A Riemannian-Geometry Approach for Control of Robotic Systems under Constraints

A Riemannian-geometry approach for control of two-dimensional object grasping and manipulation by using a pair of multi-joint planar robot fingers is presented, together with a basic discussion on stability of position and force hybrid control of redundant robotic systems under geometric constraints. Even in the case that the shape of the object is arbitrary, it is possible to see that rolling contact constraints induce the Euler equation of motion in an implicit function form, in which constraint forces appear as wrench vectors affecting on the object. The Riemannian metric can be introduced in a natural way on a constraint submanifold induced by rolling contacts. A control signal called “blind grasping” is defined and shown to be effective in stabilization of grasping without using the details of information of object shape and parameters or external sensing. The concept of stability of the closed-loop system under constraints is renewed in order to overcome the degrees-of-freedom redundancy problem. An extension of Dirichlet-Lagrange's stability theorem to a system of DOF-redundancy under constraints is presented by using a Morse-Lyapunov function.