# Dynamic Modeling and Attitude–Vibration Cooperative Control for a Large-Scale Flexible Spacecraft

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model of the LSFS

#### 2.1. Assumptions and Geometry Descriptions of the Model

#### 2.2. The Expression of the LSFS’s Solar Panel Displacement

#### 2.3. The LSFS’s Kinetic Energy

#### 2.4. The LSFS’s Potential Energy

## 3. Discrete Dynamic Model for the LSFS

**p**(t) is the generalized modal coordinate vector expressed as

## 4. The Validation Analysis for the Method

## 5. The Dynamic Response Analysis

#### 5.1. Displacements of the LSFS under the Three-Axis Attitude-Driving Torque Pulse

#### 5.2. The Effects of the Hinges for the Attitude Maneuver

## 6. The Cooperative Control for Attitude Motion and Structure Vibration

#### 6.1. The LQR Cooperative Controller

**Q**is a positive semimatrix, and R is a positive weighting scalar. The gain matrix

**G**can be obtained as $\mathit{G}={\mathit{R}}^{-1}{\mathit{B}}^{\mathrm{T}}\mathit{P},$ by solving the functional extremum problem, where

**P**is the solution of the following Riccati equation:

#### 6.2. The PD Cooperative Controller

_{d}and K

_{p}are the differential gain and proportional gain, and ${\theta}_{\mathit{d}}$ and ${\dot{\theta}}_{\mathit{d}}$ are the desired attitude angle and the desired attitude angular velocity of the spacecraft, respectively. Substituting the control torque ${\tau}_{\mathit{y}}$ into Equation (23), the desired attitude angle of the flexible spacecraft can be accurately achieved, and the vibration of the solar arrays can be suppressed synchronously.

#### 6.3. The PD + IS Cooperative Controller

_{mult}is a multimodal input shaper with the following expression:

_{is}(i = 1, 2, …, n) is the pulse sequence of the ith mode of the system, including the pulse amplitude A

_{j}, and the action time t

_{j}; “$\ast $” is a convolution symbol. During calculation, the A

_{j}of each A

_{is}is multiplied, and the corresponding tj is added.

#### 6.4. The Simulations and Discussions

_{3}and the hinge ${\mathit{B}}_{{\mathit{R}}_{1}}$ oscillate intensely. The relatively large amplitude vibration of the solar array is observed, and the maximum amplitude of w

_{3}reaches up to 8.8 mm. In addition, the maximum control torque is about 1.2 Nm.

_{d}= 48 and K

_{p}= 1.2, respectively.

_{3}and hinge ${\mathit{B}}_{{\mathit{R}}_{1}}$ also oscillate intensely at the beginning of the attitude maneuvering. However, the maximum vibration amplitude of the solar array w

_{3}reaches up to 4.8 mm. In addition, the maximum control torque is about 0.7 Nm.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**M**is as follows:

## Appendix B

**K**and

**C**are as follows:

**C**

_{77}is as follows:

