Synthesis of control of the angular motion of a spacecraft on the basis of generalization of the direct Lyapunov method
Heading:
^{1}Volosov, VV, ^{1}Shevchenko, VM ^{1}Space Research Institute of the National Academy of Sciences of Ukraine and the State Space Agency of Ukraine, Kyiv, Ukraine 
Space Sci.&Technol. 2018, 24 ;(4):0313 
https://doi.org/10.15407/knit2018.04.003 
Publication Language: Russian 
Abstract: The work deals with solving the problems of attitude control synthesis for an orbital spacecraft based on the known ZubovKrasovskii generalizations of the direct Lyapunov method for studying the stability of individual solutions of differential equations to analyze the stability of closed bounded sets in their phase space. The kinematic equations of the angular motion of a spacecraft are represented by differential equations in the vectormatrix form with a state vector in the real Euclidean space. The elements of this space are the vectors of RodriguesHamilton parameters. The given equations are characterized by the following generally known peculiarities. One of them lies in the fact that the same specified physical orientation of a spacecraft corresponds to two state vectors whose components differ only in sign. The second peculiarity of the differential equations is the presence of the geometrical motion integral – conservation of the norm of the vector of orientation parameters. It is noted that, as a consequence, solutions of the equations cannot be asymptotically stable by Lyapunov. They can be characterized only by conditional stability.
With regard for these peculiarities, the corresponding problem of control synthesis for spacecraft orientation is for the first time formulated as a problem of the conditional asymptotic stability of the solutions of the differential equations or stability on manifolds. More precisely, it is a problem of stability of twopoint sets in the phase space. A new nonsmooth piecewisequadratic Lyapunov function is proposed to solve the problem. The obtained solutions of the problems of control synthesis provide the achievement and stabilization of the given orientations of a spacecraft in the inertial and orbital coordinate systems. The effectiveness of the proposed algorithms is illustrated by computer simulation.

Keywords: control synthesis, Lyapunov function, orientation, spacecraft 
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