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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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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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Motion design with Euler-Rodrigues frames of quintic Pythagorean-hodograph curves
Marjetka Knez, Vito Vitrih, 2012, original scientific article

Abstract: The paper presents an interpolation scheme for ▫$G^{1}$▫ Hermite motion data, i.e., interpolation of data points and rotations at the points, with spatial quintic Pythagorean-hodograph curves so that the Euler-Rodrigues frame of the curve coincides with the rotations at the points. The interpolant is expressed in a closed form with three free parameters, which are computed based on minimizing the rotations of the normal plane vectors around the tangent and on controlling the length of the curve. The proposed choice of parameters is supported with the asymptotic analysis. The approximation error is of order four and the Euler-Rodrigues frame differs from the ideal rotation minimizing frame with the order three. The scheme is used for rigid body motions and swept surface construction.
Keywords: Pythagorean-hodograph, Euler-Rodrigues frame, rotation minimizing frame, motion design, quaternion, Hermite interpolation
Published in RUP: 15.10.2013; Views: 2816; Downloads: 98
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