Two Gradient-Based Control Laws on $\textit{SE}(3)$ Derived from Distance Functions

Abstract

We are interested in gradient systems on the special Euclidean group with application in the control of rigid bodies. This is motivated by the idea of lifting the gradient system to a control law for a systems with Newtonian dynamics, all in the spirit of Daniel Koditschek. In particular, we want to compute gradients of distance functions; in these flows, we can enforce stability of our reference configurations by construction. Therefore, we first outline the computation of a gradient systems on $\textit{SE}\left( 3\right)$ on the example of a distance function associated with a Riemannian metric proposed by Frank Park and Roger Brockett. Consequently, we choose a distance function that is easy to compute in camera vision systems and derive the corresponding gradient flow.

We are indebted to the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart and to Daniel Peralta-Salas for fruitful discussions.