## References

- Hu, Q.; Shi, P.; Gao, H. Adaptive variable structure and commanding shaped vibration control of flexible spacecraft. J. Guid. Control Dynam.
**2007**, 30, 804–815. [Google Scholar] [CrossRef] - Wei, J.; Cao, D.; Wang, L.; Huang, H.; Huang, W. Dynamic modeling and simulation for flexible spacecraft with flexible jointed solar panels. Int. J. Mech. Sci.
**2017**, 130, 558–570. [Google Scholar] [CrossRef] - Wei, J.; Cao, D.; Huang, H.; Wang, L.; Huang, W. Dynamics of a multi-beam structure connected with nonlinear joints: Modelling and simulation. Arch. App. Mech.
**2018**, 88, 1059–1074. [Google Scholar] [CrossRef] - Xing, Y.; Liu, B. New exact solutions for free vibrations of thin orthotropic rectangular plates. Compos. Struct.
**2009**, 89, 567–574. [Google Scholar] [CrossRef] - Li, Z.; Cao, D.; Zhang, C. Natural characteristics of rectangular plates clamped at 4 corner points. J. Vib. Shock.
**2015**, 34, 98–102. [Google Scholar] - Dong, C. Three-dimensional free vibration analysis of functionally graded annular plates using the Chebyshev-Ritz method. Mater. Design
**2008**, 29, 1518–1525. [Google Scholar] [CrossRef] - Li, Z.; Cao, D. Optimization Analysis of Supporting Point Distribution for Folding Solar Panels. J. Astronaut.
**2015**, 36, 662–666. [Google Scholar] - Cao, D.; Wang, L.; Wei, J.; Nie, Y. Natural frequencies and global mode functions for flexible jointed-panel structures. J. Aerospace Eng.
**2020**, 33, 04020018. [Google Scholar] [CrossRef] - He, G.; Cao, D.; Cao, Y.; Huang, W. Investigation on global analytic modes for a three-axis attitude stabilized spacecraft with jointed panels. Aerosp. Sci. Technol.
**2020**, 106, 106087. [Google Scholar] [CrossRef] - He, G.; Cao, D.; Wei, J.; Cao, Y.; Chen, Z. Study on analytical global modes for a multi-panel structure connected with flexible hinges. Appl. Math. Model.
**2021**, 91, 1081–1099. [Google Scholar] [CrossRef] - Bhat, R. Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh-Ritz method. J. Sound Vib.
**1985**, 102, 493–499. [Google Scholar] [CrossRef] - Zhang, W.; Chen, X.; Yang, Z.; Shen, Z. Multivariable wavelet finite element for flexible skew thin plate analysis. Sci. China Tech. Sci.
**2014**, 57, 1532–1540. [Google Scholar] [CrossRef] - Sales, T.; Rade, D.; De Souza, L. Passive vibration control of flexible spacecraft using shunted piezoelectric transducers. Aerosp. Sci. Technol.
**2013**, 29, 403–412. [Google Scholar] [CrossRef] - Liu, L.; Cao, D.; Wei, J. Rigid-flexible coupling dynamic modeling and vibration control for flexible spacecraft based on its global analytical modes. Sci. China Tech. Sci.
**2019**, 62, 608–618. [Google Scholar] [CrossRef] - Karray, F.; Grewal, A.; Glaum, M.; Modi, V. Stiffening control of a class of nonlinear affine systems. IEEE Trans. Aerosp. Electron. Syst.
**1997**, 33, 473–484. [Google Scholar] [CrossRef] - Shabana, A.; Yakoub, R. Three dimensional absolute nodal coordinate formulation for beam elements: Theory. J. Mech. Design
**2001**, 123, 614–621. [Google Scholar] [CrossRef] - Sahoo, R.; Singh, B. Assessment of dynamic instability of laminated composite-sandwich plates. Aerosp. Sci. Technol.
**2018**, 81, 41–52. [Google Scholar] [CrossRef] - Frikha, A.; Zghal, S.; Dammak, F. Dynamic analysis of functionally graded carbon nanotubes-reinforced plate and shell structures using a double directors finite shell element. Aerosp. Sci. Technol.
**2018**, 78, 438–451. [Google Scholar] [CrossRef] - Jen, C.; Johnson, D.; Dubois, F. Numerical modal analysis of structures based on a revised substructure synthesis approach. J. Sound Vib.
**1995**, 180, 185–203. [Google Scholar] [CrossRef] - Hablani, H. Modal analysis of gyroscopic flexible spacecraft: A continuum approach. J. Guid. Control Dynam.
**2015**, 5, 448–457. [Google Scholar] [CrossRef] - Hughes, P.C.; Skelton, R.E. Modal truncation for flexible spacecraft. J. Guid. Control
**1981**, 4, 291–297. [Google Scholar] - Hurty, W. Dynamic analysis of structural systems using component modes. AIAA J.
**1965**, 3, 678–685. [Google Scholar] [CrossRef] - Craig, R.; Bampton, M. Coupling of Substructures for Dynamic Analyses. AIAA J.
**1968**, 6, 1313–1319. [Google Scholar] [CrossRef] - Liu, J.; Pan, K. Rigid-flexible-thermal coupling dynamic formulation for satellite and plate multibody system. Aerosp. Sci. Technol.
**2016**, 52, 102–114. [Google Scholar] [CrossRef] - Schwertassek, R.; Wallrapp, O.; Shabana, A. Flexible multibody simulation and choice of shape functions. Nonlinear Dyn.
**1999**, 20, 361–380. [Google Scholar] [CrossRef] - Guy, N.; Alazard, D.; Cumer, C.; Charbonnel, C. Dynamic modeling and analysis of spacecraft with variable tilt of flexible appendages. J. Dyn. Syst. Meas. Control
**2014**, 136, 021020. [Google Scholar] [CrossRef] - Azimi, M.; Joubaneh, E.F. Dynamic modeling and vibration control of a coupled rigid-flexible high-order structural system: A comparative study. Aerosp. Sci. Technol.
**2020**, 102, 105875. [Google Scholar] [CrossRef] - Johnston, J.; Thornton, E. Thermally induced attitude dynamics of a spacecraft with a flexible appendage. J. Guid. Control Dynam.
**1998**, 21, 581–587. [Google Scholar] [CrossRef] - Lim, K. Method for optimal actuator and sensor placement for large flexible structures. J. Guid. Control Dynam.
**1992**, 15, 49–57. [Google Scholar] [CrossRef] - Song, G.; Kotejoshyer, B. Vibration reduction of flexible structures during slew operations. Int. J. Acoust. Vib.
**2002**, 7, 105–109. [Google Scholar] [CrossRef] - Hu, Q.; Ma, G. Vibration suppression of flexible spacecraft during attitude maneuvers. J. Guid. Control Dynam.
**2005**, 28, 377–380. [Google Scholar] [CrossRef] - Bailey, T.; Hubbard, J. Distributed piezoelectric-polymer active vibration control of a cantilever beam. J. Guid. Control Dynam.
**1985**, 8, 23. [Google Scholar] [CrossRef] - Won, C.; Sulla, J.; Sparks, D.; Belvin, W. Application of piezoelectric devices to vibration suppression. J. Guid. Control Dynam.
**1994**, 17, 1333–1338. [Google Scholar] [CrossRef] - Ge, X.; Sun, K. Optimal control of a spacecraft with deployable solar arrays using particle swarm optimization algorithm. Sci. China Tech. Sci.
**2011**, 54, 1107–1112. [Google Scholar] [CrossRef] - Li, Q.; Wang, B.; Deng, Z.; Ouyang, H.; Wei, Y. A simple orbit-attitude coupled modelling method for large solar power satellites. Acta Astronaut.
**2018**, 145, 83–92. [Google Scholar] [CrossRef] - Paik, J.; Thayamballi, A.; Kim, G. The strength characteristics of aluminum honeycomb sandwich panels. Thin Wall Struct.
**1999**, 35, 205–231. [Google Scholar] [CrossRef] - He, G.; Cao, D.; Cao, Y.; Huang, W. Dynamic modeling and orbit maneuvering response analysis for a three-axis attitude stabilized large scale flexible spacecraft installed with hinged solar arrays. Mech. Syst. Signal Pract.
**2022**, 162, 108083. [Google Scholar] [CrossRef] - Crawley, E.; O’Donnell, K. Force-state mapping identification of nonlinear joints. AIAA J.
**1987**, 25, 1003–1010. [Google Scholar] [CrossRef]

