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IssuesArchive of Issues2011-4pp.519-533

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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), 519-533 (2011)
Year 2011 Volume 46 Number 4 Pages 519-533
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 boundary-value problems which form systems of nine nonlinear differential equations. It is shown that the process of solving the differential boundary-value 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 rigid-body, angular momentum, orientation, rotation, optimal control, minimization functional, orientation quaternion
References
1.  Yu. N. Chelnokov, "On the Motion of a Heavy Symmetric Rigid Body with Movable Suspension Point," Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, No. 4, 3-10 (1990) [Mech. Solids (Engl. Transl.)].
2.  Yu. N. Chelnokov and V. A. Yurko, "Quaternions and Related Transformations in Dynamics of a Symmetric Rigid Body. I," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 6, 3-16 (1997) [Mech. Solids (Engl. Transl.) 32 (6), 1-11 (1997)].
3.  O. V. Zelepukina and Yu. N. Chelnokov, "Construction of Laws of Optimal Variation in the Angular Momentum Vector of a Dynamically Symmetric Spacecraft," in Mathematics, Mechanics, Collection of Scientific Papers, No. 6 (Izd-vo Saratov Univ., Saratov, 2004), pp. 189-192 [in Russian].
4.  V. N. Branets and I. P. Shmyglevskii, "Application of Quaternions in Rigid Body Control by the Angular Position," Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, No. 4, 24-31 (1972) [Mech. Solids (Engl. Transl.)].
5.  V. N. Branets and I. P. Shmyglevskii, "Kinematical Problems of Orientation in a Rotating Coordinate Systems," Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, No.6, 36-43 (1972) [Mech. Solids (Engl. Transl.)].
6.  V. N. Branets and I. P. Shmyglevskii, Application of Quaternions in Problems of Orientation of a Rigid Body (Nauka, Moscow, 1973) [in Russian].
7.  P. K. Plotnikov, A. N. Sergeev, and Yu. N. Chelnokov, "Kinematic Problem of Control of a Rigid Body Orientation," Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, No. 5, 9-18 (1991) [Mech. Solids (Engl. Transl.)].
8.  A. A. Pankov and Yu. N. Chelnokov, "Quaternion Laws of Kinematic Attitude Control of a Rigid Body by the Angular Velocity," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 6, 3-13 (1995) [Mech. Solids (Engl. Transl.) 30 (6), 1-10 (1995)].
9.  A. V. Molodenkov, "Quaternion Solution of the Problem of a Rigid Body Optimal Rotation in the Sense of the Energy Expenditure Minimum," in Problems of Mechanics and Control, Interuniversity Collection of Scientific Papers (Izd-vo Perm Univ., Perm, 1995), pp. 122-131 [in Russian].
10.  V. G. Biryukov and Yu. N. Chelnokov, "Kinematic Problem of Optimal Nonlinear Stabilization of the Angular Motion of a Rigid Body," in Mathematics, Mechanics, Collection of Scientific Papers, No. 4 (Izd-vo Saratov Univ., Saratov, 2002), pp. 172-174 [in Russian].
11.  V. V. Malanin and N. A. Strelkova, Optimal Control of Orientation and Helical Motion of a Rigid Body (NITS "Regular and Chaotic Mechanics," Moscow-Izhevsk, 2004) [in Russian].
12.  Yu. N. Chelnokov, Quaternion and Biquaternion Models and Methods of Mechanics of Solids and Their Applications: Geometry and Kinematics of Motion (Fizmatlit, Moscow, 2006) [in Russian].
13.  M. V. Levskii, "Control of a Spacecraft's Spatial Turn with Minimum Value of the Path Functional," Kosmich. Issled. 45 (3), 250-263 (2007) [Cosmic Res. (Engl. Transl.) 45 (3), 234-247 (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 (Izd-vo Saratov Univ., Saratov, 2004), pp. 171-173 [in Russian].
Received 10 March 2009
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