Analysis of analytical attitude propagators for spin-stabilized satellites

Abstract

In this paper, analytical attitude propagators for spin-stabilized satellites are analyzed. For this purpose, external torques are introduced in the motion equations such as solar radiation pressure torque, aerodynamic torque, gravity gradient torque, and magnetic residual and eddy current torques. For the magnetic torques, it used both the dipole and quadripole Earth’s magnetic field models. The obtained analytical solution is applied and compared with actual data for Brazilian data collection satellites SCD1 and SCD2. The results, when a daily data update is applied, based on the INPE-Brazilian Institute for Space Research supplied data, show a good agreement of all involved parameters with the actual deviations in the parameter values within the precision required for the satellite mission. This permits to conclude that the used theory is suitable for the studied problem. Thus, the propagators presented in this work can be applied to predict the rotational motion of spin-stabilized artificial satellites, particularly for SCD1 and SCD2 satellites over the considered time period.

Appendix: Mathematical model for the external torques for spin-stabilized satellite

1.1 Gravity gradient torque model

The GGT (Wertz 1978; Zanardi et al. 2011) for a spin-stabilized satellite in a body system ($\hat{i},\,\,\hat{j},\,\,\hat{k},$) can be expressed by:

$$ \vec{N}_{\text{GG}} = N_{{{\text{GG}}x}} \,\hat{i} + N_{{{\text{GG}}y}} \,\hat{j} + N_{{{\text{GG}}z}} \,\hat{k}, $$

with

$$ N_{{{\text{GG}}x}} = 3\frac{\mu }{{r^{\prime 3} }}\left[ {a_{21} \,a_{31} \left( {I_{z} - I_{y} } \right)\,\cos \theta \, - \,a_{11} \,a_{31} \left( {I_{x} - I_{z} } \right){\text{sen}}\;\theta \,} \right], $$

$$ N_{{{\text{GG}}y}} = 3\frac{\mu }{{r^{\prime 3} }}\left[ {a_{21} {\mkern 1mu} a_{31} \left( {I_{z} - I_{y} } \right){\mkern 1mu} {\text{sen}}\;\theta {\mkern 1mu} + {\mkern 1mu} a_{11} {\mkern 1mu} a_{31} \left( {I_{x} - I_{z} } \right){\mkern 1mu} { \cos }\;\theta } \right], $$

$$ N_{{{\text{GG}}z}} = 3\frac{\mu }{{r^{3} }}\left[ {a_{11} \,a_{21} \left( {I_{y} - I_{x} } \right)} \right], $$

where µ (3.986 × 10¹⁴ m³/s²) is the Earth gravitational parameter, $r^{{{\prime } }}$ is the satellite geocentric distance, a₁₁, a₂₁, and a₃₁ are the direction cosines which relate the orbital system and the satellite fixed system, I_x, I_y, and I_z are the Principal Moments of Inertia of the satellite and θ is the angle between the satellite principal axis of inertia x′ and the satellite axis x, defined in each instance by the product of the spin velocity W and the time t. The elements a₁₁, a₂₁, and a₃₁ depend on the orbital elements (orbit inclination, true anomaly, longitude of the ascending node, and argument of the perigee), the angle θ, the right ascension, and declination of the spin axis (Zanardi et al. 2003).

To compute the mean components of each considered torque in the satellite system, an average time in the fast varying orbit element, the mean anomaly, is used. This approach involves several rotation matrices, which are dependent on the orbit elements, right ascension, and declination of the satellite spin axis. The GGT mean components are presented in Zanardi et al. (2003).

