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Mathematical Problems in Engineering
Volume 2009, Article ID 242396, 18 pages
http://dx.doi.org/10.1155/2009/242396
Research Article

Spin-Stabilized Spacecrafts: Analytical Attitude Propagation Using Magnetic Torques

1Space Mechanic and Control Division, National Institute for Space Research (INPE), Sao Paulo, São José dos Campos 12227-010, Brazil
2Department of Mathematics, Group of Orbital Dynamics and Planetology, Sao Paulo State University (UNESP), Sao Paulo, Guaratinguetá 12516-410, Brazil

Received 28 July 2009; Accepted 4 November 2009

Academic Editor: Antonio Prado

Copyright © 2009 Roberta Veloso Garcia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An analytical approach for spin-stabilized satellites attitude propagation is presented, considering the influence of the residual magnetic torque and eddy currents torque. It is assumed two approaches to examine the influence of external torques acting during the motion of the satellite, with the Earth's magnetic field described by the quadripole model. In the first approach is included only the residual magnetic torque in the motion equations, with the satellites in circular or elliptical orbit. In the second approach only the eddy currents torque is analyzed, with the satellite in circular orbit. The inclusion of these torques on the dynamic equations of spin stabilized satellites yields the conditions to derive an analytical solution. The solutions show that residual torque does not affect the spin velocity magnitude, contributing only for the precession and the drift of the spacecraft's spin axis and the eddy currents torque causes an exponential decay of the angular velocity magnitude. Numerical simulations performed with data of the Brazilian Satellites (SCD1 and SCD2) show the period that analytical solution can be used to the attitude propagation, within the dispersion range of the attitude determination system performance of Satellite Control Center of Brazil National Research Institute.

1. Introduction

This paper aims at analyzing the rotational motion dynamics of spin-stabilized Earth’s artificial satellites, through derivation of an analytical attitude prediction. Emphasis is placed on modeling the torques steaming from residual magnetic and eddy currents perturbations, as well as their influences on the satellite angular velocity and space orientation. A spherical coordinated system fixed in the satellite is used to locate the spin axis of the satellite in relation to the terrestrial equatorial system. The directions of the spin axis are specified by the right ascension (𝛼) and the declination (𝛿) as represented in Figure 1. The magnetic residual torque occurs due to the interaction between the Earth magnetic field and the residual magnetic moment along the spin axis of the satellite. The eddy currents torque appears due to the interaction of such currents circulating along the satellite structure chassis and the Earth’s magnetic field.

242396.fig.001
Figure 1: Orientation of the spin axis (̂𝑠): equatorial system (𝐾𝐼,𝐽,), satellite body frame reference system (̂̂̂𝑘𝑖,𝑗,), right ascension (𝛼), and declination (𝛿) of the spin axis.

The torque analysis is performed through the quadripole model for the Earth’s magnetic field and the satellite in circular and elliptical orbits. Essentially an analytical averaging method is applied to determine the mean torque over an orbital period.

To compute the average components of both the residual magnetic and eddy current torques in the satellite body frame reference system (satellite system), an average time in the fast varying orbit element, the mean anomaly, is utilized. This approach involves several rotation matrices, which are dependent on the orbit elements, right ascension and declination of the satellite spin axis, the magnetic colatitudes, and the longitude of ascending node of the magnetic plane.

Unlike the eddy currents torques, it is observed that the residual magnetic torque does not have component along the spin axis; however, it has nonzero components in satellite body x-axis and y-axis. Afterwards, the inclusion of such torques on the rotational motion differential equations of spin-stabilized satellites yields the conditions to derive an analytical solution [1]. The theory is developed accounting also for orbit elements time variation, not restricted to circular orbits, giving rise to some hundreds of curvature integrals solved analytically.

In order to validate the analytical approach, the theory developed has been applied for the spin-stabilized Brazilian Satellites (SCD1 and SCD2), which are quite appropriated for verification and comparison of the theory with the data generated and processed by the Satellite Control Center (SCC) of Brazil National Research Institute (INPE). The oblateness of the orbital elements is taken into account.

