  Recently, many reactive trajectory planning approaches were suggested in the
literature because of their inherent immediate adaption in the ever more
demanding cluttered and unpredictable environments of robotic systems. However,
typically those approaches are only locally reactive without considering global
path planning and no guarantees for simultaneous collision avoidance and goal
convergence can be given. In this paper, we study a recently developed circular
field (CF)-based motion planner that combines local reactive control with
global trajectory generation by adapting an artificial magnetic field such that
multiple trajectories around obstacles can be evaluated. In particular, we
provide a mathematically rigorous analysis of this planner in a planar
environment to ensure safe motion of the controlled robot. Contrary to existing
results, the derived collision avoidance analysis covers the entire CF motion
planning algorithm including attractive forces for goal convergence and is not
limited to a specific choice of the rotation field, i.e., our guarantees are
not limited to a specific potentially suboptimal trajectory. Our Lyapunov-type
collision avoidance analysis is based on the definition of an (equivalent)
two-dimensional auxiliary system, which enables us to provide tight, if and
only if conditions for the case of a collision with point obstacles.
Furthermore, we show how this analysis naturally extends to multiple obstacles
and we specify sufficient conditions for goal convergence. Finally, we provide
a challenging simulation scenario with multiple non-convex point cloud
obstacles and demonstrate collision avoidance and goal convergence.
