitude control method is workable for spacecraft moving in near-Earth orbits with any inclinations, and a local decrease in its efficiency may occur in a situation where the Lorentz torque becomes zero. Such situations may occur for spacecraft in polar orbits.

4. The Earth’s Magnetic Field Induction

The potential of the Earth’s magnetic field is represented as an infinite series in spherical coordinates. In applied research, it is usual to limit the potential representation to a finite number of terms. However magnetic field induction is quite important since it is necessary to use the projections of the geomagnetic induction on the axes of the orbital coordinate system. These aspects of research are detailed in articles [28,29].

Our study, aimed at taking into account the ellipticity of the spacecraft’s orbit, does not exclude cases of small eccentricity, leading to the appearance of small additional terms in the right-hand sides of the Euler equations. In this regard, the question of the accuracy of the mathematical model should be solved by taking into account the accuracy of the representation of geomagnetic induction. This article uses a fairly accurate model of the Earth’s magnetic field, including quadrupole components.

Auxiliary coordinate systems $O_{\mathrm{E}}XYZ$ (unit vectors $\overrightarrow{{\tau }_1},\overrightarrow{{\tau }_2},\overrightarrow{{\tau }_3}$) and $O_{\mathrm{E}}{X}^0{Y}^0{Z}^0$ (unit vectors ${\overrightarrow{{\tau }_1}}^0,{\overrightarrow{{\tau }_2}}^0,{\overrightarrow{{\tau }_3}}^0$) are used for constructing analytically the vector $\overrightarrow{B}$ by taking into account two multipole components (Figure 5). The axis $O_{\mathrm{E}}X$ is directed to the point of intersection between the Earth’s equator and the Greenwich meridian; the axis $O_{\mathrm{E}}Y$ lies in the equatorial plane and rotates relative to the axis $O_{\mathrm{E}}X$ to 90 degrees to the east. The axis $O_{\mathrm{E}}{X}^0$ is directed to the ascending node of the spacecraft’s orbit; the axis $O_{\mathrm{E}}{Z}^0$ is normal to the orbital plane. Here, $\varphi =\left(\widehat{{\overrightarrow{{\tau }_1}}^0,\overrightarrow{{\tau }_1}}\right)$ is the Earth’s daily rotation angle.

A rotation matrix $\mathsf{\Gamma }$ transforms the coordinate frame $O_{\mathrm{E}}XYZ$ to $O_{\mathrm{E}}{X}^0{Y}^0{Z}^0$:

$$\mathsf{\Gamma }=\left({\gamma }_{ij}\right),(i=\overline{1,3},j=\overline{1,3})$$

(13)

such that the following equalities are fulfilled:

$${\overrightarrow{{\tau }_i}}^0=\sum _{j=1}^3{\gamma }_{ij}\overrightarrow{{\tau }_j}.$$

(14)

The components ${\gamma }_{ij}$ of matrix $\mathsf{\Gamma }$ (13) are represented by terms of the orbital inclination i and the Earth’s daily rotation angle $\varphi ={\omega }_{E}t$ as follows [29]:

$$\begin{array}{ccc}{\gamma }_{11}=cos\varphi ,\hfill & {\gamma }_{12}=-sin\varphi ,\hfill & {\gamma }_{13}=0,\hfill \\ {\gamma }_{21}=sin\varphi cosi,\hfill & {\gamma }_{22}=cos\varphi cosi,\hfill & {\gamma }_{23}=sini,\hfill \\ {\gamma }_{31}=-sin\varphi sini,\hfill & {\gamma }_{32}=-cos\varphi sini,\hfill & {\gamma }_{33}=cosi.\hfill \end{array}$$

(15)

Thus, the vector $\overrightarrow{B}$ has the following form:

$$\overrightarrow{B}=\sum _{n=1}^2{\overrightarrow{B}}^{\left(n\right)}=-grad\sum _{n=1}^2\left(R_E^{n+2}/r^{2n+1}\right){\mathbf{M}}^{\left(n\right)}\underset{n}{\underbrace{\dots }}\left({\otimes }^n\overrightarrow{r}\right).$$

(16)

Here, $\overrightarrow{r}$ is the radius-vector of the current point in the near-Earth space with respect to the Earth’s mass center, ${\mathbf{M}}^{\left(n\right)}$ are multipole tensors of the first and second ranks that are dipole and quadrupole magnetic moments, respectively.

To determine vectors ${\overrightarrow{B}}^{\left(1\right)}$ and ${\overrightarrow{B}}^{\left(2\right)}$ for the spacecraft’s center of mass, let us use the following expressions:

$$\left(\begin{array}{c}{B}_{\xi }^{\left(1\right)}\\ B_{\eta }^{\left(1\right)}\\ B_{\zeta }^{\left(1\right)}\end{array}\right)=-{\left({\displaystyle \frac{R_E}{R}}\right)}^3\left(\begin{array}{c}{\mathbf{M}}_0^{\left(1\right)}·{\mathbf{T}}_{\lambda }^{\left(1\right)}\\ {\mathbf{M}}_0^{\left(1\right)}·{\mathbf{T}}_{\vartheta }^{\left(1\right)}\\ -2{\mathbf{M}}_0^{\left(1\right)}·{\mathbf{T}}_r^{\left(1\right)}\end{array}\right);$$