The behaviors of right ascension, declination, and spin velocity of the spin axis with the time are presented and the results show the agreement between the analytical solution and the actual satellite behavior.

2. Geomagnetic Field

It is well known that the Earth’s magnetic field can be obtained by the gradient of a scalar potential 𝑉 [2]; it means that

𝐵=𝑉,(2.1) with the magnetic potential 𝑉 given by

𝑉𝑟,𝜙,𝜃=𝑟𝑇𝑘𝑛=1𝑟𝑇𝑟𝑛𝑛+1𝑚=0𝑔𝑚𝑛cos𝑚𝜃+𝑚𝑛𝑃sen𝑚𝜃𝑚𝑛(𝜙),(2.2) where 𝑟𝑇 is the Earth’s equatorial radius, 𝑔𝑚𝑛, 𝑚𝑛 are the Gaussian coefficients, 𝑃𝑚𝑛(𝜙) are the Legendre associated polynomial and 𝑟, 𝜙, 𝜃 mean the geocentric distance, the local colatitudes, and local longitude, respectively.

In terms of spherical coordinates, the geomagnetic field can be expressed by [2],

𝐵=𝐵𝑟̂𝑟+𝐵𝜙𝜙+𝐵𝜃̂𝜃,(2.3) with 𝐵𝑟=𝜕𝑉𝜕𝑟,𝐵𝜙1=𝑟𝜕𝑉𝜕𝜙,𝐵𝜃1=𝑟sen𝜙𝜕V𝜕𝜃.(2.4) For the quadripole model, it is assumed that 𝑛 equals 1 and 2 and 𝑚 equals 0, 1 and 2 in (2.2). After straightforward computations, the geomagnetic field can be expressed by [3, 4]

𝐵𝑟𝑟=2𝑇𝑟3𝑓1𝑟(𝜃,𝜙)+3𝑇𝑟4𝑓2𝐵(𝜃,𝜙),(2.5)𝜙𝑟=𝑇𝑟3𝑓3𝑟(𝜃,𝜙)𝑇𝑟4𝑓4𝐵(𝜃,𝜙),(2.6)𝜃1=𝑟sen𝜙𝑇𝑟3𝑓5(𝑟𝜃,𝜙)+𝑇𝑟4𝑓6(𝑟𝜃,𝜙)+2𝑇𝑟4𝑓7(𝜃,𝜙),(2.7) where the functions 𝑓𝑖, 𝑖=1,2,,7, are shown in [3] and depend on the Gaussian coefficients 𝑔22,11,12,22.

In the Equator reference system, the geomagnetic field is expressed by [2]

𝐵𝑋=𝐵𝑟cos𝛿+𝐵𝜙sen𝛿cos𝛼𝐵𝜃sen𝐵𝛼,(2.8)𝑌=𝐵𝑟cos𝛿+𝐵𝜙sen𝛿sen𝛼𝐵𝜃cos𝐵𝛼,(2.9)𝑍=𝐵𝑟sen𝛿+𝐵𝜙cos𝛿,(2.10) where 𝛼and𝛿are the right ascension and declination of the satellite position vector, respectively, which can be obtained in terms of the orbital elements; 𝐵𝑟, 𝐵𝜙, and 𝐵𝜃 are given by (2.5), (2.6), and (2.7), respectively.

In a satellite reference system, in which the axis z is along the spin axis, the geomagnetic field is given by [4, 5]

𝐵=𝐵𝑥̂𝑖+𝐵𝑦̂𝑗+𝐵𝑧̂𝑘,(2.11) where

𝐵𝑥=𝐵𝑋sen𝛼+𝐵𝑌𝐵cos𝛼,𝑦=𝐵𝑋sen𝛿cos𝛼𝐵𝑌sen𝛿sen𝛼+𝐵𝑍𝐵cos𝛿,𝑧=𝐵𝑋cos𝛿cos𝛼𝐵𝑌cos𝛿sen𝛼+𝐵𝑍sen𝛿,(2.12) with 𝐵𝑋, 𝐵𝑌, and 𝐵𝑍given by (2.8)–(2.10).

