Smooth Distances for Second-Order Kinematic Robot Control

In this article, we propose an algorithm for computing a smoothed version of the distance between two objects. As opposed to the traditional Euclidean distance between two objects, which may not be differentiable, this smoothed distance is guaranteed to be differentiable. Differentiability is an important property in many applications, in particular in robotics, in which obstacle-avoidance schemes often rely on the derivative/Jacobian of the distance between two objects. We prove mathematical properties of this smoothed distance and of the algorithm for computing it, and show its applicability in robotics by applying it to a second-order kinematic control framework, also proposed in this article. The control framework using smooth distances was successfully implemented on a 7 DOF manipulator.