  Real-time motion planning is a vital function of robotic systems. Different
from existing roadmap algorithms which first determine the free space and then
determine the collision-free path, researchers recently proposed several convex
relaxation based smoothing algorithms which first select an initial path to
link the starting configuration and the goal configuration and then reshape
this path to meet other requirements (e.g., collision-free conditions) by using
convex relaxation. However, convex relaxation based smoothing algorithms often
fail to give a satisfactory path, since the initial paths are selected
randomly. Moreover, the curvature constraints were not considered in the
existing convex relaxation based smoothing algorithms. In this paper, we show
that we can first grid the whole configuration space to pick a candidate path
and reshape this shortest path to meet our goal. This new algorithm inherits
the merits of the roadmap algorithms and the convex feasible set algorithm. We
further discuss how to meet the curvature constraints by using both the Beamlet
algorithm to select a better initial path and an iterative optimization
algorithm to adjust the curvature of the path. Theoretical analyzing and
numerical testing results show that it can almost surely find a feasible path
and use much less time than the recently proposed convex feasible set
algorithm.