3. Residual and Eddy Currents Torques

Magnetic residual torques result from the interaction between the spacecraft’s residual magnetic field and the Earth’s magnetic fields. If 𝑚 is the magnetic moment of the spacecraft and 𝐵 is the geomagnetic field, then the residual magnetic torques are given by [2]

𝑁𝑟=𝑚×𝐵.(3.1) For the spin-stabilized satellite, with appropriate nutation dampers, the magnetic moment is mostly aligned along the spin axis and the residual torque can be expressed by [5]

𝑁𝑟=𝑀𝑠̂𝑘×𝐵,(3.2) where 𝑀𝑠 is the satellite magnetic moment along its spin axis and ̂𝑘 is the unit vector along the spin axis of the satellite.

By substituting the geomagnetic field (2.11) in (3.1), the instantaneous residual torque is expressed by

𝑁𝑟=𝑀𝑠𝐵𝑦̂𝑖+𝐵𝑥̂𝑗.(3.3) On the other hand, the eddy currents torque is caused by the spacecraft spinning motion. If 𝑊 is the spacecraft’s angular velocity vector and 𝑝 is the Foucault parameter representing the geometry and material of the satellite chassis [2], then this torque may be modeled by [2]

𝑁𝑖𝑊=𝑝𝐵×𝐁×.(3.4) For a spin-stabilized satellite, the spacecraft’s angular velocity vector and the satellite magnetic moment, along the z-axis and induced eddy currents torque, can be expressed by [5, 6]

𝑁𝑖=𝑝𝑊𝐵𝑥𝐵𝑧̂𝑖𝐵𝑦𝐵𝑧̂𝐵𝑗+2𝑦+𝐵2𝑥̂𝑘.(3.5)

4. Mean Residual and Eddy Currents Torques

In order to obtain the mean residual and eddy currents torques, it is necessary to integrate the instantaneous torques 𝑁𝑟and 𝑁𝑖, given in (3.3) and (3.5), over one orbital period 𝑇as

𝑁𝑟𝑚=1𝑇𝑡𝑖𝑡+𝑇𝑖𝑁𝑟𝑁𝑑𝑡,𝑖𝑚=1𝑇𝑡𝑖𝑡+𝑇𝑖𝑁𝑖𝑑𝑡,(4.1) where 𝑡 is the time 𝑡𝑖 the initial time, and 𝑇 the orbital period. Changing the independent variable to the fast varying true anomaly, the mean residual and eddy currents torque can be obtained by [4]

𝑁𝑟𝑚=1𝑇𝜐𝑖𝜐+2𝜋𝑖𝑁𝑟𝑟2𝑁𝑑𝜐,𝑖𝑚=1𝑇𝜐𝑖𝜐+2𝜋𝑖𝑁𝑖𝑟2𝑑𝜐,(4.2) where 𝜐𝑖 is the true anomaly at instant 𝑡𝑖, 𝑟 is the geocentric distance, and is the specific angular moment of orbit.

To evaluate the integrals of (4.2), we can use spherical trigonometry properties, rotation matrix associated with the references systems, and the elliptic expansions of the true anomaly in terms of the mean anomaly [7], including terms up to first order in the eccentricity (e). Without losing generality, for the sake of simplification of the integrals, we consider the initial time for integration equal to the instant that the satellite passes through perigee. After extensive but simple algebraic developments, the mean residual and eddy currents torques can be expressed by [3, 6]

𝑁𝑟𝑚=𝑁𝑟𝑥𝑚̂𝑖+𝑁𝑟𝑦𝑚̂𝑁𝑗,𝑖𝑚=𝑝𝑊𝑁2𝜋𝑖𝑥𝑚̂𝑖+𝑁𝑖𝑦𝑚̂𝑗+𝑁𝑖𝑧𝑚̂𝑘,(4.3) with

𝑁𝑟𝑥𝑚=𝑀𝑠(𝑁2𝜋𝐴sen𝛿cos𝛼+𝐵sen𝛿sen𝛼𝐶sen𝛿),𝑟𝑦𝑚=𝑀𝑠2𝜋(𝐷sen𝛼+𝐸cos𝛿)(4.4) and 𝑁𝑖𝑥𝑚, 𝑁𝑖𝑦𝑚, 𝑁𝑖𝑧𝑚 as well as the coefficients 𝐴, 𝐵, 𝐶, 𝐷, and 𝐸 are presented in the appendix. It is important to observe that the mean components of these torques depend on the attitude angles (𝛿, 𝛼) and the orbital elements (orbital major semi-axis: a, orbital eccentricity: e, longitude of ascending node: Ω, argument of perigee: 𝜔, and orbital inclination: 𝑖).

