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Journal articles on the topic 'Flexible Spacecraft Control'

Author: Grafiati

Published: 4 June 2021

Last updated: 6 September 2023

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1

Yu, Ya Nan, Xiu Yun Meng, and Li Chao Ma. "PSO-Based State Feedback Control of Flexible Spacecraft for Attitude Tracking and Vibration Suppression." Applied Mechanics and Materials 229-231 (November 2012): 2161–65. http://dx.doi.org/10.4028/www.scientific.net/amm.229-231.2161.

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PD state-feedback controller has been adopted in many spacecraft for attitude tracking and presents good performance. For flexible spacecraft, the controller can be designed with a term which takes into account the flexible dynamics. However, duo to nonlinearity and coupling, how to determine state-feedback control parameters which ensure fast attitude tracking and significant vibration suppression must be considered. In this paper, the dynamics model of spacecraft with flexible appendages is derived with the hybrid coordinate method and the full state feedback controller originated from the P

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2

Wang, Guan Yu, Wei Ping Ge, Guang Wei Yang, and Sheng Chao Wang. "CSVS Method for Spacecraft with Two Flexible Appendages during Attitude Maneuver." Advanced Materials Research 1049-1050 (October 2014): 939–44. http://dx.doi.org/10.4028/www.scientific.net/amr.1049-1050.939.

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An angle maneuver control strategy of spacecraft with two flexible appendages based on component synthesis vibration suppression (CSVS) method is put forward. Unwanted flexible vibration modes can be eliminated while desired rigid motion can be achieved by this method. Jet device and Momentum wheel system are used as the actuator of the spacecraft’s angle maneuver. Several time-fuel control strategies are designed for spacecraft with flexible appendages. Simulation results validate the feasibility of the CSVS method. The DC motor control method is researched in order to combine momentum wheel

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3

Kwatny, H. G., M. J. Baek, W. H. Bennett, and G. L. Blankenship. "Attitude Control of Articulated, Flexible Spacecraft." IFAC Proceedings Volumes 25, no. 13 (1992): 463–69. http://dx.doi.org/10.1016/s1474-6670(17)52325-3.

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4

Zhang, Shuo, Yukang Zhou, and Suting Cai. "Fractional-Order PD Attitude Control for a Type of Spacecraft with Flexible Appendages." Fractal and Fractional 6, no. 10 (2022): 601. http://dx.doi.org/10.3390/fractalfract6100601.

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As large-sized spacecraft have been developed, they have been equipped with flexible appendages, such as solar cell plates and mechanical flexible arms. The attitude control of spacecraft with flexible appendages has become more complex, with higher requirements. In this paper, a fractional-order PD attitude control method for a type of spacecraft with flexible appendages is presented. Firstly, a lumped parameter model of a spacecraft with flexible appendages is constructed, which provides the transfer function of the attitude angle and external moment. Then, a design method for the fractional

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5

Nadafi, Reza, Mansour Kabganian, Ali Kamali, and Mahboobeh Hossein Nejad. "Super-twisting sliding mode control design based on Lyapunov criteria for attitude tracking control and vibration suppression of a flexible spacecraft." Measurement and Control 52, no. 7-8 (2019): 814–31. http://dx.doi.org/10.1177/0020294019847696.

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The three-axis attitude tracking manoeuvre and vibration suppression of a flexible spacecraft in the presence of external disturbances are investigated in this paper. The spacecraft consists of a rigid hub and two flexible appendages. The Euler–Bernoulli beam theory is used to model the flexible parts. The attitude dynamic equations of motion are derived using the law of conservation of angular momentum, and the flexural equations are derived. The attitude of the spacecraft is represented using the quaternion parameters. The controller is designed based on the super-twisting sliding mode contr

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6

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

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Modern spacecraft usually have larger and more flexible appendages whose vibration becomes more and more prominent, and it has a great influence on the precision of spacecraft attitude. Therefore, the cooperative control of attitude maneuvering and structural vibration of the system has become a significant issue in the spacecraft design process. We developed a low-dimensional and high-precision mathematical model for a large-scale flexible spacecraft (LSFS) equipped with a pair of hinged solar arrays in this paper. The analytic global modes are used to obtain the rigid–flexible coupling discr

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7

Huang, Wenke, Linfeng Li, Wenye Dong, Liwen He, Taoming Feng, and Jun Xiao. "Finite element dynamic modeling and attitude control for a slender flexible spacecraft." Journal of Physics: Conference Series 2472, no. 1 (2023): 012034. http://dx.doi.org/10.1088/1742-6596/2472/1/012034.

