2008 Volume 1 Issue 1 Pages 2-11
This paper extends a stability theory of 2-D object grasp to cope with 3-dimensional(3-D) object grasp 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 their eyes closed. Rolling contacts are modeled as Pfaffian constraints that can not be integrated into holonomic constraints but exert tangential constraint forces on the object surfaces. A noteworthy difference of modeling of 3-D object grasping from the 2-D case is that the instantaneous axis of rotation of the object dynamics of the overall fingers-object system are subject to non-holonomic constraints regarding a 3-D orthogonal matrix consisting of three mutually orthogonal unit-vectors fixed at the object. Lagrange's equation of motion of the overall system can be derived from the variational principle without violating the causality that governs the nonholonomic constraints. Then, a simple control signal constructed on the basis of fingers-thumb opposable forces together with an object-mass estimator is shown to accomplish stable grasp in a dynamic sense without using object information or external sensing. The closed-loop dynamics can be regarded as Lagrange's equation of motion with an artificial potential function that attains its minimum at some equilibrium state of force/torque balance. A mathematical proof of stability and asymptotic stability on a constraint manifold of the closed-loop dynamics under the nonholonomic constraints is presented.