Analytical Boundary of the Workspace for General3-DOF Mechanisms

Abstract

An analytical method is presented to obtain all surfaces en veloping the workspace of a general 3-DOF mechanism. The method is applicable to kinematic chains that can be mod eled using the Denavit-Hartenberg representation for serial kinematic chains or its modification for closed-loop kinematic chains. The method developed is based upon analytical crite ria for determining singular behavior of the mechanism. By manipulating the Jacobian of the underlying mechanism, first- order singularities are computed. These singularities are then substituted into the constraint equation to parameterize singu lar surfaces representing barriers to motion. Singular surfaces are those resultihg from a singular behavior of a joint gen eralized coordinate, allowing the manipulator to lose one or more degrees of mobility. These surfaces are then intersected to determine singular curves, which represent the manipulator losing at least two degrees of mobility. Difficulties in sepa rating singular behaviors at points along singular curves are encountered. Also, difficulties in computing tangents at the intersections of singular curves are addressed. These difficul ties are resolved using an analysis of a quadratic form of the intersection of singular surfaces. An example is presented to validate the theory. Although the methods used are numerical, the main result of this work is the ability to analytically define boundary surfaces of the workspace.