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Abstract This paper aims to study the dynamic modeling and attitude control of a kind of slender flexible spacecraft. The basic structure of this kind of spacecraft is different from the traditional structure of a center rigid body with flexible appendages, but rigid bodies at both ends, which are connected by a flexible truss structure. Firstly, the finite element modeling method is adopted to establish the dynamic model of this kind of spacecraft, and the vibration frequency and damping of the flexible structure are obtained. The unconstrained modes of the flexible spacecraft are obtained th

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8

Bennett, W. H., C. LaVigna, H. G. Kwatny, and G. Blankenship. "Nonlinear and Adaptive Control of Flexible Space Structures." Journal of Dynamic Systems, Measurement, and Control 115, no. 1 (1993): 86–94. http://dx.doi.org/10.1115/1.2897412.

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This paper addresses design of nonlinear control systems for rapid, large angle multiaxis, slewing and LOS pointing of realistic flexible space structures. The application of methods based on adaptive feedback linearization for nonlinear control design for flexible space structures is presented. A comprehensive approach to modeling the nonlinear dynamics and attitude control of multibody systems with structural flexure is considered. Adaptive feedback linearizing control laws are described based on Lagrangian dynamical system model for the spacecraft. Simulation results for attitude slewing an

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9

Joshi, S. M., and P. G. Maghami. "Robust dissipative compensators for flexible spacecraft control." IEEE Transactions on Aerospace and Electronic Systems 28, no. 3 (1992): 768–74. http://dx.doi.org/10.1109/7.256297.

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10

Ye, Dong, and Zhaowei Sun. "Variable structure tracking control for flexible spacecraft." Aircraft Engineering and Aerospace Technology 88, no. 4 (2016): 508–14. http://dx.doi.org/10.1108/aeat-04-2014-0038.

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11

Ikeda, Yuichi, Takashi Kida, and Tomoyuki Nagashio. "Passivity based Control of Nonlinear Flexible Spacecraft." IFAC Proceedings Volumes 37, no. 11 (2004): 149–54. http://dx.doi.org/10.1016/s1474-6670(17)31604-x.

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12

Janschek, K., and M. Surauer. "Decentralized/Hierarchical Control for Large Flexible Spacecraft." IFAC Proceedings Volumes 20, no. 5 (1987): 49–56. http://dx.doi.org/10.1016/s1474-6670(17)55183-6.

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13

Lee, Keum W., and Sahjendra N. Singh. "adaptive control of flexible spacecraft despite disturbances." Acta Astronautica 80 (November 2012): 24–35. http://dx.doi.org/10.1016/j.actaastro.2012.05.007.

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14

Singh, Sahjendra. "Robust Nonlinear Attitude Control of Flexible Spacecraft." IEEE Transactions on Aerospace and Electronic Systems AES-23, no. 3 (1987): 380–87. http://dx.doi.org/10.1109/taes.1987.310836.

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15

Hu, Yabo, Baolin Wu, Yunhai Geng, and Yunhua Wu. "Smooth time-optimal attitude control of spacecraft." Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 233, no. 7 (2018): 2331–43. http://dx.doi.org/10.1177/0954410018776531.

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In this paper, a trajectory optimization method for generating smooth and approximate time-optimal attitude maneuver trajectories of flexible spacecraft is proposed. Smooth attitude maneuver is highly desirable for flexible spacecraft, since vibration of flexible appendices can be suppressed. In order to obtain smooth and approximate time-optimal attitude trajectory, a novel objective function composed of two terms is developed in the problem of trajectory optimization: the first term is proportional to the total maneuver time and the other one is proportional to the integral of the squared co

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16

Pang, Aiping, Hui Zhu, Junjie Zhou, Zhen He, and Jing Yang. "Robust H∞ Control for the Spacecraft with Flexible Appendages." Complexity 2020 (December 12, 2020): 1–8. http://dx.doi.org/10.1155/2020/6652300.

