This paper, as an extension of an earlier paper, presents a geometrical representation and theoretical foundation of robot grasping that is affected by friction and by the magnitudes of normal contact forces. Grasp analysis and synthesis are based on the concepts of constraint cone (region), restraint cone and freedom cone in the force screw-space and location screw-space. In fact, the freedom cone is the aggregate of all the screws repelling and reciprocal to the grasp geometry (denoted by the restraint cone). Moreover, the constraint cone and the constraint region describe the effect of friction and the influence of the amplitudes of normal forces. The conditions of equilibrium grasp, stable grasp, form closure, force closure, and relative form closure are derived from the geometrical representation, and approaches are suggested for the design of adaptable fixtures, and for automatic grasp planning. It is shown that relative form closure is of great significance for frictional grasp, since it is possible to generate enough internal forces in the subspace spanned by the contacts and consequently produce corresponding friction forces in the reciprocal subspace.

This paper describes the development of the topological and geometric approach to the analysis and synthesis of form-closed grasps of arbitrary objects. Concepts of wrench spaces and the relevant subsets of the positive cone, the affine hull, the convex huli, and the related polar cone are used, and applied to a representative example.