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Hermite interpolation by rational G [sup] k motions of low degree
Gašper Jaklič, Bert Jüttler, Marjetka Knez, Vito Vitrih, Emil Žagar, 2013, original scientific article

Abstract: Interpolation by rational spline motions is an important issue in robotics and related fields. In this paper a new approach to rational spline motion design is described by using techniques of geometric interpolation. This enables us to reduce the discrepancy in the number of degrees of freedom of the trajectory of the origin and of the rotational part of the motion. A general approach to geometric interpolation by rational spline motions is presented and two particularly important cases are analyzed, i.e., geometric continuous quartic rational motions and second order geometrically continuous rational spline motions of degree six. In both cases sufficient conditions on the given Hermite data are found which guarantee the uniqueness of the solution. If the given data do not fulfill the solvability conditions, a method to perturb them slightly is described. Numerical examples are presented which confirm the theoretical results and provide an evidence that the obtained motions have nice shapes.
Keywords: mathematics, numerical analysis, motion design, geometric interpolation, rational spline motion, geometric continuity
Published in RUP: 03.04.2017; Views: 2142; Downloads: 39
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3.
Construction of G[sup]3 rational motion of degree eight
Karla Ferjančič, Marjetka Knez, Vito Vitrih, 2015, original scientific article

Abstract: The paper presents a construction of a rigid body motion with point trajectories being rational spline curves of degree eight joining together with ▫$G^3$▫ smoothness. The motion is determined through interpolation of positions and derivative data up to order three in the geometric sense. Nonlinearity in the spherical part of construction results in a single univariate quartic equation which yields solutions in a closed form. Sufficient conditions on the regions for the curvature data are derived, implying the existence of a real admissible solution. The algorithm how to choose appropriate data is proposed too. The theoretical results are substantiated with numerical examples.
Keywords: motion design, geometric interpolation, rational spline motion, geometric continuity
Published in RUP: 15.10.2015; Views: 2816; Downloads: 124
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