Passive velocity field control with discontinuous desired fields: Non-smooth potential gradient vector field

Abstract

A wide variety of robot tasks cannot be achieved by a smooth velocity field. This paper considers discontinuous desired velocity fields that represent robot tasks. We propose passive velocity field control (PVFC) with discontinuous desired velocity fields. The method deals with a discontinuous field produced by a generalized differential of a non-smooth potential function. The PVFC requires the calculation of a derivative of desired velocity fields. Therefore, we introduce a novel generalized differential of discontinuous vector fields, called disassembled Hessian matrix, which is a second derivative of a non-smooth potential function. Theoretical analysis shows that a closed-loop system with the method satisfies passivity in the sense of the sample-and-hold solution. The simulation result illustrates that the proposed method achieves the same properties of the conventional PVFC, even though the desired velocity field is discontinuous.