Smooth movement of a rigid body in orientational space along the shortest path through the uniform lattice of the points on SO(3) / Mityushov E.A., Misyura N.E., Berestova S.A. // Vestnik Udmurtskogo Universiteta: Matematika, Mekhanika, Komp'yuternye Nauki. - 2017. - V. 27, l. 1. - P. 138-145.

ISSN:

19949197

Type:

Article

Abstract:

Many tasks of motion control and navigation, robotics and computer graphics are related to the description of a rigid body rotation in three-dimensional space. We give a constructive solution for the smooth movement of a rigid body to solve such problems. The smooth movement in orientational space is along the shortest path. Spherical solid body motion is associated with the movement of the point on the hypersphere in four-dimensional space along the arcs of large radius through the vertices of regular four-dimensional polytope. Smooth motion is provided by the choice of a special nonlinear function of quaternion interpolation. For an analytical presentation of the law of continuous movement, we use the original algebraic representation of the Heaviside function. The Heaviside function is represented using linear, quadratic and irrational functions. The animations in the computer program MathCad illustrate smooth motion of a rigid body through the nodes of a homogeneous lattice on the group SO(3). The algorithm allows one to change in a wide range the time intervals displacements between nodes, as well as the laws of motion on these intervals.

Author keywords:

Discrete distribution on SO(3); Heaviside function; Quaternion interpolation; Regular four-dimensional polytope; Shortest paths

Index keywords:

нет данных

DOI:

10.20537/vml70112

Смотреть в Scopus:

https://www.scopus.com/inward/record.uri?eid=2-s2.0-85018747547&doi=10.20537%2fvml70112&partnerID=40&md5=d52f8f616fb794e17759bdc8b4a87ca6

Соавторы в МНС:

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Link	https://www.scopus.com/inward/record.uri?eid=2-s2.0-85018747547&doi=10.20537%2fvml70112&partnerID=40&md5=d52f8f616fb794e17759bdc8b4a87ca6
Affiliations	Department of Physics and Mathematics, Ural Federal University, ul. Mira 19, Yekaterinburg, Russian Federation; Department of Theoretical Mechanics, Ural Federal University, ul. Mira, 19, Yekaterinburg, Russian Federation
Author Keywords	Discrete distribution on SO(3); Heaviside function; Quaternion interpolation; Regular four-dimensional polytope; Shortest paths
References	Borisov, A.V., Mamaev, I.S., (2001) Dinamika Tverdogo Tela (Rigid Body Dynamics), p. 384. , Izhevsk: Regular & Chaotic Dynamics; Dubrovin, B.A., Fomenko, A.T., Novikov, S.P., (2013) Sovremennaya Geometriya. Metody i Prilozheniya. Tom I. Geometriya Poverkhnostei Grupp Preobrazovanii i Polei (Modern Geometry-methods and Applications. Part I. The Geometry of Surfaces, Transformation Groups, Fields), p. 336. , Moscow: Librokom; Kopytov, N.P., Mityushov, E.A., Uniform distribution of points on hypersurfaces: Simulation of random equiprobable rotations (2015) Vestn. Udmurt. Univ. Mat. Mekh. Komp'Yut. Nauki, 25 (1), pp. 29-35. , in Russian; Golubev, Y.F., (2013) Quaternion Algebra in Rigid Body Kinematics, (39), pp. 1-23. , Keldysh Institute of Applied Mathematics Preprint, Moscow in Russian; Wikipedia, the Free Encyclopedia, , https://en.wikipedia.org/wiki/24-cell, 24-cell; Shoemake, K., Animating rotation with quaternion curves (1985) Proceedings of the 12th Annual Conference On Computer Graphics and Interactive Techniques (SIGGRAPH'85), pp. 245-254. , ACM, New York, NY, USA; Mityushov, E.A., Misyura, N.E., (2017) Exact Representation of the Unit Step Function Through Algebraic Functions, , http://www.intellectualarchive.com, ID: 1796; Mityushov, E.A., Misyura, N.E., Zhilin, S.S., The 3D Animation of a Smooth Motion, , https://www.youtube.com/watchv=k00jJIBqWY; Mityushov, E.A., 3D Animation in the MathCad: Smooth Change of Orientations, , https://www.youtube.com/watchv=KwqQVov83jk
Publisher	Udmurt State University
Language of Original Document	Russian
Abbreviated Source Title	Vestn. Udmurt. Univ., Matematika, Mekhanika, Kompyuternye Nauki
Source	Scopus