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Aiming at the oscillation suppression of spacecraft with large flexible appendages, we propose a control strategy using H∞ control. The weighting functions are designed for the specific flexible modes of the spacecraft and the frequency of harmonic interference in its operating environment. Taking into account the structural uncertainty of systematic modeling and the comprehensive performance requirements of system bandwidth constraint and attitude stability, the H∞ comprehensive performance matrix is constructed. A space telescope with a large flexible solar array is presented as an illustrat

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17

Long, Haihui, and Jiankang Zhao. "Robust constrained fault-tolerant attitude control for flexible spacecraft." Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 232, no. 16 (2017): 3011–23. http://dx.doi.org/10.1177/0954410017733291.

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In this paper, robust constrained fault-tolerant attitude controllers are proposed for flexible spacecraft subjected to external disturbance, model uncertainty, input saturation, and actuator faults. Three types of actuator faults of spacecraft, i.e. partial loss of effectiveness, stuck fault, and outage fault, are modeled explicitly. To handle these actuator faults, a significant lemma is proposed and rigorous proof is conducted at length. By introducing two e-modification parameter update laws to online estimate the unknown parameters caused by actuator faults, constrained fault-tolerant att

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18

Zhang, Gaowang, Xueqin Chen, Ruichen Xi, and Huayi Li. "Nonsingular Integral Sliding Mode Attitude Control for Rigid-Flexible Coupled Spacecraft with High-Inertia Rotating Appendages." Complexity 2021 (February 15, 2021): 1–17. http://dx.doi.org/10.1155/2021/8812187.

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This study addresses the challenge of attitude tracking control for a rigid-flexible spacecraft with high-inertia rotating appendages. The Lagrange method was used to establish the kinematic and dynamic models of the spacecraft. The translation and rotation of the spacecraft, vibrations of solar panels, and imbalance caused by the rotating appendages, which cause a complex control problem, were considered. To address the complex control problem, a novel, fast nonsingular integral sliding mode control method is proposed to perform the attitude tracking function of spacecraft. A sliding mode con

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19

Zhang, Gaowang, Feng Wang, Jian Chen, and Huayi Li. "Fixed-time sliding mode attitude control of a flexible spacecraft with rotating appendages connected by magnetic bearing." Mathematical Biosciences and Engineering 19, no. 3 (2022): 2286–309. http://dx.doi.org/10.3934/mbe.2022106.

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<abstract> <p>This study focuses on the attitude control of a flexible spacecraft comprising rotating appendages, magnetic bearings, and a satellite platform capable of carrying flexible solar panels. The kinematic and dynamic models of the spacecraft were established using Lagrange methods to describe the translation and rotation of the spacecraft system and its connected components. A simplified model of the dynamics of a five-degrees-of-freedom (DOF) active magnetic bearing was developed using the equivalent stiffness and damping methods based on the magnetic gap variations in t

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20

Zhang, Wei, Weibing Zhu, Shijie Zhang, and Xiangtian Zhao. "Adaptive Fuzzy Control for Attitude Stabilization of Spacecraft with Deployable Composite Laminated Solar Array." Complexity 2020 (August 4, 2020): 1–26. http://dx.doi.org/10.1155/2020/3098684.

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Modern spacecraft are often equipped with large-scale, complex, and lightweight solar arrays whose deployment involves a highly dynamic movement. This paper proposed a novel adaptive proportional-derivative typed fuzzy logic control scheme for the attitude stabilization of a flexible spacecraft during the deployment of a composite laminated solar array. First, a constrained rigid-flexible coupling spacecraft model consisting of a rigid main body and a flexible solar array was proposed. The solar array, which is composed of composite laminated shells, was described by the absolute nodal coordin

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21

George, V. I., B. Ganesh Kamath, I. Thirunavukkarasu, and Ciji Pearl Kurian. "Vibration Control of Flexible Spacecraft Using Adaptive Controller." International Journal on Advanced Science, Engineering and Information Technology 2, no. 1 (2012): 34. http://dx.doi.org/10.18517/ijaseit.2.1.149.

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22

Zhang, Yuntian, Aiping Pang, Hui Zhu, and Huan Feng. "Structured H∞ Control for Spacecraft with Flexible Appendages." Entropy 23, no. 8 (2021): 930. http://dx.doi.org/10.3390/e23080930.

