Closed form Navigation Functions based on harmonic potentials

This paper proposes a new class of smooth closed form Navigation Functions that are derived from harmonic functions. The resulting functions are by construction free of local minima. Utilizing the underlying structure of harmonic functions a tuning controller is proposed to establish the non-degeneracy of critical points. The construction of this new class of Navigation Functions was made possible due to the recently introduced Navigation Transformation. In addition to the theoretical guarantees, the effectiveness of the proposed Navigation Functions is demonstrated through non-trivial computer simulations with systems with first and second order dynamics in a non-trivial workspace.