**Figure 6.**The translation of the spacecraft under the $\mathit{\tau}$($\mathit{k}=50\mathrm{N}\cdot \mathrm{m}/\mathrm{rad}$ ).

**Figure 7.**The attitude motion of the spacecraft under the $\mathit{\tau}$($\mathit{k}=50\mathrm{N}\cdot \mathrm{m}/\mathrm{rad}$ ); they are listed as: (

**a**) the displacement of ${\theta}_{\mathit{x}}$; (

**b**) the displacement of ${\theta}_{\mathit{y}}$; (

**c**) the displacement of ${\theta}_{\mathit{z}}$.

**Figure 8.**Displacements of the solar arrays under the $\mathit{\tau}$($\mathit{k}=50\mathrm{N}\cdot \mathrm{m}/\mathrm{rad}$ ).

**Figure 9.**Deflections of the solar array tip for different spring stiffnesses; they are listed as: (

**a**) $\mathit{k}=50\mathrm{N}\cdot \mathrm{m}/\mathrm{rad}$; (

**b**) $\mathit{k}=100\mathrm{N}\cdot \mathrm{m}/\mathrm{rad}$; (

**c**) $\mathit{k}=500\mathrm{N}\cdot \mathrm{m}/\mathrm{rad}$; (

**d**) $\mathit{k}=1000\mathrm{N}\cdot \mathrm{m}/\mathrm{rad}$.

**Figure 10.**The vibration response of the system with various values of c ($\mathit{k}=500\mathrm{N}\cdot \mathrm{m}/\mathrm{rad}$).

