A DIFFERENTIAL GEOMETRIC METHOD FOR KINEMATIC ANALYSIS OF TWO- AND THREE-DEGREE-OF-FREEDOM RIGID BODY MOTIONS*

ABSTRACT

In this paper, we present a novel differential geometric characterization of two- and three-degree-of-freedom rigid body kinematics, using a metric defined on dual vectors. The instantaneous angular and linear velocities of a rigid body are expressed as a dual velocity vector, and dual inner product is defined on this dual vector, resulting in a positive semi-definite and symmetric dual matrix. We show that the maximum and minimum magnitude of the dual velocity vector, for a unit speed motion, can be obtained as eigenvalues of this dual matrix. Furthermore, we show that the tip of the dual velocity vector lies on a dual ellipse for a two-degree-of-freedom motion and on a dual ellipsoid for a three-degree-of-freedom motion. In this manner, the velocity distribution of a rigid body can be studied algebraically in terms of the eigenvalues of a dual matrix or geometrically with the dual ellipse and ellipsoid. The second-order properties of the two- and three-degree-of-freedom motions of a rigid body are also obtained from the derivatives of the elements of the dual matrix. This results in a definition of the geodesic motion of a rigid body. The theoretical results are illustrated with the help of a spatial 2R and a parallel three-degree-of-freedom manipulator.

*Communicated by S. Velinsky.

Keywords:

• Instantaneous kinematics
• Dual vector
• Dual matrix
• Geodesic

ACKNOWLEDGMENT

This work was partly supported by the California Department of Transportation (Caltrans) through the basic research component of the AHMCT research center at UC-Davis. The authors also wish to thank Professor Bernard Roth of Stanford University for useful discussions and help in obtaining some of the references.

Notes

*Communicated by S. Velinsky.

*Dual numbers, first introduced by Clifford,[19]] have been used extensively in kinematics (see, for example Refs. [[20] [21] [22] [23]].

*For n = 1, 2, 3 the point trajectory is a curve, a surface, and a solid region in respectively. For n>3 we have a redundant motion.

†These are known as forward kinematics equations in manipulator kinematics.

*By the property of partial differentiation, and there are only six Christoffel symbols.

*A geodesic is the shortest distance between two points on a surface.

*The dual ellipse has been identified as a cylindroid in Ref. [[28].

*We use the symbols s _(·) and c _(·) to denote sin(·) and cos(·) respectively throughout this paper.