# Autonomous Optimal Absolute Orbit Keeping through Formation Flying Techniques

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## Abstract

**:**

## 1. Introduction

- Considering the coupled attitude/relative orbit problem in the control loop;
- Proposing an MPC scheme for a satellite with a unidirectional propulsion system where the thrust can also be provided during slew maneuvers.

## 2. Mathematical Models

- 1.
- Earth-centered inertial frame:The employed Earth-Centered-Inertial (ECI) reference frame, ${\mathbb{F}}^{i}$, is defined by True-of-Date coordinate system with its origin at the center of the Earth, its x-axis along the true vernal equinox at the current epoch, its z-axis is aligned with the true rotation axis at the current epoch, while its y-axis completes the right-handed triad. A vector expressed in the ECI frame is signified by the ${\left(\xb7\right)}^{i}$ superscript.
- 2.
- Satellite-body-fixed frame:The body-fixed reference frame, ${\mathbb{F}}^{b}$, is a reference frame attached to the spacecraft in question with the origin at the satellite’s center of mass. One common choice of the directions of the three axes is along the three principal axes of inertia of the satellite. A vector expressed in the body-fixed frame of the satellite will have the superscript ${\left(\xb7\right)}^{b}$.
- 3.
- Radial–transversal–normal frame:The radial–transversal–normal (RTN) frame, ${\mathbb{F}}^{r}$, is centered on the center of mass of the chief satellite, where the x-axis of the RTN frame is defined to be along the position vector of the chief satellite, positive towards the Zenith direction, the z-axis is directed along the chief’s orbital angular momentum vector, and the y-axis completes the right-handed set. In the sequel, a vector expressed in the RTN frame is signified by the ${\left(\xb7\right)}^{r}$ superscript.

#### 2.1. Relative Orbital Dynamics

- Neighboring orbits of the deputy and the chief,
- Near circular chief’s orbit.

#### 2.2. Attitude Kinematics

## 3. Model Predictive Control

#### 3.1. MPC Problem Formulation

**Problem 1.**

#### 3.2. Parameter Tuning

- 1.
- Driving the ROE vector to zero, which is the main goal of the MPC scheme to achieve orbit keeping. This is assessed through observing the norm of the finale of the ROE time series, denoted as $\delta {\alpha}_{\mathrm{fin}}$.
- 2.
- Enhancing the stability of the relative orbit finale which, in other words, is minimizing the Root Mean Square (RMS) of the last portion of the relative semi-major axis profile over time, denoted as $\delta {a}_{\mathrm{fin}}$.
- 3.
- Minimizing the total delta-V, $\Delta {V}_{\mathrm{tot}}$.
- 4.
- Reducing the attitude effort, which is quantified through the mean angular rate of the satellite, ${\u2225\mathsf{\omega}\u2225}_{\mathrm{mean}}$.

#### 3.3. Closing the Control Loop

- Estimation errors (e.g., ${\stackrel{\u02d8}{\mathsf{\alpha}}}_{d}$);
- Actuator errors (e.g., $\stackrel{\u02d8}{\mathfrak{q}}$);
- Physical constraints (e.g., ${\stackrel{\u02d8}{\mathit{u}}}^{b}$).

- In many applications, the tracking of the reference orbit and the absolute control of the reference orbit are handled separately;
- The proposed scheme can be directly applied to the rendezvous with an actual spacecraft.

## 4. MPC Validation

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Stability of the Surrogate Model of the ADCS

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**Figure 3.**Assumed expected values $\overline{\delta \mathsf{\alpha}}$, $\overline{u}$, and $\overline{\delta \mathfrak{q}}$ over one prediction horizon.

