# A Station-Keeping Control Strategy for a Symmetrical Spacecraft Utilizing Hybrid Low-Thrust Propulsion in the Heliocentric Displaced Orbit

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## Abstract

**:**

## 1. Introduction

## 2. Orbit Dynamic Model

#### 2.1. Definition of Coordinate System and Attitude Angle

#### 2.2. Displaced Orbital Dynamics

## 3. High-Performance Station-Keeping Controller Design

#### 3.1. RBF Neural Network Estimator Design

#### 3.2. Control Law Design

#### 3.3. Stability Analysis

**Lemma**

**1**

**[31].**

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

#### 3.4. Control Variable Conversion

## 4. Simulation Results and Discussion

#### 4.1. Simulation Conditions

#### 4.2. Simulation Results and Discussion under Different Sources of Disturbance

#### 4.2.1. When the Solar Sail Spacecraft Is Mainly Subjected to External Disturbances

#### 4.2.2. When the Solar Sail Spacecraft Is Subjected to Both External Disturbances and Internal Unmodeled Dynamic

#### 4.2.3. Comparison of Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**Change curves of position errors and angular velocity: (

**a**) change curves of orbital radius error and displaced height error; (

**b**) change curve of angular velocity.

**Figure 7.**Station-keeping control variables: (

**a**) change curves of attitude angles; (

**b**) change curves of propulsion force components of argon Hall thruster.

**Figure 9.**Change curves of position errors and angular velocity: (

**a**) change curves of orbital radius error and displaced height error; (

**b**) change curve of angular velocity.

**Figure 11.**Station-keeping control variables: (

**a**) change curves of attitude angles; (

**b**) change curves of propulsion force components of argon Hall thruster.

Orbital Radius | Displaced Height | Angular Velocity |
---|---|---|

0.7AU | 0.4AU | 1 |

Propellant | Thrust (mN) | Specific Impulse (s) | Total Efficiency | Power (kW) | Mass (kg) |
---|---|---|---|---|---|

Argon | 170 | 2500 | 50% | 4.2 | 2.1 |

Parameters of the Sliding Surface | Parameters of the Approaching Law | Parameters of the Adaptive Law |
---|---|---|

${k}_{0i}=165$; ${k}_{1i}=0.0003$; ${\epsilon}_{i}=0.00001$; ${\tau}_{i}=-0.98$ | ${k}_{2i}=1$; ${k}_{4i}=0.001$ | $\gamma =0.001$ |

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## Share and Cite

**MDPI and ACS Style**

Zhang, T.; Mu, R.; Zhou, Y.; Liao, Z.; Zhang, Z.; Liao, B.; Yao, C.
A Station-Keeping Control Strategy for a Symmetrical Spacecraft Utilizing Hybrid Low-Thrust Propulsion in the Heliocentric Displaced Orbit. *Symmetry* **2023**, *15*, 1549.
https://doi.org/10.3390/sym15081549

**AMA Style**

Zhang T, Mu R, Zhou Y, Liao Z, Zhang Z, Liao B, Yao C.
A Station-Keeping Control Strategy for a Symmetrical Spacecraft Utilizing Hybrid Low-Thrust Propulsion in the Heliocentric Displaced Orbit. *Symmetry*. 2023; 15(8):1549.
https://doi.org/10.3390/sym15081549

**Chicago/Turabian Style**

Zhang, Tengfei, Rongjun Mu, Yilin Zhou, Zizheng Liao, Zhewei Zhang, Bo Liao, and Chuang Yao.
2023. "A Station-Keeping Control Strategy for a Symmetrical Spacecraft Utilizing Hybrid Low-Thrust Propulsion in the Heliocentric Displaced Orbit" *Symmetry* 15, no. 8: 1549.
https://doi.org/10.3390/sym15081549