(17)

$$\left(\begin{array}{c}{B}_{\xi }^{\left(2\right)}\\ B_{\eta }^{\left(2\right)}\\ B_{\zeta }^{\left(2\right)}\end{array}\right)=-{\left({\displaystyle \frac{R_E}{R}}\right)}^4\left(\begin{array}{c}2{\mathbf{M}}_0^{\left(2\right)}··{\mathbf{T}}_{\lambda }^{\left(2\right)}\\ 2{\mathbf{M}}_0^{\left(2\right)}··{\mathbf{T}}_{\vartheta }^{\left(2\right)}\\ -3{\mathbf{M}}_0^{\left(2\right)}··{\mathbf{T}}_r^{\left(2\right)}\end{array}\right).$$

(18)

Here, $R={\displaystyle \frac{p_0}{1+ecos\nu }}$, the $··$ operation is called a double dot product. For tensors of the second rank, this operation is defined as follows:

$$A··B=\sum _{i_1}\sum _{i_2}{A}_{i_2{i}_1}{B}_{i_1{i}_2}.$$

(19)

Tensors included in (17) and (18) have the following form:

$$\begin{array}{c}{\mathbf{T}}_r^{\left(1\right)}={\left(cosu,sinu,0\right)}^{\top },{\mathbf{T}}_{\lambda }^{\left(1\right)}={\left(-sinu,cosu,0\right)}^{\top },{\mathbf{T}}_{\vartheta }^{\left(1\right)}={\left(0,0,1\right)}^{\top },\\ {\mathbf{T}}_r^{\left(2\right)}={\mathbf{T}}_r^{\left(1\right)}\otimes {\mathbf{T}}_r^{\left(1\right)},{\mathbf{T}}_{\lambda }^{\left(2\right)}={\mathbf{T}}_{\lambda }^{\left(1\right)}\otimes {\mathbf{T}}_r^{\left(1\right)},{\mathbf{T}}_{\vartheta }^{\left(2\right)}={\mathbf{T}}_{\vartheta }^{\left(1\right)}\otimes {\mathbf{T}}_r^{\left(1\right)}.\end{array}$$

(20)

${\mathbf{M}}_0^{\left(n\right)}$ is a multipole tensor of rank n, which is calculated by the following formula:

$${\left({\mathbf{M}}_0^{\left(n\right)}\right)}_{i_1.i_2,...,i_n}=\sum _{j_1,j_2,...,j_n=1}^3{\gamma }_{i_1,j_1}{\gamma }_{i_2,j_2}...{\gamma }_{i_n,j_n}{M}_{j_1,j_2,...,j_n}^{\left(n\right)}.$$

(21)

The multipole tensors components included in (21) have the form [29]:

$$\begin{array}{ccc}{M}_1^{\left(1\right)}=g_1^1,\hfill & M_2^{\left(1\right)}=h_1^1,\hfill & M_3^{\left(1\right)}=g_1^0,\hfill \\ M_{11}^{\left(2\right)}={\displaystyle \frac{\sqrt{3}}{2}}{g}_2^2-{\displaystyle \frac{1}{2}}{g}_2^0,\hfill & M_{22}^{\left(2\right)}=-\left(\sqrt{3}{g}_2^2+{\displaystyle \frac{1}{2}}{g}_2^0\right),\hfill & M_{33}^{\left(2\right)}=g_2^0,\hfill \\ M_{12}^{\left(2\right)}={\displaystyle \frac{\sqrt{3}}{2}}{h}_2^2,\hfill & M_{23}^{\left(2\right)}={\displaystyle \frac{\sqrt{3}}{2}}{h}_2^1,\hfill & M_{31}^{\left(2\right)}={\displaystyle \frac{\sqrt{3}}{2}}{g}_2^1.\hfill \end{array}$$

(22)

Gaussian coefficients are determined by various methods including collecting and disseminating the Earth’s magnetic field data from spacecraft and from observatories by magnetic field modelers and institutes. These spherical harmonic coefficients are approved several years in advance. For the current generation of the International Geomagnetic Reference Field, the Gaussian coefficients take the values in units of nanoTesla (nT) [30]:

$$\begin{array}{ccc}{g}_1^0=-29404.8,\hfill & g_1^1=-1450.9,\hfill & h_1^1=4652.5,\hfill \\ g_2^0=-2499.6,\hfill & g_2^1=2982.0,\hfill & g_2^2=1677.0,\hfill & h_2^1=-2991.6,\hfill & h_2^2=-734.6.\hfill \end{array}$$

(23)

In the following discussion, it is assumed that the vector $\overrightarrow{B}$ is the same at all points of the volume of the spacecraft and is equal to its value $\overrightarrow{B_C}$ in the spacecraft’s mass center.

Within the conditions of the quadrupole approximation of the Earth’s magnetic field, the projections of $\overrightarrow{B}$ on the orbital axes are determined by the following formulae:

