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Section
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Computer science
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Title
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Smooth movement of a rigid body in orientational space along the shortest path through the uniform lattice of the points on $SO(3)$
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Author(-s)
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Mityushov E.A.a,
Misyura N.E.a,
Berestova S.A.a
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Affiliations
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Ural Federal Universitya
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Abstract
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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.
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Keywords
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discrete distribution on $SO(3)$, shortest paths, regular four-dimensional polytope, quaternion interpolation, Heaviside function
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UDC
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514.8, 519.688
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MSC
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17B81, 20G20
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DOI
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10.20537/vm170112
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Received
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1 February 2017
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Language
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Russian
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Citation
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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.
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References
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- Golubev Yu.F. Quaternion algebra in rigid body kinematics, Keldysh Institute of Applied Mathematics Preprint, Moscow, 2013, no. 39, pp. 1-23 (in Russian).
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- 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
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Full text
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