Safe Vector Field for Robot Navigation in $n$-Dimensions

In this work, we propose a novel artificial vector field for robot navigation in n-dimensional path-following tasks, designed to ensure safety and convergence with a smoothed control law. Unlike previous methods based on discontinuous Euclidean distance functions, our approach uses a smooth Euclidean-like function to achieve a continuous control law formulation and a field combination to balance the objectives of avoiding obstacles and following the path. This results in a navigation method that follows a target path while preventing robots from approaching obstacles, which can be used in different applications. We provide formal proofs for safety using barrier functions concepts and path convergence via Lyapunov theory. The methodology is validated through extensive numerical simulations and real-world experiments. Those include extrapolations of the methodology in more complex cases, such as quadcopters and multi-robot systems to underline the method's advantages in achieving safe and reliable robot navigation.