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We present an algorithm for generating a twice-differentiable curve on the rotation group SO(3) that interpolates a given ordered set of rotation matrices at their specified knot times. In our approach we regard SO(3) as a Lie group with a bi-invariant Riemannian metric, and apply the coordinate-invariant methods of Riemannian geometry. The resulting rotation curve is easy to compute, invariant with respect to fixed and moving reference frames, and also approximately minimizes angular acceleration.

In this paper we present a method for obtaining first-order decoupled equations of motion for multirigid body systems. The inherent flexibility in choosing generalized velocity components as a function of generalized coordinates is used to influence the structure of the resulting dynamical equations. Initially, we describe how a congruency transformation can be formed that represents the transformation between generalized velocity components and generalized coordinate derivatives. It is shown that the proper choice for the congruency transformation will insure generation of first-order decoupled equations of motion for holonomic systems. In the case of nonholonomic systems, or holonomic systems with unreduced configuration coordinates, we incorporate an orthogonal complement in conjunction with the congruency transformation. A pair of examples illustrate the results. Finally, we discuss numerical implementation of congruency transformations to achieve first-order decoupled equations for simulation purposes.

In this article we generalize the concept of Be´zier curves to curved spaces, and illustrate this generalization with an application in kinematics. We show how De Casteljau’s algorithm for constructing Be´zier curves can be extended in a natural way to Riemannian manifolds. We then consider a special class of Riemannian manifold, the Lie groups. Because of their group structure Lie groups admit an elegant, efficient recursive algorithm for constructing Be´zier curves. Spatial displacements of a rigid body also form a Lie group, and can therefore be interpolated (in the Be´zier sense) using this recursive algorithm. We apply this alogorithm to the kinematic problem of trajectory generation or motion interpolation for a moving rigid body. The orientation trajectory of motions generated in this way have the important property of being invariant with respect to choices of inertial and body-fixed reference frames.

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This paper studies planar motion approximation problems in the context of a kinematic mapping. Since a planar displacement is determined by three parameters, it can be mapped into a point of a three-dimensional space. A (single-degree-of-freedom) planar motion can, therefore, be represented by a space curve in the space of the mapping and the problem of motion approximation becomes a curve fitting problem in this space. A mapping introduced by Blaschke is used and a general theory for planar motion approximation is developed. The theory is then applied to dimensional synthesis of four-link mechanisms. Furthermore, since the structural error (i.e., the quality of motion approximation) is dependent on the closeness of the fit in the space of the mapping, a general algebraic theory for determining closest fits to points in this space is developed. The theory is illustrated by a numerical example.

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Computer Aided Geometric Design of Motion InterpolatesThis paper studies continuous computational geometry of motion and develops a method for Computer Aided Geometric Design (CAGD) of motion interpolants. The approach uses a mapping of spatial kinematics to convert the problem of interpolating displacements to that of interpolating points in the space of the mapping. To facilitate the point interpolation, the previously unorientable mapping space is made orientable. Methods are then developed for designing spline curves in the mapping space with tangent, curvature and torsion continuities. The results have application in computer animation of three-dimensional objects used in computer graphics, computer vision and simulation of mechanical systems. 756 / Vol. 116, SEPTEMBER 1994 Transactions of the ASMS Copyright © 1994 by ASME Downloaded From: http://mechanicaldesign.asmedigitalcollection.asme.org/ on 05/25/2015 Terms of Use: http://asme.org/terms