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Spacecraft with large flexible appendages are characterized by multiple system modes. They suffer from inherent low-frequency disturbances in the operating environment that consequently result in considerable interference in the operational performance of the system. It is required that the control design ensures the system’s high pointing precision, and it is also necessary to suppress low-frequency resonant interference as well as take into account multiple performance criteria such as attitude stability and bandwidth constraints. Aiming at the comprehensive control problem of this kind of f

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23

IKEDA, Yuichi, Takashi KIDA, and Tomoyuki NAGASHIO. "Attitude Control of Flexible Spacecraft by Quaternion Feedback." Transactions of the Society of Instrument and Control Engineers 40, no. 2 (2004): 239–46. http://dx.doi.org/10.9746/sicetr1965.40.239.

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24

Zee, Robert E., and Peter C. Hughes. "Mode Localization in Flexible Spacecraft: A Control Challenge." Journal of Guidance, Control, and Dynamics 23, no. 1 (2000): 69–76. http://dx.doi.org/10.2514/2.4488.

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25

Rutkovskii, V. Yu, V. M. Sukhanov, and V. M. Glumov. "Combined relay-adaptive control of flexible spacecraft orientation." Automation and Remote Control 73, no. 12 (2012): 2049–58. http://dx.doi.org/10.1134/s0005117912120090.

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26

Grewal, A., and V. J. Modi. "Multibody dynamics and robust control of flexible spacecraft." IEEE Transactions on Aerospace and Electronic Systems 36, no. 2 (2000): 491–500. http://dx.doi.org/10.1109/7.845230.

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27

Zhong, Chenxing, Zhiyong Chen, and Yu Guo. "Attitude Control for Flexible Spacecraft With Disturbance Rejection." IEEE Transactions on Aerospace and Electronic Systems 53, no. 1 (2017): 101–10. http://dx.doi.org/10.1109/taes.2017.2649259.

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28

Anthony, Tobin C., Bong Wie, and Stanley Carroll. "Pulse-modulated control synthesis for a flexible spacecraft." Journal of Guidance, Control, and Dynamics 13, no. 6 (1990): 1014–22. http://dx.doi.org/10.2514/3.20574.

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29

Meirovitch, Leonard, and Moon Kyu Kwak. "Control of flexible spacecraft with time-varying configuration." Journal of Guidance, Control, and Dynamics 15, no. 2 (1992): 314–24. http://dx.doi.org/10.2514/3.20839.

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30

Liu, Qiang, and Bong Wie. "Robust time-optimal control of uncertain flexible spacecraft." Journal of Guidance, Control, and Dynamics 15, no. 3 (1992): 597–604. http://dx.doi.org/10.2514/3.20880.

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31

Song, Gangbing, and Brij N. Agrawal. "Vibration suppression of flexible spacecraft during attitude control." Acta Astronautica 49, no. 2 (2001): 73–83. http://dx.doi.org/10.1016/s0094-5765(00)00163-6.

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32

Sakawa, Y., and K. Sato. "Modeling and Control of a Flexible Orbiting Spacecraft." IFAC Proceedings Volumes 22, no. 4 (1989): 309–14. http://dx.doi.org/10.1016/s1474-6670(17)53562-4.

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33

Wang, Zhaohui, Ming Xu, Yinghong Jia, Shijie Xu, and Liang Tang. "Vibration suppression-based attitude control for flexible spacecraft." Aerospace Science and Technology 70 (November 2017): 487–96. http://dx.doi.org/10.1016/j.ast.2017.08.014.

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34

Zhang, Yizhe, and Xin Guan. "Active damping control of flexible appendages for spacecraft." Aerospace Science and Technology 75 (April 2018): 237–44. http://dx.doi.org/10.1016/j.ast.2017.12.027.

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35

Shahravi, Morteza, Mansour Kabganian, and Aria Alasty. "Adaptive robust attitude control of a flexible spacecraft." International Journal of Robust and Nonlinear Control 16, no. 6 (2006): 287–302. http://dx.doi.org/10.1002/rnc.1051.

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36

Nayeri, M. Reza Dehghan, Aria Alasty, and Kamran Daneshjou. "Neural optimal control of flexible spacecraft slew maneuver." Acta Astronautica 55, no. 10 (2004): 817–27. http://dx.doi.org/10.1016/j.actaastro.2004.04.002.

