Mathematical Problems in Engineering
Volume 2008 (2008), Article ID 362783, 19 pages
Research Article

On Motion of Robot End-Effector Using the Curvature Theory of Timelike Ruled Surfaces with Timelike Rulings

Cumali Ekici,1 Yasin Ünlütürk,1 Mustafa Dede,1 and B. S. Ryuh2

1Department of Mathematics, Eskişehir Osmangazi University, 26480 Eskişehir, Turkey
2Division of Mechanical Engineering, College of Engineering, Chonbuk National University, Jeonju 561-756, South Korea

Received 14 August 2008; Revised 14 November 2008; Accepted 29 December 2008

Academic Editor: Giuseppe Rega

Copyright © 2008 Cumali Ekici et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The trajectory of a robot end-effector is described by a ruled surface and a spin angle about the ruling of the ruled surface. In this way, the differential properties of motion of the end-effector are obtained from the well-known curvature theory of a ruled surface. The curvature theory of a ruled surface generated by a line fixed in the end-effector referred to as the tool line is used for more accurate motion of a robot end-effector. In the present paper, we first defined tool trihedron in which tool line is contained for timelike ruled surface with timelike ruling, and transition relations among surface trihedron: tool trihedron, generator trihedron, natural trihedron, and Darboux vectors for each trihedron, were found. Then differential properties of robot end-effector's motion were obtained by using the curvature theory of timelike ruled surfaces with timelike ruling.