Developed Motion of Robot End-Effector of Timelike Ruled Surfaces With Spacelike Rulings (The Second Case)

Abstract

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 paper, we analyzed the problem of describing trajectory of a robot end-effector by a timelike ruled surface with spacelike ruling. We obtained the developed frame {t(1), r(1), k(1),} by rotating the generator frame {r, t, k} at an Darboux angle theta = theta(s) in the plane {r, k}, which is on the striction curve beta of the timelike ruled surface X. Afterword, natural frame, tool frame and surface frame which is necessary for the movements of robot are defined derivative formulas of the frames are founded by calculating the Darboux vectors. Tool frame {O, A, N} are constituted by means of this developed frame. Thus, robot end effector motion is defined for the timelike ruled surface phi generated by the orientation vector t(1) = O. Also, by using Lancret curvature of the surface and rotation angle (Darboux angle) in the developed frame the robot end-effector motion is developed. Therefore, differential properties and movements an different surfaces in Minkowski space is analyzed by getting the relations for curvature functions which are characterized a timelike ruled surface with spacelike directix. Finally, to be able to get a member of trajectory surface family which has the same trajectory curve is shown with the examples in every different choice of the Darboux angle which is used to described the developed frame.