# Comparison of Feedback Three-Axis Magnetic Attitude Control Strategies

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Equations of Motion and Environment Models

_{1}X

_{2}X

_{3}. Its first axis lies in the orbital plane, and it is directed toward the satellite’s velocity vector in a near circular orbit. The second axis is positioned along the satellite’s normal orbit. The third axis is the radius vector of the satellite. The satellite reference frame OX

_{1}X

_{2}X

_{3}is characterized by the principal central axes of inertia.

## 3. New Sliding Surface Design (SC Construction)

- $\Lambda \left(k\right)>0$—the surface parameter is a positive definite matrix (moreover, eigenvalues should be less than a predefined threshold);
- ${M}^{T}B/\left|M\right|\left|B\right|<\mathrm{cos}\left({80}^{\circ}\right)$—the control torque deviates from the plane perpendicular to the geomagnetic induction vector by no more than 10 degrees;
- ${\Vert m\Vert}_{\infty}$<0.1 A·m
^{2}—the dipole moments are bounded.

^{5}to ensure that the expressions in conditions 1 and 2 have approximately the same order. After the surface parameter matrix in the current iteration is calculated, the control torque is also calculated according to (9). It is then used to find the control dipole moment as a projection on the plane perpendicular to the geomagnetic induction vector $m=B\times M/{\left|B\right|}^{2}$. The same procedure is used during the optimization process to find the dipole moment in condition 3. The resulting control torque ${M}_{real}=m\times B$ slightly differs from the calculated one $M$. According to condition 2, the angle between the calculated and defined torques does not exceed 10 degrees. If the optimization procedure fails to find a feasible surface parameter and therefore the control torque with the defined restrictions, the control scheme is not implemented.

## 4. Description of the Feedback Laws Utilized for Comparison

#### 4.1. Lyapunov Control—LC

^{6}T∙km

^{3}, and ${B}_{k}\left(u\right)$ are periodic components of the geomagnetic induction vector in the dipole model. The linearized equations with periodic coefficients are examined for stability using Floquet theory [36].

#### 4.2. Matrix Lyapunov Control—MLC

#### 4.3. Linear Quadratic Regulator—LQR

#### 4.4. Particle Swarm Optimization—PSO

## 5. Simulation Results for Different Control Strategies

- Satellite inertia moments of 0.15 (Case 1) or 0.2 (Case 2), 0.13, and 0.11 kg·m
^{2}(the satellite is stabilized in the unstable gravitational equilibrium position); - Orbit altitude of 550 km, inclination of 57°, and eccentricity equal to 0.01;
- Aerodynamic torque calculation:
- ○
- Satellite size is 10 × 20 × 30 cm;
- ○
- Center-of-mass displacement is 1 cm along the second satellite axis (aerodynamic torque is essentially non zero in the required attitude);
- ○
- Atmospheric density is 1.8·10
^{−13}kg/m^{3};

- Principal moments of inertia are subject to 5% error.

^{2}, ${k}_{a}$ = 150 N·m/T

^{2}for Case 1 and ${k}_{\omega}$ = 360/${\omega}_{orb}$ N·m/T

^{2}, ${k}_{a}$ = 220 N·m/T

^{2}for Case 2. Damping gain ${k}_{\omega}$ is significantly shifted to the left compared to the theoretically best positions in terms of the degree of stability. The best characteristic multipliers are shown at ${k}_{\omega}$ ≈ 900/${\omega}_{orb}$ N·m/T

^{2}in Figure 1, whereas the manually adjusted “best” value is 360/${\omega}_{orb}$ N·m/T

^{2}.