5. The Rotational Motion Equations

The variations of the angular velocity, the declination, and the ascension right of the spin axis for spin-stabilized artificial satellites are given by Euler equations in spherical coordinates [5] as

̇1𝑊=𝐼𝑧𝑁𝑧,̇1𝛿=𝐼𝑧𝑊𝑁𝑦,1̇𝛼=𝐼𝑧𝑁𝑊Cos𝛿𝑥,(5.1) where 𝐼𝑧 is the moment of inertia along the spin axis and 𝑁𝑥, 𝑁𝑦, 𝑁𝑧 are the components of the external torques in the satellite body frame reference system. By substituting 𝑁𝑟𝑚, given in (4.3), in (5.1), the equations of motion are

𝑑𝑊𝑑𝑡=0,(5.2)𝑑𝛿=𝑁𝑑𝑡𝑟𝑦𝑚𝐼𝑧𝑊,(5.3)𝑑𝛼=𝑁𝑑𝑡𝑟𝑥𝑚𝐼𝑧,𝑊cos𝛿(5.4) where it is possible to observe that the residual torque does not affect the satellite angular velocity (because its z-axis component is zero).

By substituting 𝑁𝑖𝑚, given in (4.3), in (5.1), the equations of motion are

𝑑𝑊=𝑑𝑡𝑝𝑊2𝜋𝐼𝑧𝑁𝑖𝑧𝑚,(5.5)𝑑𝛿=𝑑𝑡𝑝𝑊2𝜋𝐼𝑧𝑁𝑖𝑦𝑚,(5.6)𝑑𝛼=𝑝𝑑𝑡2𝜋𝐼𝑧𝑁cos𝛿𝑖𝑥𝑚.(5.7) The differential equations of (5.2)–(5.4) and (5.5)–(5.7) can be integrated assuming that the orbital elements (𝐼, Ω, 𝑤) are held constant over one orbital period and that all other terms on right-hand side of equations are equal to initial values.

6. Analysis of the Angular Velocity Magnitude

The variation of the angular velocity magnitude, given by (5.5), can be expressed as:

𝑑𝑊𝑊𝑁=𝑘𝑑𝑡,with𝑘=𝑖𝑧𝑚𝑝2𝜋𝐼𝑧.(6.1) If the parameter 𝑘 is considered constant for one orbital period, then the analytical solution of (6.1) is

𝑊=𝑊0𝑒𝑘𝑡,(6.2) where 𝑊0 is the initial angular velocity. If the coefficient 𝑘<0 in (6.2), then the angular velocity magnitude decays with an exponential profile.

7. Analysis of the Declination and Right Ascension of Spin Axis

For one orbit period, the analytical solutions of (5.3)-(5.4) and (5.6)-(5.7) for declination and right ascension of spin axis, respectively, can simply be expressed as,

𝛿=𝑘1𝑡+𝛿0,(7.1)𝛼=𝑘2𝑡+𝛼0,(7.2) with:

(i)for the case where the residual magnetic torque is considered in the motion equations, 𝑘1=𝑁𝑟𝑦𝑚𝐼𝑧𝑊𝑜,𝑘2=𝑁𝑟𝑦𝑚𝐼𝑧𝑊𝑜cos𝛿𝑜,(7.3)(ii)for the case where the eddy currents torque is considered in the motion equations,

𝑘1=𝑝𝑁𝑖𝑦𝑚2𝜋𝐼𝑧,𝑘2=𝑝𝑁𝑖𝑦𝑚2𝜋𝐼𝑧cos𝛿𝑜,(7.4) where 𝑊0, 𝛿0, and 𝛼0 are the initial values for spin velocity, declination, and right ascension of spin axis.