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37

Maganti, Ganesh B., and Sahjendra N. Singh. "Simplified adaptive control of an orbiting flexible spacecraft." Acta Astronautica 61, no. 7-8 (2007): 575–89. http://dx.doi.org/10.1016/j.actaastro.2007.02.004.

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38

Gu, Chengsi, and Weidong Qu. "Nonsingular Terminal Sliding Mode Control for Flexible Spacecraft Attitude Control." Journal of Physics: Conference Series 1828, no. 1 (2021): 012176. http://dx.doi.org/10.1088/1742-6596/1828/1/012176.

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39

Huang, Liya, and Zhong Wu. "Extended harmonic disturbance observer-based attitude control for flexible spacecraft with control moment gyroscopes." Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 233, no. 14 (2019): 5331–46. http://dx.doi.org/10.1177/0954410019842503.

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In the flexible spacecraft with control moment gyroscopes, there are multiple disturbances including not only internal disturbances from actuators and flexible appendages, but also external disturbances from space environment. These disturbances are characterized by a wide frequency range and may degrade attitude control performance to a great extent. In this paper, the lumped disturbance is modeled as a harmonic plus a polynomial model, and an extended harmonic disturbance observer (EHDO) is proposed to estimate the total disturbance. Since the rotor dynamic imbalance disturbance from control

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40

MoradiMaryamnegari, H., and A. M. Khoshnood. "Robust adaptive vibration control of an underactuated flexible spacecraft." Journal of Vibration and Control 25, no. 4 (2018): 834–50. http://dx.doi.org/10.1177/1077546318802431.

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Designing a controller for multi-body systems including flexible and rigid bodies has always been one of the major engineering challenges. Equations of motion of these systems comprise extremely nonlinear and coupled terms. Vibrations of flexible bodies affect sensors of rigid bodies and might make the system unstable. Introducing a new control strategy for designing control systems which do not require the rigid–flexible coupling model and can dwindle vibrations without sensors or actuators on flexible bodies is the purpose of this paper. In this study, a spacecraft comprising a rigid body an

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41

Qu, Fa Yi, Liang Kuan Zhu, and Wen Long Song. "Fuzzy Adaptive Variable Structure Active Attitude Control of Flexible Spacecraft." Applied Mechanics and Materials 44-47 (December 2010): 2070–74. http://dx.doi.org/10.4028/www.scientific.net/amm.44-47.2070.

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This paper presents a novel control system design method for the three-axis-rotational tracking and vibration stabilization of a spacecraft with flexible appendages. Based on the sliding control theory, a robust attitude control law is derived to control the attitude motion of spacecraft. For actively suppressing the induced vibration, both fuzzy methods and strain rate feedback (SRF) control methods are presented. Numerical simulations are performed to show the feasibility and effeteness of the proposed methods.

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42

Shahravi, Morteza, and Milad Azimi. "A Hybrid Scheme of Synthesized Sliding Mode/Strain Rate Feedback Control Design for Flexible Spacecraft Attitude Maneuver Using Time Scale Decomposition." International Journal of Structural Stability and Dynamics 16, no. 02 (2016): 1450101. http://dx.doi.org/10.1142/s0219455414501016.

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Presented herein is a new control approach for large angle attitude maneuver of flexible spacecraft. The singular perturbation theory (SPT) provides a useful tool for two time rate scale separation (mapping) of rigid and flexible body dynamics. The resulting slow and fast subsystems, enabling the use of two control approach for attitude (Modified Sliding Mode) and vibration Strain Rate Feedback (SRF) control of flexible spacecraft, respectively. An attractive feature of the present control approach is that the global stability of the entire system has been guaranteed while the controllers acco

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43

Zhou, Chengbao, and Di Zhou. "Robust dynamic surface sliding mode control for attitude tracking of flexible spacecraft with an extended state observer." Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 231, no. 3 (2016): 533–47. http://dx.doi.org/10.1177/0954410016640822.

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The nonlinear attitude motion equations of flexible spacecraft described by the Euler angles are expressed in the vector form. Based on dynamic surface control, a new robust dynamic surface sliding mode controller is proposed for the attitude tracking and active vibration suppression of flexible spacecraft in the presence of parameter uncertainty and external disturbances. Then, a novel robust dynamic surface finite time sliding mode controller is proposed with an extended state observer such that the uncertainties can be estimated. Lyapunov stability analyses show that the two controllers can

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44

Li, Xiong Fei, Wei Cheng, and Ming Li. "Testing and Analysis of Micro-Vibrations Generated by Control Moment Gyroscope in Different Installation Boundary." Applied Mechanics and Materials 851 (August 2016): 453–58. http://dx.doi.org/10.4028/www.scientific.net/amm.851.453.