**Figure 8.**ROE profile of the Monte Carlo run defined by Table 4.

**Figure 9.**Trajectory followed by the satellite of the Monte Carlo run defined by Table 4.

**Figure 10.**Thrust level of the Monte Carlo run defined by Table 4.

**Figure 11.**Attitude profile of the Monte Carlo run defined by Table 4.

**Figure 12.**Thrust level for the benchmark simulation: (

**a**) thrust vector in the RTN frame; (

**b**) comparison of the thrust in the normal direction.

**Figure 13.**ROE profile of the benchmark maneuver defined by Table 5.

**Figure 14.**Attitude profile of the benchmark maneuver defined by Table 5.

${\tilde{\mathit{a}}}_{\mathit{c}}$ [km] | ${\tilde{\mathit{\theta}}}_{\mathit{c}}$ [${}^{\circ}$] | ${\left.{\tilde{\mathit{e}}}_{\mathit{x}}\right|}_{\mathit{c}}$ | ${\left.{\tilde{\mathit{e}}}_{\mathit{y}}\right|}_{\mathit{c}}$ | ${\tilde{\mathit{i}}}_{\mathit{c}}$ [${}^{\circ}$] | ${\tilde{\mathbf{\Omega}}}_{\mathit{c}}$ [${}^{\circ}$] |
---|---|---|---|---|---|

7121 | 0 | ${10}^{-5}$ | 0 | 45 | 0 |

Chief’s initial orbit | ${\tilde{a}}_{c}$ [km] | ${\tilde{\theta}}_{c}$ [${}^{\circ}$] | ${\left.{\tilde{e}}_{x}\right|}_{c}$ | ${\left.{\tilde{e}}_{y}\right|}_{c}$ | ${\tilde{i}}_{c}$ [${}^{\circ}$] | ${\tilde{\Omega}}_{c}$ [${}^{\circ}$] |

7121 | 0 | ${10}^{-5}$ | 0 | 45 | 0 | |

MPC parameters | ${T}_{s}$ [s] | ${n}_{p}$ [${T}_{s}$] | ${n}_{u}$ [${T}_{s}$] | ${u}_{max}$ [$\mathrm{m}/{\mathrm{s}}^{2}$] | ${\omega}_{max}$ [${}^{\circ}/\mathrm{s}$] | |

50 s | 60 | 15 | $3.5\times {10}^{-5}$ | 2 | ||

Q | ${f}_{\mathbf{P}}$ | ${f}_{u}$ | ${f}_{\delta \mathfrak{q}}$ | ${R}_{{\delta \mathfrak{q}}_{min}}$ | ||

${10}^{5}$ | 10 | $0.0$ | $0.02$ | ${10}^{-5}$ | ||

Miscellaneous | K | ${\sigma}_{\mathit{r}}$ [m] | ${\sigma}_{\mathit{v}}$ [$\mathrm{m}/\mathrm{s}$] | ${a}_{c}{\sigma}_{\delta \mathsf{\alpha}}$ [m] | ${\zeta}_{{\mathfrak{q}}_{\mathrm{pe}}}$ [arcs] | |

$0.2$ | 10 | 0.5 | 1 | 25 |

${\mathit{a}}_{\mathit{c}}\mathit{\delta}{\mathit{\alpha}}_{\mathbf{fin}}$ [m] | ${\mathit{a}}_{\mathit{c}}\mathit{\delta}{\mathit{a}}_{\mathbf{fin}}$ [m] | $\mathbf{\Delta}{\mathit{V}}_{\mathbf{tot}}$ [$\mathit{m}/\mathit{s}$] | ${\u2225\mathit{\omega}\u2225}_{\mathbf{mean}}$ [${}^{\circ}/\mathit{s}$] | |
---|---|---|---|---|

Mean | $4.38$ | $2.05$ | $0.406$ | $0.228$ |

Median | $4.29$ | $2.02$ | $0.409$ | $0.226$ |

Max | $9.82$ | $4.49$ | $0.58$ | $0.331$ |

Initial ROE | ${a}_{c}\delta a$ [m] | ${a}_{c}\delta \lambda $ [m] | ${a}_{c}\delta {e}_{x}$ [m] | ${a}_{c}\delta {e}_{y}$ [m] | ${a}_{c}\delta {i}_{x}$ [m] | ${a}_{c}\delta {i}_{y}$ [m] |

