# Attitude-Orbit Coupled Control of Gravitational Wave Detection Spacecraft with Communication Delays

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## Abstract

**:**

## 1. Introduction

## 2. Material Background and Relative Coupled Dynamics

#### 2.1. Quaternions and Dual Quaternions

#### 2.2. Graph Theory

#### 2.3. Equations of Attitude-Orbit Coupled Relative Motion Based on Dual Quaternions

#### 2.4. Problem Statement

## 3. Control Law Design

**Assumption 1.**

**Assumption 2.**

**Assumption 3.**

**Assumption 4.**

**Lemma 1**

**Assumption 5.**

**Theorem 1.**

**Proof**

**Remark 1.**

## 4. Numerical Simulations

#### 4.1. The Maximum Available Control Force Is 100 $\mu $N

#### 4.2. The Maximum Available Control Force Is 100 mN

## 5. Conclusions

- (1)
- The spacecraft can control the position and attitude of the spacecraft and the test masses simultaneously using the microthruster during the maneuver, but it takes at least 3 days under the initial error of about 100 m;
- (2)
- Increasing the thrust shortens the control time, but the test masses need to be fixed to prevent the test masses from colliding with the cavity during the orbit transfer.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\mathit{A}$ | adjacency matrix (with entries $\left[{a}_{ij}\right]$) |

${\mathit{a}}_{gi}^{i}$ | gravitational acceleration expressed in the body-fixed frame |

${\mathit{a}}_{J2i}^{i}$ | perturbing acceleration due to Earth’s oblateness expressed in the body-fixed frame |

${\mathit{a}}_{dsi}^{i}$ | the acceleration caused by solar radiation pressure expressed in the body-fixed frame |

$\mathbb{DQ}$, ${\mathbb{DQ}}_{v}$ | set of dual quaternions and dual vector quaternions, respectively |

${\mathbb{DQ}}_{s}$, ${\mathbb{DQ}}_{r}$ | set of dual scalar quaternions and dual scalar quaternions with zero dual part, respectively |

$\mathit{E}$ | set of edges |

${\mathcal{F}}_{I}$ | the Earth-centered inertial frame |

${\mathcal{F}}_{i}$, ${\mathcal{F}}_{di}$ | the body-fixed frame and the desired body-fixed frame of the ith rigid body, respectively |

${\widehat{\mathit{F}}}_{i}^{i}$, ${\widehat{\mathit{F}}}_{di}^{di}$ | total dual force expressed in the frames ${\mathcal{F}}_{i}$ and ${\mathcal{F}}_{di}$, respectively |

${\widehat{\mathit{f}}}_{ui}^{i}$ | dual control force expressed in the body-fixed frame |

${\widehat{\mathit{f}}}_{gi}^{i}$, ${\widehat{\mathit{f}}}_{gdi}^{di}$ | dual gravitational force expressed in the frame ${\mathcal{F}}_{i}$ and ${\mathcal{F}}_{di}$, respectively |

${\widehat{\mathit{f}}}_{dsi}^{i}$ | dual solar pressure perturbation expressed in the body-fixed frame |

${\widehat{\mathit{f}}}_{J2i}^{i}$, ${\widehat{\mathit{f}}}_{J2di}^{di}$ | dual ${J}_{2}$-perturbation force expressed in the frame ${\mathcal{F}}_{i}$ and ${\mathcal{F}}_{di}$, respectively |

$\mathbb{H}$, ${\mathbb{H}}_{v}$, ${\mathbb{H}}_{s}$ | set of quaternions, set of vector quaternions, set of scalar quaternions |

${\mathit{I}}_{3}$ | the 3-by-3 identity matrix |

${J}_{i}$, ${m}_{i}$ | inertia matrix and mass of i-th rigid body |

${\widehat{k}}_{1i},{\widehat{k}}_{2i},{\widehat{k}}_{3i}$ | control gains |

${\widehat{\mathit{M}}}_{i}$ | dual inertia matrix |

${\mathit{q}}_{di}$, ${\mathit{q}}_{i}$ | quaternion of the frames ${\mathcal{F}}_{di}$ and ${\mathcal{F}}_{i}$ with respect to the frame ${\mathcal{F}}_{I}$ |