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Stickler, A.C.; Alfriend, K.T. Elementary Magnetic Attitude Control System. J. Spacecr. Rockets
**1976**, 13, 282–287. [Google Scholar] [CrossRef] - Desouky, M.A.A.; Abdelkhalik, O. A new variant of the B-dot control for spacecraft magnetic detumbling. Acta Astronaut.
**2020**, 171, 14–22. [Google Scholar] [CrossRef] - Ignatov, A.I.; Sazonov, V.V. Stabilization of the Gravitational Orientation Mode of an Artificial Earth Satellite (AES) by the Electromagnetic Control System. Cosm. Res.
**2020**, 58, 33–41. [Google Scholar] [CrossRef] - Belokonov, I.V.; Timbai, I.A.; Nikolaev, P.N. Analysis and Synthesis of Motion of Aerodynamically Stabilized Nanosatellites of the CubeSat Design. Gyroscopy Navig.
**2018**, 9, 287–300. [Google Scholar] [CrossRef] - Morozov, V.M.; Kalenova, V.I. Stabilization of Satellite Relative Equilibrium Using Magnetic Moments and Aerodynamic Forces. Cosm. Res.
**2022**, 60, 213–219. [Google Scholar] [CrossRef] - Ignatov, A.I.; Sazonov, V.V. Investigation of Steady-State Motion of an Artificial Earth Satellite in the Uniaxial Magnetic Orientation Mode. Cosm. Res.
**2021**, 59, 112–125. [Google Scholar] [CrossRef] - Avanzini, G.; de Angelis, E.L.; Giulietti, F. Spin-axis pointing of a magnetically actuated spacecraft. Acta Astronaut.
**2014**, 94, 493–501. [Google Scholar] [CrossRef] - Slavinskis, A.; Kvell, U.; Kulu, E.; Sünter, I.; Kuuste, H.; Lätt, S.; Voormansik, K.; Noorma, M. High spin rate magnetic controller for nanosatellites. Acta Astronaut.
**2014**, 95, 218–226. [Google Scholar] [CrossRef] - de Ruiter, A. A fault-tolerant magnetic spin stabilizing controller for the JC2Sat-FF mission. Acta Astronaut.
**2011**, 68, 160–171. [Google Scholar] [CrossRef] - Roldugin, D.S. Stability of a Magnetically Actuated Satellite towards the Sun on a Sun-Synchronous Orbit. Cosm. Res.
**2023**, 61, 134–142. [Google Scholar] [CrossRef] - Colagrossi, A.; Lavagna, M. A Spacecraft Attitude Determination and Control Algorithm for Solar Arrays Pointing Leveraging Sun Angle and Angular Rates Measurements. Algorithms
**2022**, 15, 29. [Google Scholar] [CrossRef] - Roldugin, D.; Tkachev, S.; Ovchinnikov, M. Asymptotic Motion of a Satellite under the Action of Sdot Magnetic Attitude Control. Aerospace
**2022**, 9, 639. [Google Scholar] [CrossRef] - Bhat, S.P. Controllability of nonlinear time-varying systems: Applications to spacecraft attitude control using magnetic actuation. IEEE Trans. Autom. Control
**2005**, 50, 1725–1735. [Google Scholar] [CrossRef] - Morozov, V.M.; Kalenova, V.I. Satellite Control Using Magnetic Moments: Controllability and Stabilization Algorithms. Cosm. Res.
**2020**, 58, 158–166. [Google Scholar] [CrossRef] - Liu, S.; Huang, Q. Controllability and observability of discretized satellite magnetic attitude control system. AIMS Math.
**2023**, 8, 7899–7916. [Google Scholar] [CrossRef] - Lovera, M.; Astolfi, A. Spacecraft attitude control using magnetic actuators. Automatica
**2004**, 40, 1405–1414. [Google Scholar] [CrossRef] - Lovera, M.; Astolfi, A. Global Magnetic Attitude Control of Inertially Pointing Spacecraft. J. Guid. Control Dyn.
**2005**, 28, 1065–1072. [Google Scholar] [CrossRef] - Wisniewski, R.; Blanke, M. Fully magnetic attitude control for spacecraft subject to gravity gradient. Automatica
**1999**, 35, 1201–1214. [Google Scholar] [CrossRef] - Damaren, C.J. Comments on “Fully magnetic attitude control for spacecraft subject to gravity gradient”. Automatica
**2002**, 38, 2189. [Google Scholar] [CrossRef] - Celani, F. Gain Selection for Attitude Stabilization of Earth-Pointing Spacecraft Using Magnetorquers. Aerotec. Missili Spaz.
**2021**, 100, 15–24. [Google Scholar] [CrossRef] - Rossa, F.D.; Bergamasco, M.; Lovera, M. Bifurcation analysis of the attitude dynamics for a magnetically controlled spacecraft. In Proceedings of the 51st IEEE Conference on Decision and Control, Maui, HI, USA, 10–13 December 2012; IEEE: Maui, HI, USA, 2012; pp. 1154–1159. [Google Scholar]
- Ivanov, D.S.; Ovchinnikov, M.Y.; Penkov, V.I.; Roldugin, D.S.; Doronin, D.M.; Ovchinnikov, A.V. Advanced numerical study of the three-axis magnetic attitude control and determination with uncertainties. Acta Astronaut.
