Epsilon-Optimal Synthesis for Unicycle-Like Vehicles With Limited Field-of-View Sensors

In this paper, we study the minimum length paths covered by the center of a unicycle equipped with a limited field-of-view (FOV) camera, which must keep a given landmark in sight. Previous works on this subject have provided the optimal synthesis for the cases in which the FOV is only limited in the horizontal directions (i.e., left and right bounds) or in the vertical directions (i.e., upper and lower bounds). In this paper, we show how to merge previous results and hence obtain, for a realistic image plane modeled as a rectangle, a finite alphabet of extremal arcs and the overall synthesis. As shown, this objective cannot be straightforwardly achieved from previous results but needs further analysis and developments. Moreover, there are initial configurations such that there exists no optimal path. Nonetheless, we are always able to provide an $varepsilon$-optimal path whose length approximates arbitrarily well any other shorter path. As final results, we provide a partition of the motion plane in regions such that the optimal or $varepsilon$ -optimal path from each point in that region is univocally determined.