# Attitude Dynamics of Spinning Magnetic LEO/VLEO Satellites

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## Abstract

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## 1. Introduction

## 2. Problem Statement

- The orbit of the satellite remains circular.
- The atmosphere is not rotating.
- The density and temperature of the incident air stream are chosen using the Jacchia–Bowman 2008 Atmosphere Model [34].
- The satellite is shaped as a body of revolution.
- The centre of mass of the satellite lies on its axis of symmetry.
- The transverse moments of inertia of the satellite are equal to each other.

#### 2.1. Coordinate Systems

- The orbital frame O is defined through an orthonormal right-hand set of unit vectors ${\widehat{\mathit{o}}}_{k}$, $k=1,2,3$ with an origin at the satellite’s centre of mass (Figure 1). The ${\widehat{\mathit{o}}}_{1}$ vector is tangential to the orbit in the flight direction, the ${\widehat{\mathit{o}}}_{3}$ vector is directed along the radius vector from the centre of the Earth to the centre of mass of the satellite.
- The body-fixed frame B is defined through a set ${\widehat{\mathit{b}}}_{k}$. These vectors coincide with the satellite’s principal axes of inertia, and the ${\widehat{\mathit{b}}}_{1}$ vector lies along the axis of symmetry. The orientation of ${\widehat{\mathit{b}}}_{k}$ relative to ${\widehat{\mathit{o}}}_{k}$ is described by a symmetric $(1,3,1)$ set of Euler angles corresponding to three successive positive rotations: first about Axis 1 by the precession angle $\psi $, then about Axis 3 by the nutation angle $\theta $ (angle of attack), $0\le \theta \le \pi $, and finally, about Axis 1 by the spin angle $\phi $, as is shown in Figure 1. The transformation matrix $\mathsf{\Theta}$ between O and B is defined by$${\mathit{v}}^{B}=\mathsf{\Theta}{\mathit{v}}^{O}$$$$\mathsf{\Theta}=\left(\right)open="("\; close=")">\begin{array}{ccc}c\theta & c\psi s\theta & s\theta s\theta \\ -c\phi s\theta & c\theta c\phi c\psi -s\phi s\psi & c\psi s\phi +c\theta c\phi s\psi \\ s\theta s\phi & -c\theta c\psi s\phi -c\phi s\psi & c\phi c\psi -c\theta s\phi s\psi \end{array}$$$c\theta =cos\theta $, $s\theta =sin\theta $, etc.
- The intermediate coordinate frames I and ${I}^{\prime}$ are defined through unit vector sets ${\widehat{\mathit{i}}}_{k}$, ${{\widehat{\mathit{i}}}^{\prime}}_{k}$, respectively. The orientations of these frames relative to the orbital frame are determined by the above-mentioned rotations: a single rotation by the angle $\psi $ for the ${\widehat{\mathit{i}}}_{k}$ frame and two successive rotations by the angles $\psi $ and $\theta $ for the ${{\widehat{\mathit{i}}}^{\prime}}_{k}$ frame.

#### 2.2. Environmental Torques

#### 2.2.1. Gravitational Torque

#### 2.2.2. Magnetic Torque

#### 2.2.3. Restoring Aerodynamic Torque

#### 2.2.4. Damping Aerodynamic Torque

## 3. Equations of Motion

#### 3.1. Unperturbed Motion

#### 3.2. Perturbed Motion

## 4. Case Study: Deployable Satellite

#### 4.1. Aerodynamic Characteristics

#### 4.2. Dynamic Potential and Equilibrium Positions in Unperturbed Motion

#### 4.3. Simulations of Perturbed Motion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Generalised Forces

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**Figure 6.**Dimensionless dynamic potential in LEO ($h=700$ km, $\beta =33,000$). Black dots: stable equilibrium; white dots: unstable equilibrium.

**Figure 7.**Dimensionless dynamic potential in VLEO ($h=250$ km, $\beta =20$). Black dots: stable equilibrium; white dots: unstable equilibrium.

**Figure 8.**Poincaré sections ($h=250$ km, $\beta =\mathrm{33,000}$); (

**a**): $\epsilon =0$, (

**b**): $\epsilon =580$, (

**c**): $\epsilon =3900$. Red lines: typical phase trajectories.

**Figure 9.**Poincaré sections ($h=700$ km, $\beta =20$); (

**a**): $\epsilon =0$, (

**b**): $\epsilon =0.58$, (

**c**): $\epsilon =1.35$. Red lines: typical phase trajectories.

Parameter | Value |
---|---|

Body length = Reference length ${L}_{r}$ | 0.3 m |

Body radius r | 0.15 m |

Nose radius | 0.2 m |

Reference area ${A}_{r}=\pi {r}^{2}$ | 0.0707 ${\mathrm{m}}^{2}$ |

Aerobrake half-cone angle | 45° |

Aerobrake diameter | 1.1 m |

Longitudinal moment of inertia A | 0.005 kg$\xb7{\mathrm{m}}^{2}$ |

Transverse moment of inertia C | 0.05 kg$\xb7{\mathrm{m}}^{2}$ |

Longitudinal shift of the centre of mass from the body’s geometric centre | $-0.125$ m |

h (km) | Order of Magnitude (N·m) | ||||
---|---|---|---|---|---|

Gravitational | Aerodynamic Restoring | Aerodynamic Damping | Magnetic | ||

250 | ${10}^{-8}$ | ${10}^{-4}$ | ${10}^{-12}$ | $\epsilon =580$:${10}^{-6}$ | $\epsilon =3900$:${10}^{-5}$ |

700 | ${10}^{-8}$ | ${10}^{-7}$ | ${10}^{-15}$ | $\epsilon =0.58$:${10}^{-9}$ | $\epsilon =1.35$:${10}^{-8}$ |

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**MDPI and ACS Style**

Aslanov, V.S.; Sizov, D.A.
Attitude Dynamics of Spinning Magnetic LEO/VLEO Satellites. *Aerospace* **2023**, *10*, 192.
https://doi.org/10.3390/aerospace10020192

**AMA Style**

Aslanov VS, Sizov DA.
Attitude Dynamics of Spinning Magnetic LEO/VLEO Satellites. *Aerospace*. 2023; 10(2):192.
https://doi.org/10.3390/aerospace10020192

**Chicago/Turabian Style**

Aslanov, Vladimir S., and Dmitry A. Sizov.
2023. "Attitude Dynamics of Spinning Magnetic LEO/VLEO Satellites" *Aerospace* 10, no. 2: 192.
https://doi.org/10.3390/aerospace10020192