Accurate Motion of a Robot End-Effector Using the Curvature Theory of Ruled Surfaces

Abstract

In robotics, there are two methods of trajectory planning: the joint interpolation method which is appropriate for fast transition of the robot end-effector; and the cartesian interpolation method which is appropriate for slower motion of the end-effector along straight path segments. Neither method, however, is sufficient to allow a smooth, differentiable, transition of position and orientation of the end-effector. In this paper, we propose a method of trajectory planning that will permit more accurate motion of a robot end-effector. The method is based on the curvature theory of a ruled surface generated by a line fixed in the end-effector, referred to as the tool line. The orientation of the end-effector about the tool line is included in the analysis to completely describe the six degree-of-freedom motion of the end-effector. The linear and angular properties of motion of the end-effector, determined from the differential properties of the ruled surface, are utilized in the trajectory planning.

The adjoint trajectory of robot end effector using the curvature theory of ruled surface

“…In [11], the authors showed that relate to the motion of body carrying the line that generates the curvature theory of line trajectories seeks to characterize the shape of the trajectory ruled surface. Ryuh and Pennock [4], made use of the curvature theory of a ruled surface to study the instantaneous motion properties of a robotic device. Also, Ryuh [5] proposed a method which geometric modeling technique is related to robot trajectory planning which is based on the curvature theory of a ruled surface.…”

Modified Roller Coaster Surface in Space

“…In differential geometry, the search for certain classes of surfaces with particular features in three-dimensional Euclidean space, such as ruled and developable surfaces which are 1-parameter smooth families of straight lines, has a significant role in the field of civil engineering and architecture (CAD) [1][2][3][4][5][6]. Moreover, the work path and workspace of robotic and/or mechanic operations are formed by these spatial curves and surfaces during the generation process [7][8][9].…”

Some Results on Point-Line Trajectories in Lorentz 3-space

“…One of these studies is presented by McCarthy and Roth [6]. Then, Ryuh and Pennock [10] carried out this theory to trajectory planing of robot end-effectors. They studied directrix and the trajectory of the tool center point.…”

“…But, the relationship and coordination between the directrix and indicatrix are given by Ting and friends [13]. For detailed information, see [2,3,4,5,6,7,8,9,10,11,12,13,1].…”