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Archive of Issues

Russia Yekaterinburg
Year
2017
Volume
27
Issue
1
Pages
138-145
 Section Computer science Title Smooth movement of a rigid body in orientational space along the shortest path through the uniform lattice of the points on $SO(3)$ Author(-s) Mityushov E.A.a, Misyura N.E.a, Berestova S.A.a Affiliations Ural Federal Universitya 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. Keywords discrete distribution on $SO(3)$, shortest paths, regular four-dimensional polytope, quaternion interpolation, Heaviside function UDC 514.8, 519.688 MSC 17B81, 20G20 DOI 10.20537/vm170112 Received 1 February 2017 Language Russian Citation Mityushov E.A., Misyura N.E., Berestova S.A. Smooth movement of a rigid body in orientational space along the shortest path through the uniform lattice of the points on $SO(3)$, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 1, pp. 138-145. References Borisov A.V., Mamaev I.S. Dinamika tverdogo tela (Rigid body dynamics), Izhevsk: Regular & Chaotic Dynamics, 2001, 384 p. Dubrovin B.A., Fomenko A.T., Novikov S.P. 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, and fields), Moscow: Librokom, 2013, 336 p. Kopytov N.P., Mityushov E.A. Uniform distribution of points on hypersurfaces: simulation of random equiprobable rotations, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2015, vol. 25, issue 1, pp. 29-35 (in Russian). DOI: 10.20537/vm150104 Golubev Yu.F. Quaternion algebra in rigid body kinematics, Keldysh Institute of Applied Mathematics Preprint, Moscow, 2013, no. 39, pp. 1-23 (in Russian). 24-cell. Wikipedia, the free encyclopedia. https://en.wikipedia.org/wiki/24-cell Shoemake K. Animating rotation with quaternion curves, Proceedings of the 12th annual conference on computer graphics and interactive techniques (SIGGRAPH'85), ACM, New York, NY, USA, 1985, pp. 245-254. DOI: 10.1145/325334.325242 Mityushov E.A., Misyura N.E. Exact representation of the unit step function through algebraic functions, 2017, ID: {1796}. http://www.intellectualarchive.com Mityushov E.A., Misyura N.E., Zhilin S.S. The 3D animation of a smooth motion. https://www.youtube.com/watch?v=_k00jJIBqWY Mityushov E.A. 3D animation in the MathCad: Smooth change of orientations. https://www.youtube.com/watch?v=KwqQVov83jk Full text