A stability theory of a manifold: concurrent realization of grasp and orientation control of an object by a pair of robot fingers

This paper is concerned with a stability theory of motion, governed by Lagrange's equation for a pair of multi-degrees of freedom robot fingers with hemi-spherical finger ends grasping a rigid object under rolling contact constraints. When a pair of dual two d.o.f. fingers is used and motion of the overall fingers-object system is confined to a plane, it is shown that the total degree of freedom of the fingers-object system is redundant for realization of stable grasping though there arise four algebraic constraints. To resolve the redundancy problem without introducing any other extra and artificial performance index, a concept of stability of motion starting from a higher dimensional manifold to a lower-dimensional manifold, expressing a set of states of stable grasp with prescribed contact force, is introduced and thereby it is proved in a rigorous way that stable grasp in a dynamic sense is realized by a sensory feedback constructed by means of measurement data of finger joint angles and the rotational angle of the object. Further, it is shown that there exists an additional sensory feedback that realizes not only stable grasp but also orientation control of the object concurrently. Results of computer simulation based on Baumgarte's method are presented, which show the effectiveness of the proposed concept and analysis.