Numerical Algorithms for Mapping Boundaries of Manipulator Workspaces

Abstract

Numerical algorithms for mapping boundaries of manipulator workspaces are developed and illustrated. Analytical criteria for boundaries of workspaces for both manipulators having the same number of input and output coordinates and redundantly controlled manipulators with a larger number of inputs than outputs are well known, but reliable numerical methods for mapping them have not been presented. In this paper, a numerical method is first developed for finding an initial point on the boundary. From this point, a continuation method that accounts for simple and multiple bifurcation of one-dimensional solution curves is developed. Second order Taylor expansions are derived for finding tangents to solution curves at simple bifurcation points of continuation equations and for characterizing barriers to control of manipulators. A recently developed method for tangent calculation at multiple bifurcation points is employed. A planar redundantly controlled serial manipulator is analyzed, determining both the exterior boundary of the accessible output set and interior curves that represent local impediments to motion control. Using these methods, more complex planar and spatial Stewart platform manipulators are analyzed in a companion paper.

A Complete Method for Workspace Boundary Determination

“…In order to see the advantages of our approach in comparison with the continuation method in [31], this method is next reviewed briefly, identifying a number of cases where it fails to produce complete maps of the workspace.…”

“…As noted in [31], [32], and [35], a workspace determination method should ideally detect all barriers in the workspace, either interior or on the boundary, as such barriers constitute true motion impediments for the end-effector. A criterion to determine whether a point q 0 = [z 0 T , u 0 T ] ∈ S corresponds to a barrier or a traversable singularity was given in [31], based on the following idea.…”

“…A point u lies on the boundary of A, denoted ∂A, if every open set of U containing u intersects the interior and the exterior of A. The following known conditions will be used to isolate such points [31].…”

“…A criterion to determine whether a point q 0 = [z 0 T , u 0 T ] ∈ S corresponds to a barrier or a traversable singularity was given in [31], based on the following idea. Let q = q(v) be a smooth parametrization of C in a neighborhood of q 0 , where v is a vector of d parameters and q 0 = q(v 0 ) for some v 0 ∈ R d .…”

“…Section II provides working definitions, assumptions, and necessary background for the paper. Section III reviews the continuation method in [31] and identifies a number of situations in which it fails to properly determine the boundary of the workspace. Section IV describes the alternative method we propose, which is robust to such situations.…”