Fine Tuning of Rational B-Splines Motions

Abstract

Abstract This paper presents two algorithms for fine-tuning rational spatial motions suitable for Computer Aided Design. The rational motions are represented by rational B-spline curves in a projective dual three-space known as the Image Space of Spatial Kinematics. The problem of fine-tuning of rational motions is studied as that of fine-tuning the corresponding rational curves in the Image Space called the image curves. The path-smoothing algorithm automatically detects and smoothes out the third order geometric discontinuities in the path of a cubic rational Bspline image curve. The speed-smoothing algorithm uses a quintic rational spline image curve to obtain a second-order geometric approximation of the path of a cubic rational B-spline image curve while allowing specification of the speed and the rate of change of speed at the key points to obtain a near constant kinetic energy parametrization. The notion of kinetic energy is used in the paper as a natural way of combining the rotational and translational speed of a spatial motion. The results have applications in trajectory generation in robotics, planing of camera movement, spatial navigation in visualization and virtual reality systems, as well as mechanical system simulation.