Arimoto, Suguru (Ritsumeikan Univ.), Yoshida, Morio (Ritsumeikan Univ.), Bae, Ji-Hun (Ritsumeikan Univ.)

Stability of 3-D Object Grasping under the Gravity and Nonholonomic Constraints

Scheduled for presentation during the Regular Session "Robotic Systems" (MoP07), Monday, July 24, 2006, 17:00−17:25, Room H

17th International Symposium on Mathematical Theory of Networks and Systems, July 24-28, 2006, Kyoto, Japan

This information is tentative and subject to change. Compiled on April 19, 2024

Abstract

This paper extends the stability theory of 2-D object grasping to cope with 3-dimensional (3-D) object grasping by a pair of multi-joint robot fingers with hemi-spheric ends. It shows that secure grasp of a 3-D object with parallel surfaces in a dynamic sense can be realized in a blind manner like human grasp an object by a pair of thumb and index finger while closing their eyes. Rolling contacts are modeled as Pfaffian constraints that can be apparently integrated into holonomic constraints, which can exert tangential constraint forces to the object surfaces. A noteworthy difference of modeling of 3-D object grasping with 2-D case is that the instantaneous axis of rotation of the object is fixed in the latter case but it is time-varying in the former case. Hence, the dynamics of the overall fingers-object system is subject to non-holonomic constraints regarding a 3-D orthogonal matrix consisting of three mutually orthogonal unit-vectors fixed at the object. A further difference arises due to the physical assumption that spinning around the opposing axis between the two contact points does no more arise, which induces another nonholonomic constraint. It is shown that Lagrange's equation of motion of the overall system can be derived from Hamilton's principle without violating the causality that governs the nonholonomic constraints. Then, a simple control signal constructed on the basis of finger-thumb opposable forces and an object-mass estimator is proposed and shown to achieve stable grasping in a dynamic sense without using object information or external sensing. This is called ``blind grasping" if in addition the overall closed-loop dynamics converge to a state of force/torque balance. A sketchy proof of stability and asymptotic stability on a constraint manifold of the closed-loop dynamics under the nonholonomic constraints is presented. A differential geometric meaning of exponential convergence of the solution trajectory is also discussed.