1.2 Aerodynamic torque model

For this paper is adopted the following model to represent the AT (Wertz 1978; Zanardi et al. 2011):

$$ \vec{N}_{\text{A}} = \left[ {D_{z} m_{ey} \, - \,D_{y} m_{ez} } \right]\,\hat{i} + \left[ {D_{x} m_{ez} \, - \,D_{z} m_{ex} } \right]\,\hat{j} + \left[ {D_{y} \,m_{ex} \, - D_{x} \,m_{ey} } \right]\,\hat{k}, $$

where m_ex, m_ey, m_ez and D_x, D_y, D_z are the components of the position vector between the center of pressure and the center of mass of the satellite and the drag force in a satellite system. In this paper, the influence of the lift force in the AT is neglected and

$$ D_{x} = - D\left[ {a_{11} \cos \,\gamma_{\text{S}} + a_{21} \,{\text{sen}}\,\gamma_{\text{S}} } \right], $$

$$ D_{y} = - D\left[ {a_{12} \cos \,\gamma_{\text{S}} + a_{22} \,{\text{sen}}\,\gamma_{\text{S}} } \right], $$

$$ D_{z} = - D\left[ {a_{13}^{{}} \cos \gamma_{\text{S}} + a_{32} {\text{sen}}\,\gamma_{\text{S}} } \right], $$

$$ D = \tfrac{1}{2}\rho \,v^{2} \,S\,C_{\text{D}} , $$

where ρ is the local density, v represents the magnitude of the satellite’s velocity relative to the atmosphere, S is a reference section area of the satellite, C_D is the drag coefficient, $\gamma_{S}$ is the angle between the position vector and the orbital velocity vector, and $a$_ij, i = 1, 2, 3, j = 1, 2, are the direction cosines which relate the orbital system and the satellite system (Zanardi et al. 2011).

To estimate the influence of the AT magnitude in the rotational motion, in this paper, the thermosphere model TD-88 is used for the atmospheric density (Sehnal and Pospísilová 1988) and some simplifications are done. The velocity v is assumed equal to the orbit velocity; the drag coefficient is fixed (equal to 2.2). In the applications for the SDC1 and SCD2 and by the analyses development in Zanardi et al. (2011), it also assumed that $m_{ex} = m_{ey } = 0$ and $m_{ez} = 0.1\;{\text{m}}$. Then in that case the z-component of AT is null. By the same way of the GGT, the AT mean components are presented by Zanardi et al. (2011).

1.3 Residual magnetic torque

The spacecraft’s magnetic moment is usually the dominant source between the disturbances torques. If $\vec{m}$ is the magnetic moment of the spacecraft and $\vec{B}$ is the geocentric magnetic field, the RMT is given by de Moraes and Zanardi (1999):

$$ \vec{N}_{\text{RM}} \, = \,\,\vec{m}\, \times \,\vec{B}\,. $$

Here the magnetic torque is developed only for a spin-stabilized satellite. In this case, the spacecraft’s spin velocity vector and the satellite magnetic moment are along z-axis and the RMT can be expressed in the satellite fixed system by Zanardi et al. (2005):

$$ \vec{N}_{\text{RM}} = \, - \,\,m\,B_{y\,\,} \hat{i} + \,m\,B_{x\,\,} \hat{j}, $$

where B_x, B_y, and B_z are the components of the geomagnetic field in the satellite fixed system (Wertz 1978). These components are obtained in terms of the geocentric inertial components of the geomagnetic field and the attitude angles of the satellite. To describe the geomagnetic field, the dipole model and quadripole model (Garcia et al. 2009) are used.

The mean components of this torque are presented in Garcia et al. (2009) and Zanardi et al. (2005).

1.4 Eddy current torque

The torque induced by eddy currents is caused by the spacecraft spinning motion. It is known (Wertz 1978) that the eddy currents produce a torque which causes the precession in the spin axis and causes an exponential decay of the spin rate. If $\vec{W}$ is the spacecraft’s spin velocity vector and p is a constant coefficient which depends on the spacecraft geometry and conductivity, this torque is given by Wertz (1978):

$$ \vec{N}_{\text{EC}} \, = \,p\,\,\vec{B}\, \times (\,\vec{B}\, \times \,\vec{W})\,. $$

For the spin-stabilized satellite the spin velocity vector and is along z-axis and the ECT can the expressed in the body system by Zanardi et al. (2003):

$$ \vec{N}_{\text{EC}} = pW\left( {B_{x} B_{z\,\,} \hat{i} + B{}_{y\,}B_{z\,\,} \hat{j}\, - (B_{x}^{2} + B_{y}^{2} )\hat{k}} \right). $$

The mean components of this torque are presented in Zanardi et al. (2011).

