# Comparative Cost Functions Analysis in the Construction of a Reference Angular Motion Implemented by Magnetorquers

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## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

#### 2.1. Equations of Motion

- $O{X}_{1}{X}_{2}{X}_{3}$ is the inertial frame J2000 (IF), the axis $O{X}_{3}$ is co-directed with the axis of rotation of the Earth, $O{X}_{1}$ is directed to the point of the vernal equinox, and the second axis completes the system to the right triple;
- $O{Y}_{1}{Y}_{2}{Y}_{3}$ is the orbital frame (OF), the $O{Y}_{3}$ axis is directed along the SC radius vector, $O{Y}_{2}$ coincides with the normal to the satellite orbit, and the third axis $O{Y}_{1}$ complements the right-hand frame;
- $Oxyz$ is the reference frame (RF), which describes the required satellite attitude trajectory;
- $O\xi \eta \varsigma $ is the satellite-fixed frame (SF); its axes coincide with the principal central axes of inertia.

**C**:

**J**is the satellite’s inertia tensor, ${M}_{grav}$ is the gravitational torque, ${M}_{aero}$ is the aerodynamic torque, ${M}_{ctrl}$ is the control torque, and ${M}_{dist}$ is the external disturbing torque. In accordance with [25], the expression for the absolute angular velocity ${\omega}_{abs}$ is

#### 2.2. Models of External Torques

#### 2.2.1. Gravity Gradient Torque

#### 2.2.2. Aerodynamic Torque

#### 2.2.3. Magnetic Torque and Geomagnetic Field

#### 2.2.4. Additional External Disturbances

#### 2.3. Control Torque

#### 2.4. Magnetic Control Torque

## 3. Reference Trajectory Construction

#### 3.1. Particle Swarm Optimization Algorithm

#### 3.2. Trajectory Parametrization

#### 3.3. Control Gains Searching

## 4. Optimization Problems Statement

#### 4.1. Reference Trajectory Optimization Problem

#### 4.2. Control Gains Optimization Problem

#### 4.3. Ideal Case Numerical Simulation

## 5. The Influence of Disturbances

- ${B}_{magn}^{oblique}\left(t\right)$ is a periodic function, with the same period as the direct dipole model;
- The difference between ${B}_{magn}^{oblique}\left(t\right)$ and ${B}_{magn}^{inclined}\left(t\right)$ in a given time interval is in some sense less than the difference between ${B}_{magn}^{direct}\left(t\right)$ and ${B}_{magn}^{inclined}\left(t\right)$.

## 6. Full Model Numerical Simulation

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Euler’s angles (2-3-1). $OXYZ\stackrel{OY,\alpha}{\to}O{X}^{\prime}{Y}^{\prime}{Z}^{\prime}\stackrel{O{Z}^{\prime},\beta}{\to}O{X}^{\u2033}{Y}^{\u2033}{Z}^{\u2033}\stackrel{O{X}^{\u2033},\gamma}{\to}O{X}^{\u2034}{Y}^{\u2034}{Z}^{\u2034}\equiv Oxyz$.

**Figure 4.**Trajectories in OF in steady state for unperturbed problems with different cost functions for different heights: (

**a**) $h=550\hspace{0.17em}\mathrm{km}$ (

**b**) $h=650\hspace{0.17em}\mathrm{km}$.

**Figure 5.**Unit vectors of the direct, inclined, and oblique dipole axes. Vectors ${k}^{direct}$ and ${k}^{oblique}$ do not change over time, and the vector ${k}^{inclined}$ rotates.

**Figure 7.**Angular trajectories (in the OF) in the inclined dipole models for various cost functions at an altitude of 650 km, 6-orbit magnetic field approximation: (

**a**) set of optimization problems for the reference trajectory construction with ${\Phi}_{1}^{trajectory}$ and different dipole models; (

**b**) set of optimization problems for the reference trajectory construction with ${\Phi}_{2}^{trajectory}$ and different dipole models; (

**c**) set of optimization problems for the reference trajectory construction with ${\Phi}_{3}^{trajectory}$ and different dipole models.

**Figure 8.**Distribution of the worst values of reference trajectory tracking accuracy for orbit height 650 km.

**Figure 9.**Reference trajectory obtained with 3 intervals with different oblique dipole geomagnetic field approximation.

