 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4760 |
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| 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," Mech. Solids. 48 (1), 31-46 (2013) |
| Year |
2013 |
Volume |
48 |
Number |
1 |
Pages |
31-46 |
| DOI |
10.3103/S0025654413010044 |
| Title |
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 |
| Author(s) |
Yu.N. Chelnokov (Institute for Precision Mechanics and Control Problems, Rabochaya 24, Saratov, 410028 Russia, ChelnokovYuN@info.sgu.ru) |
| Abstract |
The problem of reducing the body-attached coordinate system to the reference (programmed) coordinate system moving relative to the fixed coordinate system with a given instantaneous velocity screw along a given trajectory is considered in the kinematic statement. The biquaternion kinematic equations of motion of a rigid body in normalized and unnormalized finite displacement biquaternions are used as the mathematical model of motion, and the dual orthogonal
projections of the instantaneous velocity screw of the body motion onto the body coordinate axes are used as the control.
Various types of correction (stabilization), which are biquaternion analogs of position and integral corrections, are proposed. It is shown that the linear (obtained without linearization) and stationary biquaternion error equations that are invariant under any chosen programmed motion of the reference coordinate system can be obtained for the proposed types of correction and the use of unnormalized finite displacement biquaternions and four-dimensional dual controls allows one to construct globally regular control laws. The general solution of the error equation is constructed, and conditions for asymptotic stability of the programmed motion are obtained. The constructed theory of kinematic control of motion is used to solve inverse problems of robot-manipulator kinematics.
The control problem under study is a generalization of the
kinematic problem [1, 2] of reducing the body-attached coordinate system to the reference coordinate system rotating at a given (programmed) absolute angular velocity, and the presented method for solving inverse problems of robot-manipulator kinematics is a development of the method proposed in [3-5]. |
| Keywords |
rigid body, kinematic control, programmed motion and control, stabilization, correction, manipulator, inverse kinematic problem, stability of motion, biquaternion, kinematic screw |
| References |
| 1. | P. K. Plotnikov, A. N. Sergeev, Yu. N. Chelnokov,
"Kinematic Control Problem for the Orientation of a Rigid Body,"
Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela,
No. 5, 9-18 (1991)
[Mech. Solids (Engl. Transl.)
26 (5), 7-16 (1991)]. |
| 2. | 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)]. |
| 3. | A. V. Molotkov and Yu. N. Chelnokov,
"Application of the Kinematic Control Theory to Solving
the Inverse Problem of Kinetics of Robot-Manipulator with Revolute Joints,"
in Mathematics, Mechanics,
Collection of papers, No. 3
(Izd-vo Saratov Univ., Saratov, 2001),
pp. 170-172
[in Russian]. |
| 4. | Yu. N. Chelnokov,
"Some Problems of Nonlinear Dynamics and Motion Control,"
in Problems and Perspectives of Precession Mechanics and Control in Mechanical Engineering,
Collection of papers
(Izd-vo Saratov Univ., Saratov, 2002),
pp. 10-13
[in Russian]. |
| 5. | A. V. Molotkov and Yu. N. Chelnokov,
"Solution of the Inverse Problem of Robot-Manipulator "Puma" Kinematics
by Using the Biquaternion Theory of Kinematic Control,"
in Mathematics, Mechanics,
Collection of papers, No. 4
(Izd-vo Saratov Univ., Saratov, 2002),
pp. 204-206
[in Russian]. |
| 6. | 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). |
| 7. | 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). |
| 8. | V. N. Branets and I. P. Shmyglevskii,
Application of Quaternions in Problems of Orientation of a Rigid Body
(Nauka, Moscow, 1973)
[in Russian]. |
| 9. | V. N. Branets and I. P. Shmyglevskii,
Introduction to the Theory of Strapdown Inertial Navigation Systems
(Nauka, Moscow, 1992)
[in Russian]. |
| 10. | A. V. Molotkov and Yu. N. Chelnokov, "Quaternion Solution of the
Problem of Optimal Turn with Spherical Distribution of the Rigid
Body Mass," in Problems of Mechanics and Control,
Collection of papers (Izdat. Perm State Univ., Perm, 1995),
pp. 122-131 [in Russian]. |
| 11. | V. G. Biryukov and Yu. N. Chelnokov, "Kinematic Problem of Optimal
Nonlinear Stabilization of a Rigid Body Angular Motion," in
Mathematics, Mechanics, Collection of papers, No. 4
(Izdat. Saratov Univ., Saratov, 2002), pp. 172-174 [in Russian]. |
| 12. | V. V. Malanin and N. A. Strelkova,
Optimal Control of Orientation and Helical Motion of a Rigid Body
(NITs "Regular and Chaotic Dynamics," Moscow-Izhevsk, 2004)
[in Russian]. |
| 13. | 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]. |
| 14. | N. A. Strelkova, "Optimal in the Speed of Response Kinematic
Control of Screw Displacement of a Rigid Body," Izv. Akad. Nauk
SSSR. Mekh. Tverd. Tela, No. 4, 73-76 (1982) [Mech. Solids (Engl.
Transl.)]. |
| 15. | Yu. N. Chelnokov,
"On Integration of Kinematic Equations of a Rigid Body's Screw-Motion,"
Prikl. Mat. Mekh.
44 (1), 32-39 (1980)
[J. Appl. Math. Mech. (Engl. Transl.)
44 (1), 19-23 (1980)]. |
| 16. | K. S. Fu, R. C. Gonzalez, and C. S. G. Lee,
Robotics
(McGraw Hill, 1987; Mir, Moscow, 1989). |
|
| Received |
20 May 2010 |
| Link to Fulltext |
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