A Riemannian-Geometric Approach for Intelligent Control and Fingertip Design of Multi-fingered Hands

Based upon the analysis of modeling and control of two-dimensional (2-D) grasping and manipulation of arbitrary rigid objects under rolling contact constraints, geometrical conditions for the design of desired fingertip shapes of robot fingers are discussed from the Riemannian-geometric standpoint. A required condition is given as an inequality expressed in terms of quantities of the second fundamental form of fingertip contour curves, although the quantities do not enter into the Euler–Lagrange equation of motion of the overall fingers/object system. Satisfaction of the inequality is necessary for stabilization of grasping by using fingers–thumb opposable control signals without use of external sensings or knowledge of an object to be grasped. At the same time, asymptotic convergence of a solution to the closed-loop dynamics of 2-D precision prehension by a pair of multi-joint robot fingers is proved under rolling contact constraints and the existence of redundancy in the system's degrees of freedom. This is regarded as an extension of the Dirichlet–Lagrange stability theorem to a dynamical system with redundancy and geometric constraints.