$64.62$ | $-947.77$ | $-57.84$ | $23.68$ | $-80.35$ | $24.03$ | |

Reference ROE | ${a}_{c}\delta a$ [m] | ${a}_{c}\delta \lambda $ [m] | ${a}_{c}\delta {e}_{x}$ [m] | ${a}_{c}\delta {e}_{y}$ [m] | ${a}_{c}\delta {i}_{x}$ [m] | ${a}_{c}\delta {i}_{y}$ [m] |

0 | 0 | 0 | 0 | 0 | 0 |

Chief’s initial orbit | ${\tilde{a}}_{c}$ [km] | ${\tilde{\theta}}_{c}$ [${}^{\circ}$] | ${\left.{\tilde{e}}_{x}\right|}_{c}$ | ${\left.{\tilde{e}}_{y}\right|}_{c}$ | ${\tilde{i}}_{c}{[}^{\circ}]$ | ${\tilde{\Omega}}_{c}$ [${}^{\circ}$] |

6828 | 0 | ${10}^{-5}$ | 0 | 78 | 0 | |

Initial ROE | ${a}_{c}\delta a$ [m] | ${a}_{c}\delta \lambda $ [m] | ${a}_{c}\delta {e}_{x}$ [m] | ${a}_{c}\delta {e}_{y}$ [m] | ${a}_{c}\delta {i}_{x}$ [m] | ${a}_{c}\delta {i}_{y}$ [m] |

0 | 0 | 273 | 0 | 10 | 70 | |

Reference ROE | ${a}_{c}\delta a$ [m] | ${a}_{c}\delta \lambda $ [m] | ${a}_{c}\delta {e}_{x}$ [m] | ${a}_{c}\delta {e}_{y}$ [m] | ${a}_{c}\delta {i}_{x}$ [m] | ${a}_{c}\delta {i}_{y}$ [m] |

0 | 0 | 273 | 0 | 400 | 120 |

Convergence Time [orbits] | Terminal ${\mathit{a}}_{\mathit{c}}\left(\mathit{\delta}\mathit{\alpha}-\mathit{\delta}{\mathit{\alpha}}_{\mathit{r}}\right)\phantom{\rule{0.277778em}{0ex}}\left[\mathbf{m}\right]$ | $\mathbf{\Delta}{\mathit{V}}_{\mathbf{tot}}\phantom{\rule{0.277778em}{0ex}}[\mathbf{m}/\mathbf{s}]$ | |
---|---|---|---|

Proposed MPC | $4.1$ | ${\left[\begin{array}{cccccc}0.9& -0.4& 1.9& -0.33.& 0.0& -0.5\end{array}\right]}^{\u22ba}$ | $0.664$ |

Reference MPC | $6.6$ | ${\left[\begin{array}{cccccc}-3.6& 9.2& -1.4& 2.0& -2.9& 1.6\end{array}\right]}^{\u22ba}$ | $0.501$ |

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**MDPI and ACS Style**

Mahfouz, A.; Gaias, G.; Venkateswara Rao, D.M.K.K.; Voos, H.
Autonomous Optimal Absolute Orbit Keeping through Formation Flying Techniques. *Aerospace* **2023**, *10*, 959.
https://doi.org/10.3390/aerospace10110959

**AMA Style**

Mahfouz A, Gaias G, Venkateswara Rao DMKK, Voos H.
Autonomous Optimal Absolute Orbit Keeping through Formation Flying Techniques. *Aerospace*. 2023; 10(11):959.
https://doi.org/10.3390/aerospace10110959

**Chicago/Turabian Style**

Mahfouz, Ahmed, Gabriella Gaias, D. M. K. K. Venkateswara Rao, and Holger Voos.
2023. "Autonomous Optimal Absolute Orbit Keeping through Formation Flying Techniques" *Aerospace* 10, no. 11: 959.
https://doi.org/10.3390/aerospace10110959