${\widehat{\mathit{q}}}_{i}$ | dual quaternion of the frame ${\mathcal{F}}_{i}$ with respect to the frame ${\mathcal{F}}_{I}$ |

${\mathit{q}}_{i}^{\ast}$, ${\widehat{\mathit{q}}}_{i}^{\ast}$ | the conjugate of ${\mathit{q}}_{i}$ and ${\widehat{\mathit{q}}}_{i}$ |

${\widehat{\mathit{q}}}_{ei}$ | dual quaternion of the frame ${\mathcal{F}}_{i}$ with respect to the frame ${\mathcal{F}}_{di}$ |

${T}_{ij}$ | the communication delay between the j-th rigid body and the i-th rigid body |

${\omega}_{i}^{i}$ | angular velocity of ${\mathcal{F}}_{i}$ frame with respect to the ${\mathcal{F}}_{I}$ frame expressed in the ${\mathcal{F}}_{i}$ frame |

${\widehat{\omega}}_{i}^{i}$ | dual velocity of the frame ${\mathcal{F}}_{i}$ with respect to the ${\mathcal{F}}_{I}$ frame expressed in the ${\mathcal{F}}_{i}$ frame |

${\widehat{\omega}}_{ei}^{i}$ | dual velocity of the frame ${\mathcal{F}}_{i}$ with respect to the frame ${\mathcal{F}}_{di}$ frame expressed in the ${\mathcal{F}}_{i}$ frame |

${\mu}_{e}$ | Earth’s gravitational parameter |

${\mu}_{m}$ | Moon’s gravitational parameter |

${\mu}_{s}$ | sun’s gravitational parameter |

$\mathcal{V}$ | set of vertices |

$\epsilon $ | dual unit |

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**Figure 3.**Relative position errors of spacecraft SC$1\sim 3$, the max control force of spacecraft is 100 $\mathsf{\mu}$N.

**Figure 4.**Relative linear velocity errors of spacecraft SC$1\sim 3$, the max control force of spacecraft is 100 $\mathsf{\mu}$N.

**Figure 5.**Relative angular velocity errors of spacecraft SC$1\sim 3$, the max control torque of spacecraft is 100 $\mathsf{\mu}$N· m.

**Figure 6.**Relative attitude errors of spacecraft SC1 ∼ 3, the max control torque of spacecraft is 100 $\mathsf{\mu}$N· m.

**Figure 7.**Control forces and control torques about SC1 ∼ 3, the max control force of spacecraft is 100 $\mathsf{\mu}$N.

**Figure 8.**Relative position errors of TM1 ∼ 6, the max control force of test mass is 0.2 $\mathsf{\mu}$N.

**Figure 9.**Relative linear velocity errors of spacecrafts TM1 ∼ 6, the max control force of spacecraft is 100 $\mathsf{\mu}$N.

**Figure 10.**Relative angular velocity errors of spacecrafts TM1 ∼ 6, the max control torque of spacecraft is 100 $\mathsf{\mu}$N· m.

**Figure 11.**Relative attitude errors of spacecrafts TM1 ∼ 6, the max control torque of spacecraft is 100 $\mathsf{\mu}$N· m.

**Figure 12.**Control forces and control torques of TM1 ∼ 6, the max control force of spacecraft is 100 $\mathsf{\mu}$N.

**Figure 13.**Relative position errors of spacecraft SC$1\sim 3$, the max control force of spacecraft is 100 mN.

**Figure 14.**Relative linear velocity errors of spacecraft SC$1\sim 3$, the max control force of spacecraft is 100 mN.

**Figure 15.**Relative angular velocity errors of spacecraft SC$1\sim 3$, the max control torque of spacecraft is 100 mN· m.