**Figure 11.**The vibration response of the system ($\mathit{k}=500\mathrm{N}\cdot \mathrm{m}/\mathrm{rad}$); they are listed as: (

**a**) the displacement of ${\theta}_{\mathit{y}}$; (

**b**) the displacement of the 3rd-right panel tip.

**Figure 15.**Time responses of the flexible spacecraft for using the LQR control ($\mathit{k}=500\mathrm{N}\cdot \mathrm{m}/\mathrm{rad}$); they are listed as: (

**a**) the displacement of ${\theta}_{\mathit{y}}$; (

**b**) the displacement of the 3rd-right panel tip; (

**c**) the displacement of the right hinge ${\mathit{B}}_{{\mathit{R}}_{1}}$; (

**d**) the control torque ${\tau}_{\mathit{y}}$.

**Figure 16.**Time responses of the flexible spacecraft for using the PD control ($\mathit{k}=500\mathrm{N}\cdot \mathrm{m}/\mathrm{rad}$); they are listed as: (

**a**) the displacement of ${\theta}_{\mathit{y}}$; (

**b**) the displacement of the 3rd-right panel tip; (

**c**) the displacement of the right hinge ${\mathit{B}}_{{\mathit{R}}_{1}}$; (

**d**) the control torque ${\tau}_{\mathit{y}}$.

**Figure 17.**Time responses of the flexible spacecraft for using the PD + IS control ($\mathit{k}=500\mathrm{N}\cdot \mathrm{m}/\mathrm{rad}$); they are listed as: (

**a**) the displacement of ${\theta}_{\mathit{y}}$; (

**b**) the displacement of the 3rd-right panel tip; (

**c**) the displacement of the right hinge ${\mathit{B}}_{{\mathit{R}}_{1}}$; (

**d**) the control torque ${\tau}_{\mathit{y}}$.

Frequency Order | ANSYS | Proposed Method | ${\mathit{R}}_{\mathit{t}}$ (%) |
---|---|---|---|

1 | 0.102 | 0.102 | 0.000 |

2 | 0.233 | 0.235 | 0.858 |

3 | 0.630 | 0.634 | 0.635 |

4 | 0.673 | 0.677 | 0.594 |

5 | 1.664 | 1.676 | 0.721 |

6 | 1.672 | 1.684 | 0.718 |

7 | 3.300 | 3.275 | 0.758 |

8 | 3.444 | 3.420 | 0.697 |

9 | 10.220 | 10.169 | 0.499 |

10 | 10.272 | 10.220 | 0.506 |

Parameters | Values |
---|---|

The number of solar panels (N) | 3.0 |

The length of the panel a (m) | 2.0 |

The width of the panel 2b (m) | 2.0 |

Distance between the hinges A and B b_{0} (m) | 1.6 |

The thickness of the honeycomb core 2 h_{c} (m) | 0.0197 |

The thickness of the honeycomb face sheet h_{f} (m) | 0.15 × 10^{−3} |

The length of the honeycomb wall ${\mathit{l}}_{\mathit{c}}$ (m) | 6.35 × 10^{−3} |

The thickness of the honeycomb wall ${\delta}_{\mathit{c}}$ (m) | 0.0254 × 10^{−3} |

The elastic modulus of the aluminum E_{0} (Pa) | 6.89 × 10^{10} |

The mass density of the aluminum ${\rho}_{0}$ (kg m^{−3}) | 2.8 × 10^{3} |

Poisson ratio $\mathit{v}$ | 0.33 |

The size of the distance r_{0} (m) | 2.0 |

The inertial moment of the hub J_{x, y, z} (kg m^{2}) | 100,100,100 |

The mass of the hub ${\mathit{m}}_{\mathit{R}}(\mathrm{kg})$ | 150 |

The linear stiffness of the rotation spring $\mathit{k}(\mathrm{N}\cdot \mathrm{m}/\mathrm{rad})$ | 50 |

The damping coefficient of the hinges $\mathit{c}(\mathrm{N}\cdot \mathrm{m}\cdot \mathrm{s}/\mathrm{rad})$ | 10 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

He, G.; Cao, D.
Dynamic Modeling and Attitude–Vibration Cooperative Control for a Large-Scale Flexible Spacecraft. *Actuators* **2023**, *12*, 167.
https://doi.org/10.3390/act12040167

**AMA Style**

He G, Cao D.
Dynamic Modeling and Attitude–Vibration Cooperative Control for a Large-Scale Flexible Spacecraft. *Actuators*. 2023; 12(4):167.
https://doi.org/10.3390/act12040167

**Chicago/Turabian Style**

He, Guiqin, and Dengqing Cao.
2023. "Dynamic Modeling and Attitude–Vibration Cooperative Control for a Large-Scale Flexible Spacecraft" *Actuators* 12, no. 4: 167.
https://doi.org/10.3390/act12040167