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Modelling and Simulation in Engineering
Volume 2011, Article ID 684034, 8 pages
Research Article

A Mathematical and Numerically Integrable Modeling of 3D Object Grasping under Rolling Contacts between Smooth Surfaces

RIKEN-TRI Collaboration Center for Human-Interactive Robot Research, Nagoya, Aichi 463-0003, Japan

Received 31 March 2011; Accepted 25 July 2011

Academic Editor: Antonio Munjiza

Copyright © 2011 Suguru Arimoto and Morio Yoshida. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A computable model of grasping and manipulation of a 3D rigid object with arbitrary smooth surfaces by multiple robot fingers with smooth fingertip surfaces is derived under rolling contact constraints between surfaces. Geometrical conditions of pure rolling contacts are described through the moving-frame coordinates at each rolling contact point under the postulates: (1) two surfaces share a common single contact point without any mutual penetration and a common tangent plane at the contact point and (2) each path length of running of the contact point on the robot fingertip surface and the object surface is equal. It is shown that a set of Euler-Lagrange equations of motion of the fingers-object system can be derived by introducing Lagrange multipliers corresponding to geometric conditions of contacts. A set of 1st-order differential equations governing rotational motions of each fingertip and the object and updating arc-length parameters should be accompanied with the Euler-Lagrange equations. Further more, nonholonomic constraints arising from twisting between the two normal axes to each tangent plane are rewritten into a set of Frenet-Serre equations with a geometrically given normal curvature and a motion-induced geodesic curvature.