Stability on a manifold: Simultaneous realization of grasp and orientation control of an object by a pair or 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 hemispherical finger ends grasping a rigid object under rolling contact constraints. When a pair of 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 extra performance specifications, 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 on the basis 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. These results can be extended to other two cases that 1) motion of the overall system is confined to a vertical plane and therefore it is affected directly by the gravity and 2) the object has non-parallel but flat surfaces.