  In this article, we consider the problem of trajectory planning and control
for on-road driving of an autonomous ground vehicle (AGV) in presence of static
or moving obstacles. We propose a systematic approach to partition the
collision-free portion of the space-time into convex sub-regions that can be
interpreted in terms of relative positions with respect to a set of fixed or
mobile obstacles. We show that this partitioning allows decomposing the NP-hard
problem of computing an optimal collision-free trajectory, as a path-finding
problem in a well-designed graph followed by a simple (polynomial time)
optimization phase for any quadratic convex cost function. Moreover, robustness
criteria such as margin of error while executing the trajectory can easily be
taken into account at the graph-exploration phase, thus reducing the number of
paths to explore.
