Spin-Stabilized Spacecrafts: Analytical Attitude Propagation Using Magnetic Torques

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.

Figure 1 

Orientation of the spin axis (𝑠̂): equatorial system (𝐼,𝐽,𝐾), satellite body frame reference system (𝑖,̂𝑗,̂𝑘̂), 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

𝐵⃗=−∇𝑉,(2.1) with the magnetic potential 𝑉 given by

𝑉𝑟^,𝜙,𝜃=𝑟_𝑇𝑘_𝑛=1𝑟𝑟^𝑇^𝑛𝑛+1_𝑚=0𝑔^𝑚𝑛cos𝑚𝜃+ℎ^𝑚𝑛sen𝑚𝜃𝑃^𝑚𝑛(𝜙),(2.2) where 𝑟_𝑇 is the Earth’s equatorial radius, 𝑔^𝑚𝑛, ℎ^𝑚𝑛 are the Gaussian coefficients, 𝑃^𝑚𝑛(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],

𝐵⃗=𝐵_𝑟𝑟̂+𝐵_𝜙𝜙+𝐵_𝜃𝜃̂,(2.3) with 𝐵_𝑟=−𝜕𝑉𝜕𝑟,𝐵^𝜙=−1𝑟𝜕𝜕𝑉𝜙,𝐵^𝜃=−1𝑟sen𝜙𝜕𝜕V𝜃.(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]

𝐵_𝑟=2𝑟^𝑇𝑟³𝑓₁(𝜃,𝜙)+3𝑟^𝑇𝑟⁴𝐵𝑓₂(𝜃,𝜙),(2.5)_𝜙=−𝑟^𝑇𝑟³𝑓₃(𝜃,𝜙)−𝑟^𝑇𝑟⁴𝐵𝑓₄(𝜃,𝜙),(2.6)_𝜃=−1sen𝜙𝑟𝑟^𝑇³𝑓₅(𝜃,𝜙)+𝑟^𝑇𝑟⁴𝑓₆(𝜃,𝜙)+2𝑟^𝑇𝑟⁴𝑓₇(𝜃,𝜙),(2.7) where the functions 𝑓_𝑖, 𝑖=1,2,…,7, are shown in [3] and depend on the Gaussian coefficients 𝑔²²,ℎ¹¹,ℎ¹²,ℎ²².

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

𝐵𝐵_𝑋=𝐵_𝑟cos𝛿+𝐵_𝜙sen𝛿cos𝛼−𝐵_𝜃sen𝛼,(2.8)_𝑌=𝐵_𝑟𝐵cos𝛿+𝐵_𝜙sen𝛿sen𝛼−𝐵_𝜃cos𝛼,(2.9)_𝑍=𝐵_𝑟sen𝛿+𝐵_𝜙cos𝛿,(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; 𝐵_𝑟, 𝐵_𝜙, and 𝐵_𝜃 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]

𝐵⃗=𝐵_𝑥𝑖̂+𝐵_𝑦𝑗̂+𝐵_𝑧𝑘̂,(2.11) where

𝐵𝐵_𝑥=−𝐵_𝑋sen𝛼+𝐵_𝑌cos𝛼,_𝑦𝐵=−𝐵_𝑋sen𝛿cos𝛼−𝐵_𝑌sen𝛿sen𝛼+𝐵_𝑍cos𝛿,_𝑧=−𝐵_𝑋cos𝛿cos𝛼−𝐵_𝑌cos𝛿sen𝛼+𝐵_𝑍sen𝛿,(2.12) with 𝐵_𝑋, 𝐵_𝑌, and 𝐵_𝑍given 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 𝑚⃗ is the magnetic moment of the spacecraft and 𝐵⃗ is the geomagnetic field, then the residual magnetic torques are given by [2]

𝑁_𝑟=𝑚⃗×𝐵⃗.(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]

