Navigation functions in topologically complex 3-D workspaces

Navigation Functions constructed according to the Koditschek-Rimon construction require the workspace to be topologically simple, i.e. homeomorphic to a sphere world. This paper proposes the first provably correct construction of Navigation Functions in 3D workspaces that are topologically complex. To achieve this construction, an extension of the recently introduced Navigation Transformation is proposed, that can handle any workspace and obstacle topology that can be categorized under the Classification Theorem of orientable 2-manifolds. The constructed Navigation Function is based on an underlying harmonic potential, that is guaranteed by construction to be free of local minima, hence tuning free in that aspect. In addition to the theoretical guarantees, a case study is presented along with a non-trivial computer simulation to demonstrate the effectiveness of the proposed solution.