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

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.