1.5 Solar radiation torque

A solar radiation torque model was developed in de Moraes and Zanardi (1999) for the case which the illuminated surfaces of the satellite are a circular flat surface S₁, and a portion of the cylindrical surface S₂. It is given by:

$$ \vec{N}_{\text{SR}} = N_{{{\text{SR}}x }} \hat{i} + N_{{{\text{SR}}y}} \hat{j}, $$

$$ N_{{{\text{SR}}x}} = - \frac{{\bar{K}}}{{R^{4} }}\left( {\beta_{1} \gamma_{1} - \beta_{2} \gamma_{2} } \right)\frac{h}{2}\pi \sigma^{2} u_{z}^{*} \left( {u_{y}^{*} {\text{cos }}\theta + u_{x}^{*} {\text{sin }}\theta } \right), $$

$$ N_{{{\text{SR}}y}} = - \frac{{\bar{K}}}{{R^{4} }}\left( {\beta_{1} \gamma_{1} - \beta_{2} \gamma_{2} } \right)\frac{h}{2}\pi \sigma^{2} u_{z}^{*} \left( {u_{y}^{*} \sin \theta - u_{x}^{*} \cos \theta } \right), $$

with

$$ u_{x}^{*} = \left( {a_{\text{s}} R_{x} + r^{{\prime }} a_{11} } \right), $$

$$ u_{y}^{*} = \left( {a_{\text{s}} R_{y} + r^{{\prime }} a_{12} } \right), $$

$$ u_{z}^{*} = \left( {a_{\text{s}} R_{z} + r^{{\prime }} a_{13} } \right), $$

where $\bar{K}$ is a solar parameter with assumed value given by 1.01 × 10¹⁷ kgm/s², a_s is Sun–Earth distance and here assume the value 1.49597870 × 10¹¹ m, r′ is the satellite geocentric distance, R is the Sun satellite distance, β_i, γ_i, i = 1, 2, are specular and total reflection coefficients, respectively, for each satellite surface (which assume constant values), a₁₁, a₁₂, and a₁₃ are direction cosines which relate the Orbital System and the principal system, given in Motta et al. (2013) and R_x, R_y, and R_z give the Sun direction in the body and are obtained by:

$$ R_{x} = b_{1} \cos \delta_{s} \cos \alpha_{s} + b_{2} \cos \delta_{\text{s}} { \sin } \,\alpha_{\text{s}} + b_{3} \sin \delta_{\text{s}}, $$

$$ R_{y} = b_{4} \cos \delta_{\text{s}} \cos \alpha_{\text{s}} + b_{5} \cos \delta_{\text{s}} { \sin }\, \alpha_{\text{s}} + b_{6} \sin \delta_{\text{s}}, $$

$$ R_{z} = b_{7} \cos \delta_{\text{s}} { \cos }\,\alpha_{\text{s}} + b_{8} \cos \delta_{s} {\text{sen}} \,\alpha_{\text{s}} + b_{9}\, {\text{sen }}\delta_{\text{s}}, $$

with being b₁, b₂,…, b₉ the direction cosines which relate the Equatorial System and satellite system (given in terms of satellite’s rotation angle, right ascension, and declination of the spin axis (Motta et al. 2013).

The Sun–satellite distance can be obtained by de Moraes and Zanardi (1999):

$$ R^{2} = a_{\text{s}}^{2} + r^{\prime 2} + 2r^{\prime } {\text{a}}_{\text{s}} \left( {a_{1} R_{x} + a_{2} R_{y} + a_{3} R_{z} } \right). $$

The mean components of SRT are presented in Motta et al. (2013).