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3-parametric robot manipulator with intersecting axes

Jerzy Gądek (1995)

Applications of Mathematics

A p -parametric robot manipulator is a mapping g of ℝ p into the homogeneous space P = ( C 6 × C 6 ) / Diag ( C 6 × C 6 ) represented by the formula g ( u 1 , u 2 , ⋯ , u p ) = exp ( u 1 X 1 ) · ⋯ · exp ( u p X p ) , where C 6 is the Lie group of all congruences of E 3 and X 1 , X 2 , ⋯ , X p are fixed vectors from the Lie algebra of C 6 . In this paper the 3 -parametric robot manipulator will be expressed as a function of rotations around its axes and an invariant of the motion of this robot manipulator will be given. Most of the results presented here have been obtained during the author’s stay at Charles University in Prague....

An algorithm based on rolling to generate smooth interpolating curves on ellipsoids

Krzysztof Krakowski, Fátima Silva Leite (2014)

Kybernetika

We present an algorithm to generate a smooth curve interpolating a set of data on an n -dimensional ellipsoid, which is given in closed form. This is inspired by an algorithm based on a rolling and wrapping technique, described in [11] for data on a general manifold embedded in Euclidean space. Since the ellipsoid can be embedded in an Euclidean space, this algorithm can be implemented, at least theoretically. However, one of the basic steps of that algorithm consists in rolling the ellipsoid, over...

An example of a non-degenerate precession possessing two distinct pairs of axes

Giancarlo Cantarelli, Corrado Risito (2003)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In the present paper we provide an interesting example of a non-degenerate precession possessing two distinct pairs p , f , p ′ , f ′ of axes of precession and figure. Thus the problem arises of the existence of classes of precessions possessing a unique axis of precession and a unique axis of figure. In the fourth section we show that the class of non-degenerate regular precessions enjoys this property.

Approximation of Jacobian inverse kinematics algorithms

Krzysztof Tchoń, Joanna Karpińska, Mariusz Janiak (2009)

International Journal of Applied Mathematics and Computer Science

This paper addresses the synthesis problem of Jacobian inverse kinematics algorithms for stationary manipulators and mobile robots. Special attention is paid to the design of extended Jacobian algorithms that approximate the Jacobian pseudoinverse algorithm. Two approaches to the approximation problem are developed: one relies on variational calculus, the other is differential geometric. Example designs of the extended Jacobian inverse kinematics algorithm for 3DOF manipulators as well as for the...

Carathéodory balls and norm balls in H p , n = z ∈ ℂ n : ∥ z ∥ p < 1

Binyamin Schwarz, Uri Srebro (1996)

Banach Center Publications

It is shown that for n ≥ 2 and p > 2, where p is not an even integer, the only balls in the Carathéodory distance on H p , n = z ∈ ℂ n : ∥ z ∥ p < 1 which are balls with respect to the complex l p norm in ℂ n are those centered at the origin.

Critical configurations of planar robot arms

Giorgi Khimshiashvili, Gaiane Panina, Dirk Siersma, Alena Zhukova (2013)

Open Mathematics

It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, P can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or robot arms. We introduce the notion of the oriented area for an open polygonal chain, prove that critical points are exactly the cyclic configurations with antipodal endpoints and derive...

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