**Figure 16.**Relative attitude errors of spacecraft SC1 ∼ 3, the max control torque of spacecraft is 100 mN· m.

**Figure 17.**Control forces and control torques about SC1 ∼ 3, the max control force of spacecraft is 100 mN.

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

Perigee altitude | $9.999\times {10}^{7}$ | m |

Eccentricity | 0.00043 | - |

Inclination | 74.5362 | |

Argument of perigee | 346.5528 | |

RAAN | 211.6003 | |

True anomaly (SC1) | 61.3296 | |

True anomaly (SC2) | 181.3296 | |

True anomaly (SC3) | 301.3296 |

Initial Position Error (m) | Initial Velocity Error ($\mathbf{m}\xb7{\mathbf{s}}^{-1}$) | Initial Angular Velocity Error ($\mathbf{Rad}\xb7{\mathbf{s}}^{-1}$) | Initial Quaternion Error (−) | |
---|---|---|---|---|

1(SC1) | [−100 80 −120]${}^{\mathrm{T}}$ | [3 −2 1]${}^{\mathrm{T}}\times {10}^{-3}$ | [0.8 −2 1] ${}^{\mathrm{T}}\times {10}^{-5}$ | [0.9972 0.0416 0.0454 0.0416] |

2(SC2) | [160 100 −40]${}^{\mathrm{T}}$ | [−1 −2 1] ${}^{\mathrm{T}}\times {10}^{-3}$ | [0.7 −2 2] ${}^{\mathrm{T}}\times {10}^{-5}$ | [0.9976 0.0515 0.0445 0.0151] |

3(SC3) | [−80 120 100]${}^{\mathrm{T}}$ | [4 0 −1] ${}^{\mathrm{T}}\times {10}^{-3}$ | [0.9 −1 1] ${}^{\mathrm{T}}\times {10}^{-5}$ | [0.9977 0.0447 −0.0424 0.0280] |

4(TM1) | [−3 2 −2]${}^{\mathrm{T}}\times {10}^{-5}$ | [3 −2 −1]${}^{\mathrm{T}}\times {10}^{-8}$ | [0.8 −2 1]${}^{\mathrm{T}}\times {10}^{-9}$ | [0.9972 0.0416 0.0454 0.0416] |

5(TM2) | [5 −2 1]${}^{\mathrm{T}}\times {10}^{-5}$ | [−2 −1 -1]${}^{\mathrm{T}}\times {10}^{-8}$ | [1 −3 2]${}^{\mathrm{T}}\times {10}^{-9}$ | [0.9976 0.0515 0.0445 0.0151] |

6(TM3) | [2 7 5]${}^{\mathrm{T}}\times {10}^{-5}$ | [3 1 −5]${}^{\mathrm{T}}\times {10}^{-8}$ | [3 −1 5]${}^{\mathrm{T}}\times {10}^{-9}$ | [0.9977 0.0447 −0.0424 0.0280] |

7(TM4) | [−5 6 −2]${}^{\mathrm{T}}\times {10}^{-5}$ | [−5 6 0]${}^{\mathrm{T}}\times {10}^{-8}$ | [0.8 −3 4]${}^{\mathrm{T}}\times {10}^{-9}$ | [0.5000 0 0 0.8660] |

8(TM5) | [−2 −3 −1]${}^{\mathrm{T}}\times {10}^{-5}$ | [−2 −3 −1]${}^{\mathrm{T}}\times {10}^{-8}$ | [2 −5 3]${}^{\mathrm{T}}\times {10}^{-9}$ | [0.8660 0 0 0.5000] |

9(TM6) | [−2 2 1]${}^{\mathrm{T}}\times {10}^{-5}$ | [−2 2 1]${}^{\mathrm{T}}\times {10}^{-8}$ | [0.7 −6 8]${}^{\mathrm{T}}\times {10}^{-9}$ | [0.9848 0 0.1736 0] |

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## Share and Cite

**MDPI and ACS Style**

Zhang, Y.; Liu, Y.; Yang, J.; Lu, Z.; Zhang, J. Attitude-Orbit Coupled Control of Gravitational Wave Detection Spacecraft with Communication Delays. *Sensors* **2023**, *23*, 3233.
https://doi.org/10.3390/s23063233

**AMA Style**

Zhang Y, Liu Y, Yang J, Lu Z, Zhang J. Attitude-Orbit Coupled Control of Gravitational Wave Detection Spacecraft with Communication Delays. *Sensors*. 2023; 23(6):3233.
https://doi.org/10.3390/s23063233

**Chicago/Turabian Style**

Zhang, Yu, Yuan Liu, Jikun Yang, Zhenkun Lu, and Juzheng Zhang. 2023. "Attitude-Orbit Coupled Control of Gravitational Wave Detection Spacecraft with Communication Delays" *Sensors* 23, no. 6: 3233.
https://doi.org/10.3390/s23063233