**2017**, 132, 103–110. [Google Scholar] [CrossRef] - Celani, F. Robust three-axis attitude stabilization for inertial pointing spacecraft using magnetorquers. Acta Astronaut.
**2015**, 107, 87–96. [Google Scholar] [CrossRef] - Mashtakov, Y.V.; Ovchinnikov, M.Y.; Wöske, F.; Rievers, B.; List, M. Attitude determination & control system design for gravity recovery missions like GRACE. Acta Astronaut.
**2020**, 173, 172–182. [Google Scholar] [CrossRef] - Wang, P.; Shtessel, Y.; Wang, Y.-Q. Satellite attitude control using only magnetorquers. In Proceedings of the 1998 American Control Conference, ACC, Philadelphia, PA, USA, 26–26 June 1998; pp. 500–504. [Google Scholar]
- Wisniewski, R. Sliding Mode Attitude Control for Magnetic Actuated Satellite. IFAC Proc. Vol.
**1998**, 31, 179–184. [Google Scholar] [CrossRef] - Sofyalı, A.; Jafarov, E.M. Integral Sliding Mode Control of Small Satellite Attitude Motion by Purely Magnetic Actuation. IFAC Proc. Vol.
**2014**, 47, 7947–7953. [Google Scholar] [CrossRef] - Sofyali, A.; Jafarov, E.M.; Wisniewski, R. Robust and global attitude stabilization of magnetically actuated spacecraft through sliding mode. Aerosp. Sci. Technol.
**2018**, 76, 91–104. [Google Scholar] [CrossRef] - Ovchinnikov, M.Y.; Roldugin, D.S.; Penkov, V.I.; Tkachev, S.S.; Mashtakov, Y.V. Fully magnetic sliding mode control for acquiring three-axis attitude. Acta Astronaut.
**2016**, 121, 59–62. [Google Scholar] [CrossRef] - Okhitina, A.; Roldugin, D.; Tkachev, S. Application of the PSO for the construction of a 3-axis stable magnetically actuated satellite angular motion. Acta Astronaut.
**2022**, 195, 86–97. [Google Scholar] [CrossRef] - Okhitina, A.; Roldugin, D.; Tkachev, S.; Ovchinnikov, M. Academy transaction note “closed form solution for a minimum deviation magnetically controllable satellite angular trajectory”. Acta Astronaut.
**2023**, 203, 60–64. [Google Scholar] [CrossRef] - Alken, P.; Thébault, E.; Beggan, C.D.; Amit, H.; Aubert, J.; Baerenzung, J.; Bondar, T.N.; Brown, W.J.; Califf, S.; Chambodut, A.; et al. International Geomagnetic Reference Field: The thirteenth generation. Earth Planets Space
**2021**, 73, 49. [Google Scholar] [CrossRef] - Tikhonov, A.A.; Petrov, K.G. Multipole models of the earth’s magnetic field. Cosm. Res.
**2002**, 40, 203–212. [Google Scholar] [CrossRef] - Utkin, V.I. Sliding Modes in Control and Optimization; Springer-Verlag: Berlin/Heidelberg, Germany, 1992. [Google Scholar]
- Ovchinnikov, M.Y.; Roldugin, D.S.; Ivanov, D.S.; Penkov, V.I. Choosing control parameters for three axis magnetic stabilization in orbital frame. Acta Astronaut.
**2015**, 116, 74–77. [Google Scholar] [CrossRef] - Malkin, I.G. Theory of Stability of Motion; U.S. Atomic Energy Commission, Technical Information Service: Oak Ridge, TN, USA, 1952. [Google Scholar]
- Kennedy, J.; Eberhart, R. Particle swarm optimization. In Proceedings of the International Conference on Neural Networks, Perth, WA, Australia, 27 November–1 December 1995; Volume 4, pp. 1942–1948. [Google Scholar]

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**MDPI and ACS Style**

Roldugin, D.; Okhitina, A.; Monakhova, U.; Ovchinnikov, M.
Comparison of Feedback Three-Axis Magnetic Attitude Control Strategies. *Aerospace* **2023**, *10*, 975.
https://doi.org/10.3390/aerospace10120975

**AMA Style**

Roldugin D, Okhitina A, Monakhova U, Ovchinnikov M.
Comparison of Feedback Three-Axis Magnetic Attitude Control Strategies. *Aerospace*. 2023; 10(12):975.
https://doi.org/10.3390/aerospace10120975

**Chicago/Turabian Style**

Roldugin, Dmitry, Anna Okhitina, Uliana Monakhova, and Mikhail Ovchinnikov.
2023. "Comparison of Feedback Three-Axis Magnetic Attitude Control Strategies" *Aerospace* 10, no. 12: 975.
https://doi.org/10.3390/aerospace10120975