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 University^{a}

Abstract

Many tasks of motion control and navigation, robotics and computer graphics are related to the description of a rigid body rotation in threedimensional 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 fourdimensional space along the arcs of large radius through the vertices of regular fourdimensional 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 fourdimensional 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. 138145.

References

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