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A general formulation for determining complex sweeps comprising a multiple of parameters has been presented by the authors in recent work. This paper investigates the boundaries to swept volumes, and in specific, addresses the problem of determination of voids in the volume. The determination of voids has become of major concern in CAD software, where the accurate calculation of the swept volume is used in computing solid properties such as mass and moments of inertia. A mathematical formulation based on the concept of a normal acceleration function on singular surfaces is introduced. Criteria are derived regarding the identification of a boundary from the definiteness properties of the normal acceleration function. Numerical examples are illustrated in detail and represent the first treatment of void identification in complex sweeps. ᭧

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.

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Algorithms for identifying closed form surface patches on the boundary of 5DOF manipulator workspaces are developed and illustrated. Numerical algorithms for the determination of three-and four-DOF manipulator workspaces are available, but formulations for determining equations of surface patches bounding the workspace of five-DOF manipulators were never presented. In this work, constant singular sets in terms of the generalized variables are determined. When substituted into the vector function yield hyperentities that exist internal and external to the workspace envelope. The appearance of surfaces parametrized in three variables within the workspace requires further analysis pertaining to a coupled singular behavior and is also addressed. Previous results pertaining to bifurcation points that were unexplained are now addressed and clarified. Numerous examples are presented.

Analytical methods for identifying the boundary to the workspace of serial mechanical manipulators and the boundary to voids in the workspace are presented. The determination of parametric equations of surface patches that envelop the workspace of serial manipulators was presented elsewhere and is extended in this paper to an analytical method for void identification. Because of the ability to identify closed-form surface patches that exist internal and external to the workspace, a mathematical formulation based on the concept of a normal acceleration function is introduced. Admissible motion in the normal direction to a point on a singular surface is delineated and characterized by definiteness properties of a quadratic form. An enclosure bound by surface patches that do not admit normal motion is identified as a void. Several examples are treated using this formulation to illustrate the method.