$$\begin{array}{cc}\hfill B_{\xi }& =-{\displaystyle \frac{R_E^3{\left(1+ecos\nu \right)}^3}{p_0^3}}\left(M_1\left(f_{0,1,0,1,1,0}-f_{1,0,0,0,0,1}\right)+M_2\left(f_{1,0,0,0,1,0}+f_{0,1,0,1,0,1}\right)\right.\hfill \\ \hfill & \left.+M_3{f}_{0,1,1,0,0,0}\right)-2{\displaystyle \frac{R_E^4{\left(1+ecos\nu \right)}^4}{p_0^4}}\left(2M_{13}{f}_{0,2,1,0,0,1}+\left(M_{22}-M_{11}\right)f_{1,1,0,0,0,2}\right.\hfill \\ \hfill & +\left(M_{22}-M_{11}\right)f_{0,0,0,1,1,1}+\left(-M_{22}+M_{33}\right)f_{1,1,0,0,0,0}+M_{23}{f}_{0,0,1,0,1,0}\hfill \\ \hfill & +M_{12}{f}_{0,0,0,1,0,0}-M_{13}{f}_{0,0,1,0,0,1}+\left(2M_{11}-2M_{22}\right)f_{0,2,0,1,1,1}+2M_{12}{f}_{1,1,0,0,1,1}\hfill \\ \hfill & +2M_{12}{f}_{1,1,0,2,1,1}+2M_{13}{f}_{1,1,1,1,1,0}+\left(M_{22}-M_{11}\right)f_{1,1,0,2,0,2}+2M_{23}{f}_{1,1,1,1,0,1}\hfill \\ \hfill & -2M_{12}{f}_{0,2,0,1,0,0}-2M_{12}{f}_{0,0,0,1,0,2}+\left(M_{11}-M_{33}\right)f_{1,1,0,2,0,0}+4M_{12}{f}_{0,2,0,1,0,2}\hfill \\ \hfill & \left.-2M_{23}{f}_{0,2,1,0,1,0}\right),\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill B_{\eta }& ={\displaystyle \frac{R_E^3{\left(1+ecos\nu \right)}^3}{p_0^3}}\left(f_{0,0,1,0,1,0}{M}_1+f_{0,0,1,0,0,1}{M}_2-f_{0,0,0,1,0,0}{M}_3\right)\hfill \\ \hfill & -2{\displaystyle \frac{R_E^4{\left(1+ecos\nu \right)}^4}{p_0^4}}\left(\left(M_{22}-M_{11}\right)f_{0,1,1,0,1,1}-2M_{12}{f}_{0,1,1,0,0,2}+M_{13}{f}_{0,1,0,1,0,1}\right.\hfill \\ \hfill & +M_{12}{f}_{0,1,1,0,0,0}-M_{23}{f}_{0,1,0,1,1,0}+\left(M_{33}-M_{11}\right)f_{1,0,1,1,0,0}\hfill \\ \hfill & +\left(M_{11}-M_{22}\right)f_{1,0,1,1,0,2}-2M_{12}{f}_{1,0,1,1,1,1}+2M_{13}{f}_{1,0,0,2,1,0}\hfill \\ \hfill & \left.+2M_{23}{f}_{1,0,0,2,0,1}-M_{13}{f}_{1,0,0,0,1,0}-M_{23}{f}_{1,0,0,0,0,1}\right),\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill B_{\zeta }& =2{\displaystyle \frac{R_E^3{\left(1+ecos\nu \right)}^3}{p_0^3}}\left(f_{0,1,0,0,0,1}{M}_1-f_{0,1,0,0,1,0}{M}_2+f_{1,0,0,1,1,0}{M}_1+f_{1,0,0,1,0,1}{M}_2\right.\hfill \\ \hfill & \left.+M_3{f}_{1,0,1,0,0,0}\right)+3{\displaystyle \frac{R_E^4{\left(1+ecos\nu \right)}^4}{p_0^4}}\left(M_{33}{f}_{0,0,0,0,0,0}+2M_{13}{f}_{0,0,1,1,1,0}\right.\hfill \\ \hfill & +\left(-M_{33}+M_{11}\right)f_{0,0,0,2,0,0}+2M_{23}{f}_{0,0,1,1,0,1}-2M_{12}{f}_{0,2,0,0,1,1}\hfill \\ \hfill & +\left(M_{22}-M_{11}\right)f_{0,0,0,2,0,2}+2M_{12}{f}_{0,0,0,2,1,1}+\left(-M_{11}+M_{33}\right)f_{0,2,0,2,0,0}\hfill \\ \hfill & +\left(M_{11}-M_{22}\right)f_{0,2,0,0,0,2}-2M_{12}{f}_{1,1,0,1,0,0}+\left(M_{11}-M_{22}\right)f_{0,2,0,2,0,2}\hfill \\ \hfill & +\left(2M_{11}-2M_{22}\right)f_{1,1,0,1,1,1}+4M_{12}{f}_{1,1,0,1,0,2}+2M_{13}{f}_{1,1,1,0,0,1}\hfill \\ \hfill & -2M_{23}{f}_{1,1,1,0,1,0}-2M_{12}{f}_{0,2,0,2,1,1}-2M_{13}{f}_{0,2,1,1,1,0}-2M_{23}{f}_{0,2,1,1,0,1}\hfill \\ \hfill & \left.+\left(M_{22}-M_{33}\right)f_{0,2,0,0,0,0}\right),\hfill \end{array}$$

(26)

where $f_{i_1,i_2,i_3,i_4,i_5,i_6}={sin}^{i_1}\left(u\right){cos}^{i_2}\left(u\right){sin}^{i_3}\left(i\right){cos}^{i_4}\left(i\right){sin}^{i_5}\left(\varphi \right){cos}^{i_6}\left(\varphi \right)$.