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The micro-vibrations caused by CMG (Control Moment Gyroscope) can seriously degrade the pointing accuracy and imaging quality of spacecraft. To test the micro-vibrations generated by CMG and obtain the corresponding frequency spectrum characteristics is the key to design the vibration restraining and analyze the vibration transfer properties. In order to simulate the actual installation boundary of CMG on spacecraft, a flexible interface is designed to compare with the rigid interface in traditional testing methods. The results show that: due to the existence of coupling effect between the CMG

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45

Lu, Kunfeng, Tianya Li, and Lijun Zhang. "Active attitude fault-tolerant tracking control of flexible spacecraft via the Chebyshev neural network." Transactions of the Institute of Measurement and Control 41, no. 4 (2018): 925–33. http://dx.doi.org/10.1177/0142331218803410.

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This paper describes a novel finite-time attitude tracking control approach for flexible spacecraft. This is achieved by integrating sliding-mode control and the active real-time fault-tolerant reconfiguration method. In this approach, the attitude error dynamics and the kinematics of the flexible spacecraft are first established. Then, a nonsingular terminal sliding-mode surface is designed, based on finite-time control theory. Applying the Chebyshev neural network, the uncertain dynamics induced by external disturbances and uncertain inertia parameters are approximated and estimated. The nom

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46

Wang, Yan. "A Real-time Solar Array Monitoring System Architecture Design Based on FBG." Journal of Physics: Conference Series 2428, no. 1 (2023): 012030. http://dx.doi.org/10.1088/1742-6596/2428/1/012030.

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Abstract The solar array is a key part of spacecraft, which is the main power supply. With the trend of spacecraft bus development, the size of the solar array panel attached to the spacecraft becomes larger and larger to provide more load power. Large scale, low stiffness, and flexibility affect the attitude control of the spacecraft directly. Monitoring and analyzing the solar array dynamics behavior will be the key part of the overall design of the spacecraft’s control system. The purpose of monitoring is to verify the dynamic modeling and health diagnosis for a typical spacecraft flexible

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47

de Souza, Alain G., and Luiz C. G. de Souza. "Satellite Attitude Control System Design Taking into Account the Fuel Slosh and Flexible Dynamics." Mathematical Problems in Engineering 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/820586.

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The design of the spacecraft Attitude Control System (ACS) becomes more complex when the spacecraft has different type of components like, flexible solar panels, antennas, mechanical manipulators and tanks with fuel. The interaction between the fuel slosh motion, the panel’s flexible motion and the satellite rigid motion during translational and/or rotational manoeuvre can change the spacecraft center of mass position damaging the ACS pointing accuracy. This type of problem can be considered as a Fluid-Structure Interaction (FSI) where some movable or deformable structure interacts with an int

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48

Di Gennaro, S. "Output attitude tracking for flexible spacecraft." Automatica 38, no. 10 (2002): 1719–26. http://dx.doi.org/10.1016/s0005-1098(02)00082-1.

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49

Wei, Jin, Wei Liu, Jia Liu, and Tao Yu. "Dynamic Modeling and Analysis of Spacecraft with Multiple Large Flexible Structures." Actuators 12, no. 7 (2023): 286. http://dx.doi.org/10.3390/act12070286.

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An analytical dynamic model is presented for a spacecraft with multiple large flexible structures. Based on the partial differential equations (PDEs) of the motion of the solar panel and deployable arm, the governing equations of the main-body and deployable antenna and the boundary conditions at each end point are used to obtain the frequency and mode shapes of the system. Then, the ordinary differential equations (ODEs) of the system can be obtained from the orthogonality relations and mode shape. The influence of the deployable antenna on the frequencies and mode shapes of the spacecraft is

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50

Bang, Hyochoong, and Choong-Seok Oh. "Attitude maneuver control of flexible spacecraft by observer-based tracking control." KSME International Journal 18, no. 1 (2004): 122–31. http://dx.doi.org/10.1007/bf03028797.

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