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.