The solutions presented in (7.1) and (7.2), for the spin velocity magnitude, declination and right ascension of the spin axis, respectively, are valid for one orbital period. Thus, for every orbital period, the orbital data must be updated, taking into account at least the main influences of the Earth’s oblateness. With this approach, the analytical theory will be close to the real attitude behavior of the satellite.

8. Applications

The theory developed has been applied to the spin-stabilized Brazilian Satellites (SCD1 and SCD2) for verification and comparison of the theory against data generated by the Satellite Control Center (SCC) of INPE. Operationally, SCC attitude determination comprises [8, 9] sensors data preprocessing, preliminary attitude determination, and fine attitude determination. The preprocessing is applied to each set of data of the attitude sensors that collected every satellite that passes over the ground station. Afterwards, from the whole preprocessed data, the preliminary attitude determination produces estimates to the spin velocity vector from every satellite that passes over a given ground station. The fine attitude determination takes (one week) a set of angular velocity vector and estimates dynamical parameters (angular velocity vector, residual magnetic moment, and Foucault parameter). Those parameters are further used in the attitude propagation to predict the need of attitude corrections. Over the test period, there are not attitude corrections. The numerical comparison is shown considering the quadripole model for the geomagnetic field and the results of the circular and elliptical orbits. It is important to observe that, by analytical theory that included the residual torque, the spin velocity is considered constant during 24 hours. In all numerical simulations, the orbital elements are updated, taking into account the main influences of the Earth’s oblateness.

9. Results for SCD1 Satellite

The initial conditions of attitude had been taken on 22 of August of 1993 to the 00:00:00 GMT, supplied by the INPE’s Satellite Control Center (SCC). Tables 1, 2, and 3 show the results with the data from SCC and computed values by the present analytical theory, considering the quadripole model for the geomagnetic field and the satellite in circular and elliptical orbit, under influence of the residual and eddy currents torques.

tab1
Table 1: INPE’s Satellite Control Center Data (index SCC) and computed results with the satellite in elliptical orbit and under the influence of the residual magnetic torque (index QER) for declination and right ascension and SCD1 (in degrees).
tab2
Table 2: INPE’s Satellite Control Center Data (index SCC) and computed results with the satellite in circular orbit and under the influence of the residual magnetic torque (index QCR) for declination and right ascension and SCD1 (in degrees).
tab3
Table 3: INPE’s Satellite Control Center Data (index SCC) and computed results with the satellite in circular orbit and under the influence of the eddy current torque (index QCI) for declination and right ascension and SCD1 (in degrees).

The mean deviation errors for the right ascension and declination are shown in Table 4 for different time simulations. The behavior of the SCD1 attitude over 11 days is shown in Figure 2. It is possible to note that mean error increases with the time simulation. For more than 3 days, the mean error is bigger than the required dispersion range of SCC.

tab4
Table 4: Mean deviations for different time simulations for declination and right ascension and SCD1 (in degrees).
fig2
Figure 2: Evolution of the declination (𝛿) and right ascension (𝛼) of satellite spin axis for SCD1 and its mean deviation error.

Over the 3 days of test period, better results are obtained for the satellite in circular orbit with the residual torque. In this case, the difference between theory and SCC data has mean deviation error in right ascension of 0.2444° and −0.2241° for the declination. Both are within the dispersion range of the attitude determination system performance of INPE’s Control Center.

In Table 5 is shown the computed results to spin velocity when the satellite is under influence of the eddy currents torque, and its behavior over 11 days is shown in Figure 3. The mean error deviation for the spin velocity is shown in Table 6 for different time simulation. For the test period of 3 days, the mean deviation error in spin velocity was of −0.0312 rpm and is within the dispersion range of the attitude determination system performance of INPE’s Control Center.

tab5
Table 5: INPE’s Satellite Control Center Data (index SCC) and computed results for spin velocity, with the satellite in circular orbit and under the influence of the eddy currents torque (index QCI) (in rpm).
tab6
Table 6: Mean deviations for different time simulations for spin velocity and SCD1 (in degrees).
242396.fig.003
Figure 3: Evolution of the spin velocity (𝑊) for SCD1.