5. The Disturbing Torque

To analyze the problem of spacecraft stabilization in the orbital frame and compensation of the disturbing torque, consider the Euler–Poisson differential equations [31]:

$$\left\{\begin{array}{c}A{\dot{\omega }}_x+\left(C-B\right){\omega }_y{\omega }_z=M_{\mathrm{Lx}}+M_{\mathrm{Mx}}+M_{\mathrm{Gx}},\hfill \\ B{\dot{\omega }}_y+\left(A-C\right){\omega }_z{\omega }_x=M_{\mathrm{Ly}}+M_{\mathrm{My}}+M_{\mathrm{Gy}},\hfill \\ C{\dot{\omega }}_z+\left(B-A\right){\omega }_x{\omega }_y=M_{\mathrm{Lz}}+M_{\mathrm{Mz}}+M_{\mathrm{Gz}},\hfill \end{array}\right.$$

(27)

$$\left\{\begin{array}{c}\dot{{\alpha }_1}+{\omega }_y{\alpha }_3-{\omega }_z{\alpha }_2=-{\omega }_{*}{\gamma }_1,\dot{{\beta }_1}+{\omega }_y{\beta }_3-{\omega }_z{\beta }_2=0,\dot{{\gamma }_1}+{\omega }_y{\gamma }_3-{\omega }_z{\gamma }_2={\omega }_{*}{\alpha }_1,\hfill \\ \dot{{\alpha }_2}+{\omega }_z{\alpha }_1-{\omega }_x{\alpha }_3=-{\omega }_{*}{\gamma }_2,\dot{{\beta }_2}+{\omega }_z{\beta }_1-{\omega }_x{\beta }_3=0,\dot{{\gamma }_2}+{\omega }_z{\gamma }_1-{\omega }_x{\gamma }_3={\omega }_{*}{\alpha }_2,\hfill \\ \dot{{\alpha }_3}+{\omega }_x{\alpha }_2-{\omega }_y{\alpha }_1=-{\omega }_{*}{\gamma }_3,\dot{{\beta }_3}+{\omega }_x{\beta }_2-{\omega }_y{\beta }_1=0,\dot{{\gamma }_3}+{\omega }_x{\gamma }_2-{\omega }_y{\gamma }_1={\omega }_{*}{\alpha }_3,\hfill \end{array}\right.$$

(28)

or in the vector form:

$${\displaystyle \frac{d}{dt}}\left(\mathbf{J}\overrightarrow{\omega }\right)+\overrightarrow{\omega }\times \left(\mathbf{J}\overrightarrow{\omega }\right)={\overrightarrow{M}}_L+{\overrightarrow{M}}_M+{\overrightarrow{M}}_G,$$

(29)

$${\displaystyle \frac{d{\overrightarrow{\xi }}_0}{dt}}={\overrightarrow{\xi }}_0\times \overrightarrow{\omega }-{\omega }_{*}{\overrightarrow{\zeta }}_0,{\displaystyle \frac{d\overrightarrow{{\eta }_0}}{dt}}=\overrightarrow{{\eta }_0}\times \overrightarrow{\omega },{\displaystyle \frac{d\overrightarrow{{\zeta }_0}}{dt}}=\overrightarrow{{\zeta }_0}\times \overrightarrow{\omega }+{\omega }_{*}{\overrightarrow{\xi }}_0.$$

(30)

The gravitational torque $\overrightarrow{M_G}={\left(M_{Gx},M_{Gx},M_{Gx}\right)}^{\top }$ in the dynamic Euler Equation (27) has the following components [31]:

$${\overrightarrow{M}}_{\mathrm{G}}={\left({\displaystyle \frac{3\mu }{R^3}}(C-B){\gamma }_2{\gamma }_3,{\displaystyle \frac{3\mu }{R^3}}(A-C){\gamma }_3{\gamma }_1,{\displaystyle \frac{3\mu }{R^3}}(B-A){\gamma }_1{\gamma }_2\right)}^{\top }.$$

(31)

The absolute angular velocity $\overrightarrow{\omega }$ is expressed by the equality $\overrightarrow{\omega }={\overrightarrow{\omega }}_{*}+{\overrightarrow{\omega }}^{\prime }$, which in projections on the $Cxyz$ axes has the following form:

$${\omega }_x={\omega }_{*}{\beta }_1+p,{\omega }_y={\omega }_{*}{\beta }_2+q,{\omega }_z={\omega }_{*}{\beta }_3+r,$$

(32)

where ${\overrightarrow{\omega }}_{*}={\omega }_{*}{\overrightarrow{\eta }}_0$ is the angular velocity of the orbital frame relative to the inertial coordinate system.

The orbital angular velocity is equal to the time derivative of the true anomaly:

$${\omega }_{*}=\dot{\nu }=\sqrt{{\displaystyle \frac{\mu }{p_0^3}}}{\left(1+ecos\nu \right)}^2.$$

(33)

Let us call the direct equilibrium position of the spacecraft in the orbital frame such as the $x,y$, and z axes coincide with the $\xi ,\eta$, and $\zeta$ axes, respectively. For the direct equilibrium position, the following equalities are fulfilled:

$$p=q=r=0,{\alpha }_1=1,{\beta }_2=1,{\gamma }_3=1.$$

(34)

The remaining components of the matrix $\mathbf{A}$ (2) are equal to zero.

The disturbing torque is determined by substituting expressions for torques (11), (12) and (31) on the right side of the Euler Equation (27), and also taking into account that the equality is fulfilled in the equilibrium position (34). It has the following form:

$$\overrightarrow{g}={\left(0,-2{\displaystyle \frac{Be\mu }{p_0^3}}sin\nu {\left(1+ecos\nu \right)}^3,0\right)}^{\top }.$$

(35)

It is important to note that this type of disturbing torque occurs only for elliptical orbits of any inclination i and is completely absent in the case of circular orbits, earlier considered in papers [8,9,10,11,12,13,14,15,16,17,18,19].

