Constructive Time-Varying Vector Fields for Robot Navigation
Adriano M. C. Rezende, Vinícius Mariano Gonçalves, Luciano C. A. Pimenta
Abstract
In this work, we present a methodology to compute an artificial time-varying vector field in <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> dimensions that defines trajectories that converge to and follow a given desired curve. The Euclidean distance function is used to construct the field, which is easily computed from a parametric representation of the curve. The computation of the time feedforward term to compensate for time dependence is such that its norm is limited by the maximum velocity of the curve. This fact allows the normalization of the time-varying vector field such that it has a constant norm without any negative effect for convergence. We present convergence proofs for the proposed normalized time-varying vector field and demonstrate the existence of ultimate bounds in the case bounded disturbances are present. Finally, we show several simulations and experiments with an actual quadrotor to validate our methodology.