  This paper brings together the concepts of navigation transformation and
harmonic functions to form navigation functions that are
correct-by-construction in the sense that no tuning is required. The form of
the navigation function is explicitly related to the number of obstacles in the
environment. This enables application of navigation functions for autonomous
robot navigation in partially or fully unknown environments, with the
capability of on-the-fly adjustment of the navigation function when new
obstacles are discovered by the robot. Appropriate navigation controllers,
applicable to robots with local, sector bounded sensing, are presented and
analyzed for a~kinematic point-mass robot and then for the dynamic point-mass
robot system. The closed form nature of the proposed navigation scheme provides
for online, fast-feedback based navigation. In addition to the analytic
guarantees, simulation studies are presented to verify the effectiveness of the
methodology.