6. Compensation of the Disturbing Torque

Let us solve the problem of disturbance torque $\overrightarrow{g}$ (35) using control torques $\overrightarrow{M_L},\overrightarrow{M_M}$ (5) by choosing the vectors ${\overrightarrow{P}}_{\mathrm{comp}}={\left(P_x,P_y,P_z\right)}^{\top }$ and ${\overrightarrow{I}}_{\mathrm{comp}}={\left(I_x,I_y,I_z\right)}^{\top }$. The vectors ${\overrightarrow{P}}_{\mathrm{comp}}$ and ${\overrightarrow{I}}_{\mathrm{comp}}$ must satisfy the equation:

$${\overrightarrow{P}}_{\mathrm{comp}}\times \overrightarrow{T}+{\overrightarrow{I}}_{\mathrm{comp}}\times \overrightarrow{B}+\overrightarrow{g}=0.$$

(36)

At the same time, the following equalities allow us to minimize the magnitudes $\parallel {\overrightarrow{P}}_{\mathrm{comp}}\parallel$ and $\parallel {\overrightarrow{I}}_{\mathrm{comp}}\parallel$ and, accordingly, the energy which is consumed for their implementation:

$${\overrightarrow{P}}_{\mathrm{comp}}·\overrightarrow{T}=0,{\overrightarrow{I}}_{\mathrm{comp}}·\overrightarrow{B}=0.$$

(37)

The system of linear equations to determine the components of compensating vectors is obtained by substituting the vector $\overrightarrow{g}$ (35) in the Equation (36). In addition, one of the components of the compensating vectors may be arbitrary (in particular, $P_y=0$ is accepted). Thus, the system of equations has the form

$$\left\{\begin{array}{c}{P}_y{T}_z-P_z{T}_y+I_y{B}_z-I_z{B}_y=0,\hfill \\ P_z{T}_x-P_x{T}_z+I_z{B}_x-I_x{B}_z=B{\dot{\omega }}_y,\hfill \\ P_x{T}_y-P_y{T}_x+I_x{B}_y-I_y{B}_x=0,\hfill \\ P_x{T}_x+P_y{T}_y+P_z{T}_z=0,\hfill \\ I_x{B}_x+I_y{B}_y+I_z{B}_z=0,\hfill \\ P_y=0.\hfill \end{array}\right.$$

(38)

The solution of the system (38) includes components of controlled vectors ${\overrightarrow{P}}_{\mathrm{comp}}$ and ${\overrightarrow{I}}_{\mathrm{comp}}$ in the following form:

$$\begin{array}{c}\hfill {\overrightarrow{P}}_{comp}=\left({\displaystyle \frac{B{B}_y{T}_z{\dot{\omega }}_y}{T_x{T}_y{B}_x+T_y{T}_z{B}_z-T_x^2{B}_y-T_z^2{B}_y}},0,\right.\\ \hfill {\left.-{\displaystyle \frac{B{B}_y{T}_x{\dot{\omega }}_y}{T_x{T}_y{B}_x+T_y{T}_z{B}_z-T_x^2{B}_y-T_z^2{B}_y}}\right)}^{\top },\end{array}$$

(39)

$$\begin{array}{c}\hfill {\overrightarrow{I}}_{comp}=\left(-{\displaystyle \frac{B{T}_y{\dot{\omega }}_y\left(B_y^2{T}_z+B_z{T}_x{B}_x+B_z^2{T}_z\right)}{\left(T_x{T}_y{B}_x+T_y{T}_z{B}_z-T_x^2{B}_y-T_z^2{B}_y\right)\left(B_x^2+B_y^2+B_z^2\right)}},\right.\\ \hfill {\displaystyle \frac{B{T}_y{\dot{\omega }}_y{B}_y\left(B_x{T}_z-B_z{T}_x\right)}{\left(T_x{T}_y{B}_x+T_y{T}_z{B}_z-T_x^2{B}_y-T_z^2{B}_y\right)\left(B_x^2+B_y^2+B_z^2\right)}},\\ \hfill {\left.{\displaystyle \frac{B{T}_y{\dot{\omega }}_y\left(B_y^2{T}_x+B_z{T}_z{B}_x+B_x^2{T}_x\right)}{\left(T_x{T}_y{B}_x+T_y{T}_z{B}_z-T_x^2{B}_y-T_z^2{B}_y\right)\left(B_x^2+B_y^2+B_z^2\right)}}\right)}^{\top }.\end{array}$$

(40)

Thus, the problems of compensating the disturbing torque and creating the restoring and dissipative torques are simultaneously solved by taking the vectors $\overrightarrow{P}$ and $\overrightarrow{I}$ as

$$\overrightarrow{P}={\overrightarrow{P}}_{\mathrm{comp}}+{\overrightarrow{P}}_{\mathrm{rest}}+{\overrightarrow{P}}_{\mathrm{diss}},\overrightarrow{I}={\overrightarrow{I}}_{\mathrm{comp}}+{\overrightarrow{I}}_{\mathrm{rest}}+{\overrightarrow{I}}_{\mathrm{diss}}.$$

(41)

As a result, the expression for the control torques have the following form:

$${\overrightarrow{M}}_L=\left({\overrightarrow{P}}_{\mathrm{comp}}+{\overrightarrow{P}}_{\mathrm{rest}}+{\overrightarrow{P}}_{\mathrm{diss}}\right)\times \overrightarrow{T},{\overrightarrow{M}}_M=\left({\overrightarrow{I}}_{\mathrm{comp}}+{\overrightarrow{I}}_{\mathrm{rest}}+{\overrightarrow{I}}_{\mathrm{diss}}\right)\times \overrightarrow{B}.$$

(42)

The differential equations of the controlled attitude motion of the spacecraft are obtained by substituting (42) into the dynamic Euler Equation (29):

