Proceedings of DETC’98 1998 ASME Design Engineering Technical Conferences September 13-16, 1998, Atlanta, Georgia, USA DETC98/MECH-5881 ALGEBRAIC MOTION APPROXIMATION WITH NURBS MOTIONS AND ITS APPLICATION TO SPHERICAL MECHANISM SYNTHESIS Q. Jeffrey Ge Department of Mechanical Engineering State University of New York at Stony Brook Stony Brook, NY 11794-2300, USA ge@design.eng.sunysb.edu Pierre M. Larochelle Mechanical Engineering Program Florida Institute of Technology Melbourne, FL 32901-6988, USA pierre@gretzky.me.fit.edu ABSTRACT In this work we bring together classical mechanism theory with recent works in the area of Computer Aided Geometric De- sign(CAGD) of rational motions as well as curve approximation techniques in CAGD to study the problem of mechanism mo- tion approximation from a computational geometric viewpoint. We present a framework for approximating algebraic motions of spherical mechanisms with rational B-Spline spherical motions. Algebraic spherical motions and rational B-spline spherical mo- tions are represented as algebraic curves and rational B-Spline curves in the space of quaternions (or the image space). Thus the problem of motion approximation is transformed into a curve approximation problem, where concepts and techniques in the field of Computer Aided Geometric Design and Computational Geometry may be applied. An example is included at the end to show how a NURBS motion can be used for synthesizing spher- ical four-bar linkages. 1 INTRODUCTION Non-Uniform Rational B-Splines, commonly referred to as NURBS, have become the de facto industry standard for the rep- resentation, design, and data exchange of geometric information processed by computers. Recently, it has become apparent that NURBS geometry can be extended to kinematic domain for syn- thesizing NURBS motions of rigid bodies in Euclidean three- space (Ge and Ravani, 1994; J¨ uttler, 1994; J¨ uttler and Wagner, 1996; Ge and Kang, 1996; Ge et al., 1997). The purpose of the present paper is to present a framework for algebraic motion approximation using spherical NURBS motion by combining NURBS geometry with kinematic geometry of spherical mech- anisms. From the viewpoint of mechanism synthesis, the ideas presented in this paper are extensions of the work of Gupta and Roth (1975) on kinematic approximation of circles and straight lines, the series of work of Ravani and Roth (1983, 1984), Bod- duluri and McCarthy (1992), Larochelle and McCarthy (1994) on algebraic motion synthesis using kinematic mapping, as well as the work of Liu and Angeles (1992a, 1992b) on planning global properties of a mechanism motion for optimization of function generating mechanisms. The paper is organized as follows. Section 1 reviews how spherical displacements can also be represented projectively us- ing homogeneous quaternions. Section 2 presents spherical ratio- nal B´ ezier and B-spline motions as B´ ezier and B-spline quater- nion curves. Section 3 presents algebraic motions of spherical mechanisms. Section 4 discusses three motion approximation problems and presents an example to demonstrate the feasibility of our approach. 2 SPHERICAL DISPLACEMENTS Quaternion algebra allows for an elegant treatment for spherical kinematics (Yang and Freudenstein, 1964; Ravani and Roth, 1984; Bottema and Roth, 1990; McCarthy, 1990). A quaternion is a hypercomplex number of the form q q 1 i q 2 j q 3 k q 4 where ijk are quaternion units. The components q i are 1 Copyright  1998 by ASME