  Navigation functions provide both path and motion planning, which can be used
to ensure obstacle avoidance and convergence in the sphere world. When dealing
with complex and realistic scenarios, constructing a transformation to the
sphere world is essential and, at the same time, challenging. This work
proposes a novel transformation termed the conformal navigation transformation
to achieve collision-free navigation of a robot in a workspace populated with
obstacles of arbitrary shapes. The properties of the conformal navigation
transformation, including uniqueness, invariance of navigation properties, and
no angular deformation, are investigated, which contribute to the solution of
the robot navigation problem in complex environments. Based on navigation
functions and the proposed transformation, feedback controllers are derived for
the automatic guidance and motion control of kinematic and dynamic mobile
robots. Moreover, an iterative method is proposed to construct the conformal
navigation transformation in a multiply-connected workspace, which transforms
the multiply-connected problem into multiple simply-connected problems to
achieve fast convergence. In addition to the analytic guarantees, simulation
studies verify the effectiveness of the proposed methodology in workspaces with
non-trivial obstacles.