10. Results for SCD2 Satellite

The initial conditions of attitude had been taken on 12 February 2002 at 00:00:00 GMT, supplied by the SCC. In the same way for SCD1, Tables 7, 8, and 9 presented the results with the data from SCC and computed values by circular and elliptical orbits with the satellite under the influence of the residual magnetic torque and eddy currents torque.

tab7
Table 7: INPE’s Satellite Control Center Data (index SCC) and computed results with the satellite in elliptical orbit and under the influence of the residual magnetic torque (index QER) for declination and right ascension and SCD2 (in degrees).
tab8
Table 8: INPE’s Satellite Control Center Data (index SCC) and computed results with the satellite in circular orbit and under the influence of the residual magnetic torque (index QCR) for declination and right ascension and SCD2 (in degrees).
tab9
Table 9: INPE’s Satellite Control Center Data (index SCC) and computed results with the satellite in circular orbit and under the influence of the eddy currents torque (index QCI) for declination and right ascension and SCD2 (in degrees).

The mean deviation errors are shown in Table 10 for different time simulations. For this satellite, there is no significant difference between the circular and elliptical orbits when considering the residual magnetic torque. The behavior of the SCD2 attitude over 12 days is shown in Figure 4.

tab10
Table 10: Mean deviations for different time simulation for declination and right ascension and SCD2 (in degrees).
fig4
Figure 4: Evolution of the declination (𝛿) and right ascension (𝛼) of satellite spin axis for SCD2 and its mean deviation error.

Over the test period of the 12 days with the satellite in elliptical orbit and considering the residual magnetic torque, the difference between theory and SCC data has mean deviation error in right ascension of −0.1266 and −0.1358 in the declination. Both torques are within the dispersion range of the attitude determination system performance of INPE’s Control Center, and the solution can be used for more than 12 days.

In Table 11 the computed results to spin velocity are shown when the satellite is under the influence of the eddy currents torque. The mean deviation error for the spin velocity is shown in Table 12 for different time simulation. For the test period, the mean deviation error in spin velocity was of 0.0253 rpm and it is within the dispersion range of the attitude determination system performance of INPE’s Control Center. The behavior of the spin velocity is shown in Figure 5.

tab11
Table 11: INPE’s Satellite Control Center Data (index SCC) and computed results of spin velocity, with the satellite in circular orbit and under the influence of the eddy currents torque (index QCI) (in rpm).
tab12
Table 12: Mean deviations for different time simulations for spin velocity and SCD1 (in degrees).
242396.fig.005
Figure 5: Evolution of the spin velocity magnitude (𝑊) for SCD2.

11. Mean Pointing Deviation

For the tests, it is important to observe the deviation between the actual SCC supplied and the analytically computed attitude, for each satellite. It can be computed by

𝜃=cos1̂𝑖̂𝑖𝑐+𝑗̂𝑗𝑐+̂𝑘̂𝑘𝑐,(11.1) where (̂̂̂𝑘𝑖,𝑗,) indicates the unity vectors computed by SCC and (̂𝑖𝑐,̂𝑗𝑐,̂𝑘𝑐) indicates the unity vector computed by the presented theory.

Figures 6 and 7 present the pointing deviations for the test period. The mean pointing deviation for the SCD1 for different time simulations are presented in Table 13. Over the test period of 11 days, the mean pointing deviation with the residual magnetic torque and elliptical orbit was 1.1553°, circular orbit was 1.2003°, and eddy currents torque with circular orbit was 1.1306°. The test period of SCD1 shows that the pointing deviation is higher than the precision required for SCC. Therefore for SCD1, this analytical approach should be evaluated by a time less than 11 days.

tab13
Table 13: Mean pointing deviation for SCD1.
242396.fig.006
Figure 6: Pointing deviation evolution (in degrees) for SCD1.
242396.fig.007
Figure 7: Pointing deviation evolution (in degrees) for SCD2.

