Epsilon-Optimal Synthesis for Vehicles With Vertically Bounded Field-Of-View

This paper presents a contribution to the problem of obtaining an optimal synthesis for shortest paths for a unicycle guided by an on-board limited Field-Of-View (FOV) sensor, which must keep a given landmark in sight. Previous works on this subject have provided an optimal synthesis for the case in which the FOV is limited in the horizontal directions (H-FOV, i.e., left and right boundaries). In this paper we study the complementary case in which the FOV is limited only in the vertical direction (V-FOV, i.e., upper and lower boundaries). With respect to the H-FOV case, the vertical limitation is all but a simple extension. Indeed, not only the geometry of extremal arcs is different, but also a more complex structure of the synthesis is revealed by analysis. We will indeed show that there exist initial configurations for which the optimal path does not exist. In such cases, we provide an ε-optimal path whose length approximates arbitrarily well any other shorter path. Finally, we provide a partition of the motion plane in regions such that the optimal or ε-optimal path from each point in that region is univocally determined.