**Table 1.**Expressions for the cost functions in the problem of the reference trajectory construction.

${\varphi}_{1}$ | ${\varphi}_{2}$ | ${\varphi}_{3}$ |

${\Vert \frac{C{B}_{magn}}{{B}_{magn}}\left(\frac{C{B}_{magn}^{T}{M}_{ctrl}^{0}}{{B}_{magn}}\right)\Vert}_{2}$ | $C{B}_{magn}^{T}{M}_{ctrl}^{0}$ | ${\Vert \frac{C{B}_{magn}}{{B}_{magn}}\left(\frac{C{B}_{magn}^{T}{M}_{ctrl}^{0}}{{B}_{magn}}\right)\Vert}_{\infty}$ |

$\delta {\mathsf{\beta}}_{1}$ | $\delta {\mathsf{\beta}}_{2}$ | $\delta {\mathsf{\beta}}_{3}$ | $\delta {\mathsf{\beta}}_{4}$ |

$\frac{{\Vert {B}_{magn}^{inclined}-{B}_{magn}^{oblique}\Vert}_{2}}{{\Vert {B}_{magn}^{inclined}\Vert}_{2}}$ | $\frac{{\Vert {B}_{magn}^{inclined}-{B}_{magn}^{oblique}\Vert}_{\infty}}{{\Vert {B}_{magn}^{inclined}\Vert}_{\infty}}$ | ${\Vert {B}_{magn}^{inclined}-{B}_{magn}^{oblique}\Vert}_{2}$ | $\frac{{\Vert {B}_{magn}^{inclined}-{B}_{magn}^{oblique}\Vert}_{\infty}}{{\Vert {B}_{magn}^{inclined}\Vert}_{2}}$ |

${\Phi}_{1}^{trajectory}$ | ${\Phi}_{2}^{trajectory}$ | ${\Phi}_{3}^{trajectory}$ | |

Direct dipole | № 1.0 | № 2.0 | № 3.0 |

${\Phi}_{1}^{dipole}$ | № 1.1 | № 2.1 | № 3.1 |

${\Phi}_{2}^{dipole}$ | № 1.2 | № 2.2 | № 3.2 |

${\Phi}_{3}^{dipole}$ | № 1.3 | № 2.3 | № 3.3 |

${\Phi}_{4}^{dipole}$ | № 1.4 | № 2.4 | № 3.4 |

Name | Value |
---|---|

Simulation time | $T=18{T}_{0}\approx 30\hspace{0.17em}\mathrm{h},$ |

SC initial angular velocity | ${\omega}_{abs}^{}={(1,\hspace{0.17em}2,\hspace{0.17em}3)}^{T}\cdot {10}^{-3}\hspace{0.17em}\hspace{0.17em}\mathrm{rad}/\mathrm{s}$ |

SC initial attitude | $\begin{array}{l}{\alpha}_{rel}={55}^{\circ}\\ {\beta}_{rel}={55}^{\circ}\\ {\gamma}_{rel}={55}^{\circ}\end{array}$ |

Magnetic field model | IGRF |

Inaccuracy of knowledge of the density of the atmosphere | 20% |

Inaccuracy of knowledge of the SC inertia tensor | 5% |

External random disturbances | $\left|{M}_{dist}\right|={10}^{-9}\hspace{0.17em}\hspace{0.17em}\mathrm{N}\cdot \mathrm{m}$ |

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**MDPI and ACS Style**

Okhitina, A.; Tkachev, S.; Roldugin, D.
Comparative Cost Functions Analysis in the Construction of a Reference Angular Motion Implemented by Magnetorquers. *Aerospace* **2023**, *10*, 468.
https://doi.org/10.3390/aerospace10050468

**AMA Style**

Okhitina A, Tkachev S, Roldugin D.
Comparative Cost Functions Analysis in the Construction of a Reference Angular Motion Implemented by Magnetorquers. *Aerospace*. 2023; 10(5):468.
https://doi.org/10.3390/aerospace10050468

**Chicago/Turabian Style**

Okhitina, Anna, Stepan Tkachev, and Dmitry Roldugin.
2023. "Comparative Cost Functions Analysis in the Construction of a Reference Angular Motion Implemented by Magnetorquers" *Aerospace* 10, no. 5: 468.
https://doi.org/10.3390/aerospace10050468