Dynamics of Grasping a Rigid Object with Arbitrary Smooth Surfaces under Rolling Contacts

摘要

Modeling and control of dynamics of three-dimensional object grasping by using a pair or triplet of multi-joints robot fingers are investigated under rolling contact constraints and arbitrary shapes of the object and fingertips. It is assumed that a rolling contact between two rigid bodies with smooth surfaces is governed by the condition that the two surfaces coincide at the common contact point and share the same tangent plane. The contact constraint expressing the contact of the two surfaces at a single point in the 3-dimensional Euclidean space is reinterpreted by a set of three Pfaffian constraints, one of which is a velocity equivalence in the normal to the tangent plane at the contact point and the other two are velocity equivalence relations on the tangent plane. The Euler-Lagrange equation of motion of the overall fingers/object system is derived by introducing Lagrange multipliers corresponding to those Pfaffian constraints together with update laws of length parameters of loci of each contact point on the fingertips and object surfaces. It is shown that the Euler-Lagrange equation is parameterized by the length parameters through quantities of the first fundamental form of the surfaces, but the update laws of length parameters are governed by quantities of the second fundamental form. Furthermore, it is shown that the rolling constraints are directionally integrable. In accordance with this result, a control signal for maintaining contacts with the object is suggested from the standpoint of fingers-thumb opposability.