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}\left[\mathbf{J}\left({\overrightarrow{\omega }}_{*}{\mathbf{A}}^{\top }{\overrightarrow{\eta }}_0+{\overrightarrow{\omega }}^{\prime }\right)\right]+\left[{\overrightarrow{\omega }}_{*}{\mathbf{A}}^{\top }{\overrightarrow{\eta }}_0+{\overrightarrow{\omega }}^{\prime }\right]\times \left[\mathbf{J}\left({\overrightarrow{\omega }}_{*}{\mathbf{A}}^{\top }{\overrightarrow{\eta }}_0+{\overrightarrow{\omega }}^{\prime }\right)\right]\\ \hfill =3{\omega }_0^2\left({\mathbf{A}}^{\top }{\overrightarrow{\zeta }}_0\right)\times \left(\mathbf{J}{\mathbf{A}}^{\top }{\overrightarrow{\zeta }}_0\right){\left(1+ecos\left(\nu \right)\right)}^3+\left(Q{k}_L{\overrightarrow{T}}_0+Q{h}_L{\overrightarrow{\omega }}^{\prime }\times \overrightarrow{T}\right.\\ \hfill \left.-{\displaystyle \frac{B{B}_y{T}_z{\dot{\omega }}_y}{{\left[\overrightarrow{T}\times \left(\overrightarrow{B}\times \overrightarrow{T}\right)\right]}_y}}\overrightarrow{i}+{\displaystyle \frac{B{B}_y{T}_x{\dot{\omega }}_y}{{\left[\overrightarrow{T}\times \left(\overrightarrow{B}\times \overrightarrow{T}\right)\right]}_y}}\overrightarrow{k}\right)\times \overrightarrow{T}\\ \hfill +\left(k_M{\overrightarrow{B}}_0+h_M{\overrightarrow{\omega }}^{\prime }\times \overrightarrow{B}-{\displaystyle \frac{B{T}_y{\dot{\omega }}_y\left(B_y^2{T}_z+B_z{T}_x{B}_x+B_z^2{T}_z\right)}{{\left[\left(\overrightarrow{B}\times \overrightarrow{T}\right)\times \overrightarrow{T}\right]}_y∥\overrightarrow{B}∥}}\overrightarrow{i}\right.\\ \hfill \left.-{\displaystyle \frac{B{T}_y{\dot{\omega }}_y{B}_y\left(B_x{T}_z-B_z{T}_x\right)}{{\left[\overrightarrow{T}\times \left(\overrightarrow{B}\times \overrightarrow{T}\right)\right]}_y∥\overrightarrow{B}∥}}\overrightarrow{j}-{\displaystyle \frac{B{T}_y{\dot{\omega }}_y\left(B_y^2{T}_x+B_z{T}_z{B}_x+B_x^2{T}_x\right)}{{\left[\overrightarrow{T}\times \left(\overrightarrow{B}\times \overrightarrow{T}\right)\right]}_y∥\overrightarrow{B}∥}}\overrightarrow{k}\right)\times \overrightarrow{B}.\end{array}$$

(43)

The Equation (43) must be combined with the kinematic Poisson Equation (30). In such a case, the position (34) is the direct equilibrium position of the system of differential Equations (30) and (43). The process of spacecraft stabilization in this position is actually implemented using the feedforward method, the block diagram of which is shown in Figure 6.

7. Computer Modeling and Numerical Integration Results

The Equations (27) and (28) should be transformed into a convenient form for numerical integration by using Rodrigues–Hamilton parameters, which are components of the quaternion $\Lambda ={\left({\lambda }_0,{\lambda }_1,{\lambda }_2,{\lambda }_3\right)}^{\top }$. Rodrigues–Hamilton parameters satisfy the normalization condition:

$${\Lambda }^2={\lambda }_0^2+{\lambda }_1^2+{\lambda }_2^2+{\lambda }_3^2=1.$$

(44)

The components ${\alpha }_i,{\beta }_i,{\gamma }_i$ of the rotation matrix $\mathbf{A}$ (3) are expressed in terms of Rodrigues–Hamilton parameters:

$$\begin{array}{ccc}{\alpha }_1={\lambda }_0^2+{\lambda }_1^2-{\lambda }_2^2-{\lambda }_3^2,\hfill & {\alpha }_2=2\left({\lambda }_1{\lambda }_2-{\lambda }_0{\lambda }_3\right),\hfill & {\alpha }_3=2\left({\lambda }_1{\lambda }_3+{\lambda }_0{\lambda }_2\right),\hfill \\ {\beta }_1=2\left({\lambda }_1{\lambda }_2+{\lambda }_0{\lambda }_3\right),\hfill & {\beta }_2={\lambda }_0^2+{\lambda }_2^2-{\lambda }_1^2-{\lambda }_3^2,\hfill & {\beta }_3=2\left({\lambda }_2{\lambda }_3-{\lambda }_0{\lambda }_1\right),\hfill \\ {\gamma }_1=2\left({\lambda }_1{\lambda }_3-{\lambda }_0{\lambda }_2\right),\hfill & {\gamma }_2=2\left({\lambda }_2{\lambda }_3+{\lambda }_0{\lambda }_1\right),\hfill & {\gamma }_3={\lambda }_0^2+{\lambda }_3^2-{\lambda }_1^2-{\lambda }_2^2.\hfill \end{array}$$

(45)

