Analysis of robot navigation schemes using Rantzer’s Dual Lyapunov Theorem

When robots are driven by the negative gradient of a potential field that consists of the sum of an attractive and a repulsive term, convergence to the desired configuration cannot be guaranteed by traditional Lyapunov techniques. In this paper, sufficient conditions for convergence of such systems are provided instead with the use of Rantzer´s dual Lyapunov theorem. In particular, a condition that involves the trace of the Hessian matrix of the potential function is derived and then applied to the cases of navigation of a single robot and of multi-robot formation stabilization. The main result of the paper states that a sufficient condition for convergence to a desired configuration in both cases is that the attractive potential admits a sufficiently large gain. A lower bound on the attractive potential is computed. Computer simulations that support the new results are provided.