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Stabilization of a large flexible spacecraft using robust adaptive sliding hypersurface and finite element approach

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Abstract

A novel adaptive sliding hypersurface with an adjustable dead zone scheme for attitude control of a spacecraft with large flexible sun oriented solar panels is employed to overcome the difficulties arising from the measurement of flexible dynamics coordinates. Equations of motion for multi-axis attitude maneuver of a large rigid-flexible system are derived utilizing Lagrangian formulation based on a hybrid system of coordinates. The flexible appendages are modeled as elastic plates to examine flexural deformations employing the finite element method. A proposed synthesized hybrid-sliding surface in conjunction with an adaptation law can perform desired mission properly, in which the excitation of high-frequency flexible modes, external disturbances, and system parameter uncertainties are taken into account. It is noteworthy that the design of the proposed controller does not require the exact model of the system’s dynamic and can be performed for every general nonlinear time-varying system in the real environment. Within the Lyapunov analysis and Barbalat’s lemma, the stability of the global closed-loop system is guaranteed. The comparative results expressed as flexible structures deflection shows significant improvements in terms of robustness of the highly flexible system to the vibration energy excitation.

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Correspondence to Milad Azimi.

Appendix A

Appendix A

To implement the FEM discretization technique, a series of assumptions needed in this study; (i) the appendages considered as bending plates and divided into n-rectangular elements, (ii) the solar panels consist of an isotropic material with uniform mass density and (iii) only out-of-plane deformations of the panels are considered. According to the aforementioned assumptions, as shown in Fig. 12, each element has 12 DOF.

Fig. 12
figure 12

Rectangular plate bending element

Defining:

$$ {\mathbf{R}}_{j} = \frac{{Et^{3} }}{{12\,(1 - \nu^{2} )}}\,\,\,\left[ {\begin{array}{*{20}c} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & {\frac{1 - \nu }{2}} \\ \end{array} } \right] $$
(52)

as an isotropic plate elasticity matrix, where \( \nu \), E, and t are the jth element Poisson’s ratio, Young’s modulus, and thickness, respectively. By choosing the following shape functions defined by the Bogner et al. [36], the continuity of all the nodal slopes and deflections can be guaranteed for the jth element:

$$ {\mathbf{C}}_{j}^{T} = \left( {\frac{1}{8}} \right)\left[ \begin{aligned} (1 - \xi )(1 - \eta )\,(2 - \xi - \eta - \xi^{2} - \eta^{2} ) \hfill \\ (1 - \xi )(\eta^{2} - 1)( - 1 + \eta )b \hfill \\ - (\xi^{2} - 1)\,(\xi - 1)(1 - \eta )\,a \hfill \\ (1 + \xi )(1 - \eta )\,(2 + \xi - \eta - \xi^{2} - \eta^{2} ) \hfill \\ (1 + \xi )(\eta^{2} - 1)\,(\eta - 1)b \hfill \\ - (1 + \xi )\,(1 - \eta )\,(\xi^{2} - 1)\,a \hfill \\ (1 + \xi )\,(1 + \eta )\,(2 + \xi + \eta - \xi^{2} - \eta^{2} ) \hfill \\ (1 + \xi )\,(1 + \eta )\,(\eta^{2} - 1)\,b \hfill \\ - (1 + \xi )\,(\xi^{2} - 1)\,(1 + \eta )a \hfill \\ (1 - \xi )\,(2 + \xi + \eta - \xi^{2} - \eta^{2} )(1 + \eta ) \hfill \\ (1 - \xi )\,(\eta^{2} - 1)\,(1 + \eta )b \hfill \\ - ( - 1 + \xi )(1 + \eta )(\xi^{2} - 1)a \hfill \\ \end{aligned} \right] $$
(53)