The kinematic Poisson Equation (28) using Rodrigues–Hamilton parameters take the following form:

$$\left\{\begin{array}{c}2{\dot{\lambda }}_0=-{\lambda }_1{\omega }_x-{\lambda }_2{\omega }_y-{\lambda }_3{\omega }_z+{\lambda }_2{\omega }_{*},\hfill \\ 2{\dot{\lambda }}_1={\lambda }_0{\omega }_x+{\lambda }_2{\omega }_z-{\lambda }_3{\omega }_y-{\lambda }_3{\omega }_{*},\hfill \\ 2{\dot{\lambda }}_2={\lambda }_0{\omega }_y+{\lambda }_3{\omega }_x-{\lambda }_1{\omega }_z-{\lambda }_0{\omega }_{*},\hfill \\ 2{\dot{\lambda }}_3={\lambda }_0{\omega }_z+{\lambda }_1{\omega }_y-{\lambda }_2{\omega }_x+{\lambda }_1{\omega }_{*}.\hfill \end{array}\right.$$

(46)

Let us rewrite the system (46) to a dimensionless form in order to perform numerical integration on a computer. To conduct this, we pass from the time variable t to the dimensionless variable $\nu$ (true anomaly) and also introduce a dimensionless angular velocity as

$$\overrightarrow{\mathrm{\Omega }}={\omega }_0^{-1}\overrightarrow{\omega }.$$

(47)

Finally, kinematic Poisson equations that are convenient for numerical integration have the following form:

$$\left\{\begin{array}{c}{\lambda }_0^{\prime }={\displaystyle \frac{1}{2}}\left[{\left(1+ecos\nu \right)}^{-2}\left(-{\lambda }_1{\mathrm{\Omega }}_x-{\lambda }_2{\mathrm{\Omega }}_y-{\lambda }_3{\mathrm{\Omega }}_z\right)+{\lambda }_2-k{\lambda }_0\left({\Lambda }^2-1\right)\right],\hfill \\ {\lambda }_1^{\prime }={\displaystyle \frac{1}{2}}\left[{\left(1+ecos\nu \right)}^{-2}\left({\lambda }_0{\mathrm{\Omega }}_x+{\lambda }_2{\mathrm{\Omega }}_z-{\lambda }_3{\mathrm{\Omega }}_y\right)-{\lambda }_3-k{\lambda }_1\left({\Lambda }^2-1\right)\right],\hfill \\ {\lambda }_2^{\prime }={\displaystyle \frac{1}{2}}\left[{\left(1+ecos\nu \right)}^{-2}\left({\lambda }_0{\mathrm{\Omega }}_y+{\lambda }_3{\mathrm{\Omega }}_x-{\lambda }_1{\mathrm{\Omega }}_z\right)-{\lambda }_0-k{\lambda }_2\left({\Lambda }^2-1\right)\right],\hfill \\ {\lambda }_3^{\prime }={\displaystyle \frac{1}{2}}\left[{\left(1+ecos\nu \right)}^{-2}\left({\lambda }_0{\mathrm{\Omega }}_z+{\lambda }_1{\mathrm{\Omega }}_y-{\lambda }_2{\mathrm{\Omega }}_x\right)+{\lambda }_1-k{\lambda }_3\left({\Lambda }^2-1\right)\right],\hfill \end{array}\right.$$

(48)

where k is a positive constant. Similarly, let us rewrite the dynamic Euler Equation (27) to a dimensionless form and divide both parts of the system (27) by $A{\omega }_0^2$, $B{\omega }_0^2$ and $C{\omega }_0^2$, respectively:

$$\left\{\begin{array}{c}{\mathrm{\Omega }}_x^{\prime }{\left(1+ecos\nu \right)}^2+\left(\epsilon -\delta \right){\mathrm{\Omega }}_y{\mathrm{\Omega }}_z=3\left(\epsilon -\delta \right){\left(1+ecos\nu \right)}^3{\gamma }_2{\gamma }_3+{\tilde{M}}_x,\hfill \\ {\mathrm{\Omega }}_y^{\prime }{\left(1+ecos\nu \right)}^2+{\displaystyle \frac{1-\epsilon }{\delta }}{\mathrm{\Omega }}_z{\mathrm{\Omega }}_x=3{\displaystyle \frac{1-\epsilon }{\delta }}{\left(1+ecos\nu \right)}^3{\gamma }_3{\gamma }_1+{\tilde{M}}_y,\hfill \\ {\mathrm{\Omega }}_z^{\prime }{\left(1+ecos\nu \right)}^2+{\displaystyle \frac{\delta -1}{\epsilon }}{\mathrm{\Omega }}_y{\mathrm{\Omega }}_z=3{\displaystyle \frac{\delta -1}{\epsilon }}{\left(1+ecos\nu \right)}^3{\gamma }_1{\gamma }_2+{\tilde{M}}_z,\hfill \end{array}\right.$$

(49)

where dimensionless inertial parameters are defined as $\delta =B/A$, $\epsilon =C/A$ and ${\tilde{M}}_x=M_x/\left(A{\omega }_0^2\right),{\tilde{M}}_y=M_y/\left(\delta A{\omega }_0^2\right),{\tilde{M}}_z=M_z/\left(\epsilon A{\omega }_0^2\right)$ are dimensionless projections of the main torque of non-gravitational forces, and ${\left(\phantom{n}\right)}^{\prime }$ denote the derivative with respect to dimensionless variable $\nu$.

