Modeling of 3-D Object Manipulation by Multi-Joint Robot Fingers under Non-Holonomic Constraints and Stable Blind Grasping

Abstract

This paper derives a mathematical model that expresses motion of a pair of multi-joint robot fingers with hemi-spherical rigid ends grasping and manipulating a 3-D rigid object with parallel flat surfaces. Rolling contacts arising between finger-ends and object surfaces are taken into consideration and modeled as Pfaffian constraints from which constraint forces emerge tangentially to the object surfaces. Another noteworthy difference of modeling of motion of a 3-D object from that of a 2-D object is that the instantaneous axis of rotation of the object is fixed in the 2-D case but that is time-varying in the 3-D case. A further difficulty that has prevented us to model 3-D physical interactions between a pair of fingers and a rigid object lies in the problem of treating spinning motion that may arise around the opposing axis from a contact point between one finger-end with one side of the object to another contact point. This paper shows that, once such spinning motion stops as the object mass center approaches just beneath the opposition axis, then this cease of spinning evokes a further nonholonomic constraint. Hence, the multi-body dynamics of the overall fingers-object system is subject to non-holonomic constraints concerning a 3-D orthogonal matrix expressing three mutually orthogonal unit vectors fixed at the object together with an extra non-holonomic constraint that the instantaneous axis of rotation of the object is always orthogonal to the opposing axis. It is shown that Lagrange's equation of motion of the overall system can be derived without violating the causality that governs the non-holonomic constraints. This immediately suggests possible construction of a numerical simulator of multi-body dynamics that can express motion of the fingers and object physically interactive to each other. By referring to the fact that human grasp an object in the form of precision prehension dynamically and stably by using opposable force between the thumb and another finger (index or middle finger), a simple control signal constructed from finger-thumb opposition and an object-mass estimator is proposed and shown to realize stable grasping in a dynamic sense without using object information or external sensing (this is called “blind grasping”).