For SCD2, the mean pointing deviation considering the residual magnetic torque and elliptical orbit was 0.1538, residual magnetic torque and circular orbit was 0.1507, and eddy current torque was 0.2160. All the results for SCD2 are within the dispersion range of the attitude determination system performance of INPE’s Control Center of 0.5°.

12. Summary

In this paper an analytical approach was presented to the spin-stabilized satellite attitude propagation taking into account the residual and eddy currents torque. The mean components of these torques in the satellite body reference system have been obtained and the theory shows that, unlike the eddy currents torque, there is no residual torque component along the spin axis (z-axis). Therefore this torque does not affect the spin velocity magnitude, but it can cause a drift in the satellite spin axis.

The theory was applied to the spin-stabilized Brazilian satellites SCD1 and SCD2 in order to validate the analytical approach, using quadripole model for geomagnetic field and the satellite in circular and elliptical orbits.

The result of the 3 days of simulations of SCD1, considering the residual magnetic torque, shows a good agreement between the analytical solution and the actual satellite behavior. For more than 3 days, the pointing deviation is higher than the precision required for SCC (0.5°).

For the satellite SCD2, over the test period of the 12 days, the difference between theory (when considering the residual or eddy currents torque) and SCC data is within the dispersion range of the attitude determination system performance of INPE’s Control Center.

Thus the procedure is useful for modeling the dynamics of spin-stabilized satellite attitude perturbed by residual or eddy currents torques but the time simulation depends on the precision required for satellite mission.

Appendix

The coefficients of the mean components of the residual magnetic torques, given by (2.9), are expressed by

𝐴=7𝑖=1𝑎𝑖𝑎+7𝑖=1𝑎𝑖𝑏,𝐵=7𝑖=1𝑏𝑖𝑎+7𝑖=1𝑏𝑖𝑏,𝐶=7𝑖=1𝑐𝑖𝑎+7𝑖=1𝑐𝑖𝑏,𝐷=7𝑖=1𝑎𝑖𝑎+7𝑖=1𝑎𝑖𝑏,𝐸=7𝑖=1𝑏𝑖𝑎+7𝑖=1𝑏𝑖𝑏,(A.1) where 𝑎𝑖𝑏, 𝑏𝑖𝑏, 𝑐𝑗𝑏, 𝑖=1,2,,7; 𝑗=1,,4, can be got by Garcia in [3]. It is important to note that the parcel 𝑏𝑖𝑏 is associated with the quadripole model and the satellite in an elliptical orbit. For circular, orbit, 𝑏𝑖𝑏 is zero.

The mean components 𝑁𝑖𝑥𝑚, 𝑁𝑖𝑦𝑚, 𝑁𝑖𝑧𝑚 of the eddy currents torque are expressed by

𝑁𝑖𝑥𝑚=14712𝑖=1tr𝑥(𝑖)+18426𝑖=1𝑁𝑁𝑥(𝑖),𝑖𝑦𝑚=14712𝑖=1tr𝑦(𝑖)+53765𝑖=1𝑁𝑁𝑦(𝑖),𝑖𝑧𝑚=7350𝑖=1tr𝑧(𝑖)+21435𝑖=1𝑁𝑧(𝑖),(A.3) where tr𝑥(𝑖), tr𝑦(𝑖), tr𝑧(𝑖), 𝑁𝑥(𝑖), 𝑁𝑦(𝑖), and 𝑁𝑧(𝑖) are presented by Pereira [6].

The terms 𝑎𝑖𝑏, 𝑏𝑖𝑏, 𝑐𝑗𝑏, tr𝑥(𝑖), tr𝑦(𝑖), tr𝑧(𝑖), 𝑁𝑥(𝑖), 𝑁𝑦(𝑖), and 𝑁𝑧(𝑖) depend on orbital elements (𝑎, 𝑒, 𝐼, Ω, 𝑤) and attitude angles (𝛿, 𝛼).

Acknowledgment

This present work was supported by CNPq (National Counsel of Technological and Scientific Development).

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