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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....

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...

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...

Envelope construction of two-parameteric system of curves in the technological practice

Bartoň, Stanislav, Petřík, Michal (2015)

Programs and Algorithms of Numerical Mathematics

A two-parametric system of close planar curves is defined in the introduction of the presented article. Next a theorem stating the existence of the envelope is presented and proved. A mathematical model of the collecting mechanism of the Horal forage trailer is developed and used for practical demonstrations. The collecting mechanism is a double joint system composed of three rods. An equation describing the trajectory of a random point of the working rod is derived using Maple. The trajectories...

On asymptotic motions of robot-manipulator in homogeneous space

Anton Dekrét, Ján Bakša (2008)

Applications of Mathematics

In this paper the notion of robot-manipulators in the Euclidean space is generalized to the case in a general homogeneous space with the Lie group G of motions. Some kinematic subspaces of the Lie algebra 𝒢 (the subspaces of velocity operators, of Coriolis acceleration operators, asymptotic subspaces) are introduced and by them asymptotic and geodesic motions are described.

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