Automatic Fairing of Two-Parameter Rational B-Spline Motion

Abstract

This paper deals with the problem of automatic fairing of two-parameter B-Spline spherical and spatial motions. The concept of two-parameter freeform motions brings together the notion of the analytically determined two-parameter motions in Theoretical Kinematics and the concept of freeform surfaces in the field of Computer Aided Geometric Design (CAGD). A dual quaternion representation of spatial displacements is used and the problem of fairing two-parameter motions is studied as a surface fairing problem in the space of dual quaternions. By combining the latest results in surface fairing from the field of CAGD and computer aided synthesis of freeform rational motions, smoother (C3 continuous) two-parameter rational B-Spline motions are generated. The results presented in this paper are extensions of previous results on fine-tuning of one-parameter B-spline motions. The problem of motion smoothing has important applications in the Cartesian motion planning, camera motion synthesis, spatial navigation in visualization, and virtual reality systems. Several examples are presented to illustrate the effectiveness of the proposed method.