Understanding Voids in the Workspace of Serial Robot Manipulators

Analytical methods for identifying the boundary to the workspace of 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 towards its internal is identified as a void. Several examples are treated using this formulation to illustrate the method. INTRODUCTION Numerical methods for determining workspace boundaries of manipulators have been developed in recent years. Exact computation of the workspace and its boundary is of significant importance because of its impact on manipulator design, manipulator placement in an environment, and manipulator dexterity. Although there has been many reports in this field, there are only a few that have addressed the existence of voids in the workspace. The work by Cecarelli and Vinciquerra (1995) using an algebraic formulation to determine the workspace of 4DOF-revolute manipulators exhibits a capability for determining holes and voids in the workspace. This method uses geometric envelopes of the union of all toroidal surfaces generated by revolute joints. Cross sections at various elevations are then introduced. A hole in the workspace is identified as a region of unreachable points containing at least a straight line that passes through it without making contact with the workspace. Two types of voids are identified, those inside the boundary (called ring voids) and those formed by the rotational motion of the boundary and include the main zo axis (so-called apple voids). In this paper, holes and voids are equally treated since the differentiation between an apple and a ring void has no significance using the proposed mathematical formulation. The meaning of boundary suggests boundary to the workspace, regardless whether it is external or internal. Internal boundary denotes enclosing a volume that cannot be reached. It is noted here that the importance of the analysis of voids in manipulator workspaces extends to many fields, most importantly to CAD (Abdel-Malek and Yeh 1997), where swept volume techniques can be used to generate solid models. Early investigations of manipulator workspaces were reported by Kumar and Waldron (1981), Tsai and Soni (1981), Gupta and Roth (1982), Sugimoto and Duffy (1982), Spanos and Kohli (1985), Kumar (1985), Gupta (1986), and Davidson and Hunt (1987). The consideration of joint limits in the study of manipulator workspaces was presented in a recent report by Delmas and Bidard (1995). Other works that have dealt with manipulator workspace are reported by Agrawal (1990), Gosselin and Angeles (1990), and Emiris (1993).