For this shape function matrix, the coupling matrix \( A_{j} \) can be obtained as follows:

$$ {\varvec{\Lambda}}_{j}^{T} = \frac{\rho tab}{24}\left[ {\begin{array}{*{20}l} {6y_{0j} + \frac{9}{5}b} \hfill & { - 6x_{oj} - \frac{9}{5}a} \hfill & 0 \hfill \\ {by_{0j} + \frac{2}{5}b^{2} } \hfill & { - bx_{oj} - \frac{3}{10}ab} \hfill & 0 \hfill \\ { - ay_{0j} + \frac{3}{10}ab} \hfill & { - ax_{oj} + \frac{2}{5}a^{2} } \hfill & 0 \hfill \\ {6y_{0j} + \frac{21}{5}b} \hfill & { - 6x_{oj} - \frac{9}{5}a} \hfill & 0 \hfill \\ { - by_{0j} + \frac{3}{5}b^{2} } \hfill & {bx_{oj} + \frac{3}{10}ab} \hfill & 0 \hfill \\ { - ay_{0j} + \frac{7}{10}ab} \hfill & {ax_{oj} - \frac{2}{5}a^{2} } \hfill & 0 \hfill \\ {6y_{0j} + \frac{21}{5}b} \hfill & { - 6x_{oj} - \frac{21}{5}a} \hfill & 0 \hfill \\ { - by_{0j} - \frac{3}{5}b^{2} } \hfill & {bx_{oj} + \frac{7}{10}ab} \hfill & 0 \hfill \\ {ay_{0j} + \frac{7}{10}ab} \hfill & { - ax_{oj} - \frac{3}{5}a^{2} } \hfill & 0 \hfill \\ {6y_{0j} + \frac{9}{5}b} \hfill & { - 6x_{oj} - \frac{21}{5}a} \hfill & 0 \hfill \\ {by_{0j} + \frac{2}{5}b^{2} } \hfill & { - bx_{oj} - \frac{7}{10}ab} \hfill & 0 \hfill \\ {ay_{0j} + \frac{3}{10}ab} \hfill & { - ax_{oj} - \frac{3}{5}a^{2} } \hfill & 0 \hfill \\ \end{array} } \right] $$
(54)

where a and b are the element length and width respectively, \( \xi = x/a \), \( \eta = y/b \), \( x_{oj} \) and \( y_{oj} \) are components of the vector from \( O_{b} \) to \( O_{j} \) in the \( X_{j} \)-axis and \( Y_{j} \)-axis directions, respectively. The inertia matrix for the \( j \)th element can be written as follows:

$$ {\mathbf{I}}_{j} = \rho abt\left[ {\begin{array}{*{20}c} {I_{j11} \,} & {I_{j12} \,} & {I_{j13} } \\ {} & {I_{j22} \,} & {I_{j23} } \\ {\,sym} & {} & {I_{j33} \,} \\ \end{array} } \right] $$
(55)

where

$$ \begin{aligned} I_{j\,11} & = y_{0j}^{2} + \frac{1}{12}t^{2} + \frac{1}{3}b^{2} + y_{oj} b\,\, \\ I_{j12} & = \frac{1}{12}\left( {\frac{1}{2}ab + y_{0j} a + x_{oj} b} \right) - x_{0j} y_{0j} \,\, \\ I_{j13} & = I_{j\,23} = 0\,\, \\ I_{j\,22} & = x_{0j}^{2} + \frac{1}{12}t^{2} + \frac{1}{3}a^{2} + x_{oj} a \\ I_{j\,33} & = x_{0j}^{2} + y_{0j}^{2} + \frac{1}{3}(a^{2} + b^{2} ) + x_{0j} a + y_{0j} b\, \\ \end{aligned} $$
(56)

The main body rotational displacement and the jth element displacement coupling matrix is defined as:

$$ {\mathbf{W}}_{j} = \frac{\rho tab}{24}\left[ {\begin{array}{*{20}l} 0 \hfill & {\,\,0} \hfill & {\,\,\,\,\,\,0} \hfill & 0 \hfill & {\,\,\,\,\,\,\,0} \hfill & {\,\,\,\,\,\,0} \hfill & 0 \hfill & {\,\,\,\,\,\,0} \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {\,\,0} \hfill & {\,\,\,\,\,\,0} \hfill & 0 \hfill & {\,\,\,\,\,\,\,0} \hfill & {\,\,\,\,\,\,0} \hfill & 0 \hfill & {\,\,\,\,\,\,0} \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 6 \hfill & {\,b} \hfill & { - a} \hfill & 6 \hfill & { - b} \hfill & { - a} \hfill & 6 \hfill & { - b} \hfill & a \hfill & 6 \hfill & b \hfill & a \hfill \\ \end{array} } \right] $$
(57)

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Azimi, M., Shahravi, M. Stabilization of a large flexible spacecraft using robust adaptive sliding hypersurface and finite element approach. Int. J. Dynam. Control 8, 644–655 (2020). https://doi.org/10.1007/s40435-019-00582-1

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