Crossable Surfaces of Robotic Manipulators With Joint Limits

Abstract

A broadly applicable formulation for determining the crossability of singular surfaces and curves in the workspace of serial robotic manipulators is presented. Singular surfaces and curves are analytically determined using nullspace rank-deficiency criteria of the mechanism’s constraint position Jacobian. Imposed joint limits in terms of inequality constraints are taken into account in the formulation. Directions of admissible normal movements on a surface or curve are established from the analysis of the normal curvature of singular surfaces. The normal curvature of a parametric surface used in this formulation, is determined from the first and second fundamental forms adapted from differential geometry. Definiteness properties of a quadratic form developed from the acceleration analysis determine admissible normal movements. For singular surfaces resulting from active joint constraints, definiteness properties are not enough and supplementary criteria are necessary. Two additional criteria are derived. This paper is a complete treatment of the problem of determining whether a singular surface/curve in the workspace is crossable. Planar and spatial numerical examples are presented to illustrate the formulation. [S1050-0472(00)01301-5]