𝑁_𝑟=𝑀_𝑠𝑘̂×𝐵⃗,(3.2) where 𝑀_𝑠 is the satellite magnetic moment along its spin axis and 𝑘̂ 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

𝑁_𝑟=𝑀_𝑠−𝐵_𝑦𝑖̂+𝐵_𝑥𝑗̂.(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]

𝑁_𝑖=𝑝𝐵⃗×𝐁⃗×𝑊.(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]

𝑁_𝑖=𝑝𝑊−𝐵_𝑥𝐵_𝑧𝑖̂−𝐵_𝑦𝐵_𝑧𝑗̂+𝐵^2𝑦+𝐵^2𝑥𝑘̂.(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 𝑁_𝑟and 𝑁_𝑖, given in (3.3) and (3.5), over one orbital period 𝑇as

𝑁_𝑟𝑚=1𝑇^{𝑡𝑖𝑡+𝑇}_𝑖𝑁_𝑟𝑑𝑡,𝑁_𝑖𝑚=1𝑇^{𝑡𝑖𝑡+𝑇}_𝑖𝑁_𝑖𝑑𝑡,(4.1) where 𝑡 is the time 𝑡_𝑖 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]

𝑁_𝑟𝑚=1𝑇^{𝜐𝑖𝜐+2𝜋}_𝑖𝑁_𝑟𝑟²ℎ𝑑𝜐,𝑁^𝑖𝑚=1𝑇^{𝜐𝑖𝜐+2𝜋}_𝑖𝑁_𝑖𝑟²ℎ𝑑𝜐,(4.2) where 𝜐_𝑖 is the true anomaly at instant 𝑡_𝑖, 𝑟 is the geocentric distance, and ℎ 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]

𝑁_𝑟𝑚=𝑁_𝑟𝑥𝑚𝑖̂+𝑁_𝑟𝑦𝑚𝑗,̂𝑁_𝑖𝑚=𝑝𝑊2𝜋𝑁^𝑖𝑥𝑚𝑖̂+𝑁_𝑖𝑦𝑚𝑗̂+𝑁_𝑖𝑧𝑚𝑘̂,(4.3) with

𝑁_𝑟𝑥𝑚=𝑀^𝑠2𝑁𝜋(𝐴sen𝛿cos𝛼+𝐵sen𝛿sen𝛼−𝐶sen𝛿),_𝑟𝑦𝑚=𝑀^𝑠2𝜋(−𝐷sen𝛼+𝐸cos𝛿)(4.4) and 𝑁_𝑖𝑥𝑚, 𝑁_𝑖𝑦𝑚, 𝑁_𝑖𝑧𝑚 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

𝑊̇=1𝐼_𝑧𝛿̇=1𝑁_𝑧,𝐼_𝑧𝑊𝑁^𝑦𝛼̇=1,𝐼_𝑧𝑊Cos𝛿𝑁^𝑥,(5.1) where 𝐼_𝑧 is the moment of inertia along the spin axis and 𝑁_𝑥, 𝑁_𝑦, 𝑁_𝑧 are the components of the external torques in the satellite body frame reference system. By substituting 𝑁_𝑟𝑚, given in (4.3), in (5.1), the equations of motion are

𝑑𝑊𝑑𝛿𝑑𝑡=0,(5.2)𝑑𝛼𝑑𝑡=𝑁𝐼_𝑧𝑊^𝑟𝑦𝑚,(5.3)𝑑𝑡=𝐼_𝑧𝑊𝑁c^𝑟𝑥𝑚os𝛿,(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 𝑁_𝑖𝑚, given in (4.3), in (5.1), the equations of motion are

𝑑𝑊𝑑𝑡=2𝑝𝜋𝑊𝐼_𝑧𝑑𝛿𝑁_𝑖𝑧𝑚,(5.5)𝑑𝑡=2𝑝𝜋𝑊𝐼_𝑧𝑑𝛼𝑁_𝑖𝑦𝑚,(5.6)𝑑𝑡=2𝜋𝐼_𝑧𝑝cos𝛿𝑁^𝑖𝑥𝑚.(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:

𝑑𝑊𝑊=𝑘𝑑𝑡,with𝑘=𝑁^𝑖𝑧𝑚𝑝2𝜋𝐼_𝑧.(6.1) If the parameter 𝑘 is considered constant for one orbital period, then the analytical solution of (6.1) is

𝑊=𝑊₀𝑒^𝑘𝑡,(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,

𝛼𝛿==𝑘𝑘₁₂𝑡+𝛿₀,(7.1)𝑡+𝛼₀,(7.2) with:

(i)for the case where the residual magnetic torque is considered in the motion equations, 𝑘₁=𝑁^𝑟𝑦𝑚𝐼_𝑧𝑊_𝑜𝑘₂,=𝑁^𝑟𝑦𝑚𝐼_𝑧𝑊_𝑜cos𝛿_𝑜,(7.3)(ii)for the case where the eddy currents torque is considered in the motion equations,

𝑘₁=𝑝𝑁^𝑖𝑦𝑚2𝜋𝐼_𝑧𝑘₂,=𝑝𝑁^𝑖𝑦𝑚2𝜋𝐼_𝑧cos𝛿_𝑜,(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.

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).

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).

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.

Table 4 

Mean deviations for different time simulations for declination and right ascension and SCD1 (in degrees).

(a)

(b)

(c)

(d)

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.

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).

Table 6 

Mean deviations for different time simulations for spin velocity and SCD1 (in degrees).

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.

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).

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).

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.

Table 10 

Mean deviations for different time simulation for declination and right ascension and SCD2 (in degrees).

(a)

(b)

(c)

(d)

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.

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).

Table 12 

Mean deviations for different time simulations for spin velocity and SCD1 (in degrees).

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=cos⁻¹𝑖̂𝑖_𝑐̂+𝑗𝑗_𝑐̂+𝑘̂𝑘̂_𝑐,(11.1) where (𝑖,̂𝑗,̂𝑘̂) indicates the unity vectors computed by SCC and (𝑖_𝑐̂,𝑗_𝑐̂,𝑘̂_𝑐) 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.

Table 13 

Mean pointing deviation for SCD1.

Figure 6 

Pointing deviation evolution (in degrees) for SCD1.

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=1𝑎_𝑖𝑎+𝐷=^7𝑖=1𝑎_𝑖𝑏,_7𝑖=1𝐵=^7𝑖=1𝑏_𝑖𝑎+^7𝑖=1𝑏_𝑖𝑏,𝐶=^7𝑖=1𝑐_𝑖𝑎+^7𝑖=1𝑐_𝑖𝑏,𝑎_𝑖𝑎+^7𝑖=1𝑎_𝑖𝑏,𝐸=^7𝑖=1𝑏_𝑖𝑎+^7𝑖=1𝑏_𝑖𝑏,(A.1) where 𝑎_𝑖𝑏, 𝑏_𝑖𝑏, 𝑐_𝑗𝑏, 𝑖=1,2,…,7; 𝑗=1,…,4, can be got by Garcia in [3]. It is important to note that the parcel 𝑏_𝑖𝑏 is associated with the quadripole model and the satellite in an elliptical orbit. For circular, orbit, 𝑏_𝑖𝑏 is zero.

The mean components 𝑁_𝑖𝑥𝑚, 𝑁_𝑖𝑦𝑚, 𝑁_𝑖𝑧𝑚 of the eddy currents torque are expressed by

𝑁𝑁_𝑖𝑥𝑚=^14712𝑖=1tr𝑥(𝑖)+^18426𝑖=1𝑁𝑥(𝑖),_𝑖𝑦𝑚𝑁=^14712𝑖=1tr𝑦(𝑖)+^53765𝑖=1𝑁𝑦(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 𝑎_𝑖𝑏, 𝑏_𝑖𝑏, 𝑐_𝑗𝑏, 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).