Плавное перемещение твердого тела в пространстве ориентаций по кратчайшей траектории через узлы однородной решетки на группе SO(3)

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.