 | | 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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| S.E. Perelyaev, "New Kinematic Parameters of Finite Rotation of a Solid," Mech. Solids. 48 (2), 147-155 (2013) |
| Year |
2013 |
Volume |
48 |
Number |
2 |
Pages |
147-155 |
| DOI |
10.3103/S0025654413020052 |
| Title |
New Kinematic Parameters of Finite Rotation of a Solid |
| Author(s) |
S.E. Perelyaev (Moscow Institute of Electrical Mechanics and Automation, Aviatsionnyy per. 5, Moscow, 125319 Russia, pers2030@yandex.ru) |
| Abstract |
A new family of kinematic (global and local) orientation parameters of a solid is presented and described.
All kinematic parameters are obtained by the method of mapping the variables onto the corresponding oriented subspace (hyperplane). In particular, the method of stereographic projection of a point of a five-dimensional sphere S5⊂R6 onto the oriented hyperplane R5 is presented for the classical direction cosines of the angles determining the orientation of two coordinate
systems.
A family of global kinematic parameters obtained by the method of mapping of five-dimensional kinematic Hopf parameters given in the space R5 onto the four-dimensional oriented subspace R4 is described.
The theorem of the homeomorphism of two topological spaces (the four-dimensional sphere S4⊂R5 with one deleted point and the oriented hyperplane R4) is used to establish the correspondence between five- and four-dimensional kinematic parameters defined in the corresponding spaces.
It is also shown what global four-dimensional orientation
parameters, quaternions defined in the subspace R4 are
associated with the classical local parameters, i.e., the Rodrigues and Gibbs three-dimensional finite rotation vectors.
The projection method is used to obtain the kinematic differential equations (KDE) of rotation corresponding to the five- and four-dimensional orientation parameters. All above-introduced kinematic orientation parameters of a solid permit efficiently solving the classical Darboux problem by using the corresponding KDE, i.e., determining the body current angular position in the space R3 from the known (measured) angular velocity of the object rotation and its initial position in space. |
| Keywords |
kinematic parameters, direction cosines, stereographic projection, five-dimensional Hopf parameters, quaternions, kinematic differential equations (KDE) |
| References |
| 1. | V. Ph. Zhuravlev,
Foundations of Theoretical Mechanics
(Nauka, Moscow, 1997)
[in Russian]. |
| 2. | S. N. Kirpichnikov and V. S. Novoselov,
Mathematical Aspects of Kinematics of Solids
(Izdat. LGU, Leningrad, 1986)
[in Russian]. |
| 3. | S. E. Perelyaev,
"On the Global Parametrizations of a Group of 3D Rotations,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 3, 30-44 (2006)
[Mech. Solids (Engl. Transl.)
41 (3), 23-33 (2006)]. |
| 4. | J. Stuelpnagel,
"On the Parametrization of the Three-Dimensional Rotation Group,"
SIAM Rev.
6 (4), 422-429 (1964). |
| 5. | L. Pars,
A Treatise on Analytic Mechanics
(Wiley, New York, 1965; Nauka, Moscow, 1971). |
| 6. | G. Korn and T. Korn,
Mathematical Handbook for Scientists and Engineers
(McGraw-Hill, New York, 1968; Nauka, Moscow, 1977). |
| 7. | V. N. Branets and I. P. Shmyglevskii,
Application of Quaternions in Problems of Orientation of a Rigid Body
(Nauka, Moscow, 1973)
[in Russian] |
| 8. | Yu. N. Chelnokov,
Quaternion and Biquaternion Models and Methods of Mechanics of Solids and Their Applications. Geometry and Kinematics of Motion
(Nauka, Fizmatlit, Moscow, 2006)
[in Russian]. |
| 9. | S. E. Perelyaev,
"On the Correspondence between the Three- and Four-Dimensional Parameters
of the Three-Dimensional Rotation Group,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 2, 30-44 (2009)
[Mech. Solids (Engl. Transl.)
44 (2), 204-213 (2009)]. |
| 10. | F. R. Gantmakher,
Theory of Matrices
(Nauka, Moscow, 1967)
[in Russian]. |
| 11. | S. E. Perelyaev and Yu. N. Chelnokov,
"New Equations and Algorithms for SINS Orientation in Four-Dimensional
Skew-Symmetric Operators,"
in Theses of 14th International Scientific Conference
"System Analysis, Control, and Navigation"
(Izdat. MAI-PRINT, Moscow, 2009),
pp. 35-36
[in Russian]. |
|
| Received |
31 October 2010 |
| Link to Fulltext |
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