To set the initial conditions, aircraft angles $\phi ,\vartheta ,\psi$ (roll, pitch and yaw) are used, which are related to Rodrigues–Hamilton parameters in accordance with the following formulae:

$$\begin{array}{c}{\lambda }_0=cos\left({\displaystyle \frac{\phi }{2}}\right)cos\left({\displaystyle \frac{\vartheta }{2}}\right)cos\left({\displaystyle \frac{\psi }{2}}\right)+sin\left({\displaystyle \frac{\phi }{2}}\right)sin\left({\displaystyle \frac{\vartheta }{2}}\right)sin\left({\displaystyle \frac{\psi }{2}}\right),\\ {\lambda }_1=sin\left({\displaystyle \frac{\phi }{2}}\right)cos\left({\displaystyle \frac{\vartheta }{2}}\right)cos\left({\displaystyle \frac{\psi }{2}}\right)-cos\left({\displaystyle \frac{\phi }{2}}\right)sin\left({\displaystyle \frac{\vartheta }{2}}\right)sin\left({\displaystyle \frac{\psi }{2}}\right),\\ {\lambda }_2=cos\left({\displaystyle \frac{\phi }{2}}\right)sin\left({\displaystyle \frac{\vartheta }{2}}\right)cos\left({\displaystyle \frac{\psi }{2}}\right)+sin\left({\displaystyle \frac{\phi }{2}}\right)cos\left({\displaystyle \frac{\vartheta }{2}}\right)sin\left({\displaystyle \frac{\psi }{2}}\right),\\ {\lambda }_3=cos\left({\displaystyle \frac{\phi }{2}}\right)cos\left({\displaystyle \frac{\vartheta }{2}}\right)sin\left({\displaystyle \frac{\psi }{2}}\right)-sin\left({\displaystyle \frac{\phi }{2}}\right)sin\left({\displaystyle \frac{\vartheta }{2}}\right)cos\left({\displaystyle \frac{\psi }{2}}\right).\end{array}$$

(50)

A series of systematic numerical experiments using various parameters of the spacecraft and initial conditions of motion has been organized. Further, as an example, the results of numerical modeling of the angular stabilization of the spacecraft for the following parameter values and initial conditions are presented as follows:

• orbit parameters: eccentricity $e=0.1$, orbital inclination $i={\displaystyle \frac{\pi }{30}}$, focal parameter $p_0=7·{10}^6$;
• spacecraft parameters: moments of inertia $A=1000$ $\mathrm{kg}·{\mathrm{m}}^2$, $\delta =B/A=0.7$, $\epsilon =C/A=0.8$, total charge $Q=5·{10}^{-3}$ C;
• control parameters: $k_M=2·{10}^6$, $k_L=10$, $h_M={10}^9$, $h_L=8·{10}^2$;
• initial deviation: ${\phi }_0=0,{\psi }_0=-0.2,{\vartheta }_0=0.1$;
• initial angular velocity components: ${\mathrm{\Omega }}_{x0}=0.1,{\mathrm{\Omega }}_{y0}=-0.1,{\mathrm{\Omega }}_{z0}=0.1$.

The Rodrigues–Hamilton parameters’ time history is presented in Figure 7. Let us note that the programmed attitude motion of spacecraft (34) is in accordance with the following set of aircraft angles and angular velocity: $\phi =0,\psi =0,\vartheta =0$, ${\overrightarrow{\omega }}^{\prime }=\overrightarrow{0}$. Corresponding graphs presenting the convergence of the stabilization process are shown in Figure 8 and Figure 9. It can be seen that the convergence process is smooth and fast enough.

The analysis of the stabilization process and the selection of control coefficients $k_M$, $k_L$, $h_M$, $h_L$ was carried out taking into account the possibilities of the technical implementation of the electrodynamic control system. The graphs below for comparing the magnitudes of the control torques and the disturbing gravitational torque indicate that the control torques have values of the same order of magnitude as the disturbing torque. The results presented in Figure 10 and Figure 11 confirm the feasibility of the proposed method of spacecraft attitude stabilization.

Also, the results of a series of numerical experiments confirm the suitability and operability of electrodynamic attitude control for spacecraft in an elliptic orbit.

8. Conclusions

In this paper, we have studied the problem of attitude control for a spacecraft in an elliptical near-Earth Keplerian orbit. It is known that the ellipticity of the spacecraft’s orbit is a disturbing factor that complicates the solution of the problem of spacecraft attitude stabilization and, consequently, complicates the fulfillment of spacecraft missions. Therefore, spacecraft operation in circular orbits is more preferable than in elliptical orbits. In some cases, with small values of eccentricity, the ellipticity of the orbit can be neglected. However, there are space missions in which it is not possible to neglect the ellipticity of the orbit, and therefore, it is necessary to develop spacecraft attitude stabilization systems that are capable of overcoming the noted difficulty. In this paper, it is shown that the electrodynamic attitude control system, previously used to stabilize various modes of spacecraft angular motion in circular orbits, can also be used for the angular stabilization of a spacecraft moving in an elliptical orbit. An electrodynamic attitude stabilization system based on simultaneously applied Lorentz and magnetic torques has been used. A mathematical model describing the attitude dynamics of the spacecraft under the action of the Lorentz torque, the magnetic interaction torque, and the disturbing gravitational torque is constructed. The possibility of electrodynamic attitude control for the spacecraft’s angular stabilization in the orbital frame is analyzed based on the Euler–Poisson differential equations. An electrodynamic control algorithm is proposed that makes it possible to cancel out the perturbing torque caused by the ellipticity of the orbit and acting on the spacecraft in the direct equilibrium position in the orbital coordinate system. Numerical integration of the system of equations confirming the attitude stabilization of the spacecraft in an elliptical orbit is performed. Thus, the possibility and advisability of using the electrodynamic method for the spacecraft attitude control and its angular stabilization in the orbital coordinate system in an elliptical orbit is shown.