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V.G. Biryukov and Yu.N. Chelnokov, "Kinematic Problem of Optimal Nonlinear Stabilization of Angular Motion of a Rigid Body," Mech. Solids. 52 (2), 119-127 (2017)
Year 2017 Volume 52 Number 2 Pages 119-127
DOI 10.3103/S0025654417020017
Title Kinematic Problem of Optimal Nonlinear Stabilization of Angular Motion of a Rigid Body
Author(s) V.G. Biryukov (Institute for Precision Mechanics and Control Problems of the Russian Academy of Sciences, ul. Rabochaya 24, Saratov, 410028 Russia)
Yu.N. Chelnokov (Institute for Precision Mechanics and Control Problems of the Russian Academy of Sciences, ul. Rabochaya 24, Saratov, 410028 Russia; Chernyshevskii Saratov State University, ul. Astrakhanskaya 83, Saratov, 410012 Russia, chelnokovyun@gmail.com)
Abstract The problem of optimal transfer of a rigid body to a prescribed trajectory of preset angular motion is considered in the nonlinear statement. (The control is the vector of absolute angular velocity of the rigid body.) The functional to be minimized is a mixed integral quadratic performance criterion characterizing the general energy expenditure on the control and deviations in the state coordinates.

Pontryagin's maximum principle is used to construct the general analytic solution of the problem in question which satisfies the necessary optimality condition and ensures the asymptotically stable transfer of the rigid body to any chosen trajectory of preset angular motion. It is shown that the obtained solution also satisfies Krasovskii's optimal stabilization theorem.
Keywords optimal control, rigid body, angular motion, quaternion, stabilization
References
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2.  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.)].
3.  B. Wie and P. M. Barda, "Quaternion Feedback for Spacecraft Large Angle Maneuvers," J. Guid. Contr. and Dynam. 8 (3), 360-365 (1985).
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5.  P. K. Plotnikov, A. N. Sergeev, Yu. N. Chelnokov, "Kinematic Problem of Attitude Control of a Rigid Body," Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, No. 5, 9-18 (1991) [Mech. Solids (Engl. Transl.)].
6.  Yu. N. Chelnokov, "Quaternion Solution of Kinematic Problems in Rigid Body Attitude Control - Equations of Motion, Problem Statement, Preset Motion, and Control," Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, No. 4, 7-14 (1993) [Mech. Solids (Engl. Transl.)].
7.  Yu. N. Chelnokov, "Quaternion Solution of Kinematic Problems of Solid Attitude Control: Equations of Errors, Correction Laws and Algorithms," Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 4, 3-12 (1994) [Mech. Solids (Engl. Transl.)].
8.  Yu. N. Chelnokov, "Construction of Attitude Control Laws for a Rigid Body Using Quaternions and Standard Forms of Equations Governing Transient Processes. Part 2," Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 3-17 (2002) [Mech. Solids (Engl. Transl.) 37 (2), 1-12 (2002)].
9.  Yu. N. Chelnokov, "Construction of Attitude Control Laws for a Rigid Body Using Quaternions and Standard Forms of Equations Governing Transient Processes. Part 1," Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, No. 1, 3-17 (2002) [Mech. Solids (Engl. Transl.) 37 (1), 1-12 (2002)].
10.  V. G. Biryukov and Yu. N. Chelnokov, "Kinematic Problem of Optimal Nonlinear Stabilization of a Rigid Body Angular Motion," in Mathematics, Mechanics, Collection of Scientific Papers, No. 4 (Izdat. Saratov Univ., Saratov, 2002), pp. 172-174 [in Russian].
11.  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].
12.  Yu. N. Chelnokov, "Biquaternion Solution of the Kinematic Control Problem for the Motion of a Rigid Body and Its Application to the Solution of Inverse Problems of Robot-Manipulator Kinematics,' Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, No. 1, 38-58 (2013) [Mech. Solids (Engl. Transl.) 48 (1), 31-46 (2013)].
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15.  I. G. Malkin, Theory of Stability of Motion (Nauka, Moscow, 1966) [in Russian].
Received 30 March 2015
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