Explicit parametrizations of the configuration spaces of anthropomorphic multi-linkage systems

Multi-fingered manipulation systems are important in the study of robotics. These are also challenging systems, in part because of the loop closure constraints required of several (virtual) loops each formed by two fingers and the grasped object. Most existing work describes system configurations using joint parameters, in which loop closure constraints are expressed by highly nonlinear equations. Such a formulation amounts to an implicit parametrization of the configuration space (CSpace) as a lower-dimensional semi-algebraic subset embedded in a higher-dimensional ambient joint parameter space. The non-zero difference between the two dimensions is the codimension of CSpace as seen in the given parametrization. In this paper, we point out that, quite generally, parametrizations leading to lower codimensional configuration spaces provide multi-faceted advantages over those producing higher codimensions. For two example manipulation system—a 3-fingered hand and a planar star-manipulator with any number of fingers— we present explicit parameterizations, which are effectively of codimension 0. We base these parametrizations on our recently developed construction trees of simplices (such as triangles and tetrahedra) for multi-object systems; such a tree gives simplexbased parametrers on CSpace , in which loop closure constraints become simplex formation constraints (such as triangle inequalities and Cayley–Menger determinant constraints). Both example systems are very difficult to deal with using joint angle parameters. Our results here further demonstrate the generality and effectiveness of the simplex-based approach.