  Mechanics of Solids A Journal of Russian Academy of Sciences   Founded
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O.V. Zelepukina and Yu.N. Chelnokov, "Construction of Optimal Laws of Variation in the Angular Momentum Vector of a Dynamically Symmetric Rigid Body," Mech. Solids. 46 (4), 519533 (2011) 
Year 
2011 
Volume 
46 
Number 
4 
Pages 
519533 
DOI 
10.3103/S0025654411040030 
Title 
Construction of Optimal Laws of Variation in the Angular Momentum Vector of a Dynamically Symmetric Rigid Body 
Author(s) 
O.V. Zelepukina (Institute for Precision Mechanics and Control Problems, Rabochaya 24, Saratov, 410028 Russia)
Yu.N. Chelnokov (Institute for Precision Mechanics and Control Problems, Rabochaya 24, Saratov, 410028 Russia, ChelnokovYuN@info.sgu.ru) 
Abstract 
We consider the problem of construction of optimal laws of variation in the angular momentum vector of a dynamically symmetric rigid body so as to ensure the transition of the rigid body from an arbitrary initial angular position to the required final angular position. For the functionals to be minimized, we use combined performance functionals, one of which characterizes the expenditure of time and of the squared modulus of the angular momentum vector in a given proportion, while the other characterizes the expenditure of time and momentum of the modulus of the angular momentum vector necessary to change the rigid body orientation. The control (the vector of the rigid body angular momentum) is assumed to be bounded in the modulus. The problem is solved by using Pontryagin's maximum principle and the quaternion differential equation [1, 2] relating the vector of the dynamically symmetric rigid body angular momentum to the quaternion of orientation of the coordinate system rotating with respect to the rigid body about its dynamical symmetry axis at an angular velocity proportional to the angular momentum vector projection on the axis. The use of such a model of rotational motion leads to the problem of optimal control with the moving right end of the trajectory and significantly simplifies the analytic study of the problem of construction of optimal laws of variation in the angular momentum vector, because this model explicitly exploits the body angular momentum quaternion (control) instead of the rigid body absolute angular velocity quaternion.
We construct general analytic solutions of the differential equations for the boundaryvalue problems which form systems of nine nonlinear differential equations. It is shown that the process of solving the differential boundaryvalue problems is reduced to solving two scalar algebraic transcendental equations. We obtain, as explicit functions of time, dependencies for the orientation quaternion, the absolute angular velocity vector, and the vector of the rigid body angular momentum, which describe the program optimal controlled motion of the rigid body. We construct the corresponding laws of variation in the program controlling moments for the rigid body (spacecraft) and geometrically interpret the controlled angular motion of the rigid body.
This paper generalizes and develops the results obtained in [3]. 
Keywords 
rigidbody, angular momentum, orientation, rotation, optimal control, minimization functional, orientation quaternion 
References 
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in Mathematics, Mechanics, Collection of Scientific Papers, No. 6
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No. 6, 313 (1995)
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30 (6), 110 (1995)]. 
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in Problems of Mechanics and Control, Interuniversity Collection of Scientific Papers
(Izdvo Perm Univ., Perm, 1995),
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[in Russian]. 
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[in Russian]. 
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(NITS "Regular and Chaotic Mechanics," MoscowIzhevsk, 2004)
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12.  Yu. N. Chelnokov,
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(Fizmatlit, Moscow, 2006)
[in Russian]. 
13.  M. V. Levskii,
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45 (3), 250263 (2007)
[Cosmic Res. (Engl. Transl.)
45 (3), 234247 (2007)]. 
14.  V. G. Biryukov, A. V. Molodenkov, and Yu. N. Chelnokov,
"Optimal Control of the Spacecraft Orientation by
Using the Angular Momentum Vector as the Control,"
in Mathematics, Mechanics, Collection of Scientific Papers, No. 6
(Izdvo Saratov Univ., Saratov, 2004),
pp. 171173 [in Russian]. 

Received 
10 March 2009 
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