This paper presents a study of analysis of minimum-time trajectories for a differential drive robot equipped with a fixed and limited field-of-view camera, which must keep a given landmark in view during maneuvers. Previous works have considered the same physical problem and provided a complete analysis/synthesis for the problem of determining the shortest paths. The main difference in the two cost functions (length vs. time) lays on the rotation on the spot. Indeed, this maneuver has zero cost in terms of length and hence leads to a 2D shortest path synthesis. On the other hand, in case of minimum time, the synthesis depends also on the orientations of the vehicle. In other words, the not zero cost of the rotation on the spot maneuvers leads to a 3D minimum-time synthesis. Moreover, the shortest paths have been obtained by exploiting the geometric properties of the extremal arcs, i.e., straight lines, rotations on the spot, logarithmic spirals and involute of circles. Conversely, in terms of time, even if the extremal arcs of the minimum-time control problem are exactly the same, the geometric properties of these arcs change, leading to a completely different analysis and characterization of optimal paths. In this paper, after proving the existence of optimal trajectories and showing the extremal arcs of the problem at hand, we provide the control laws that steer the vehicle along these arcs and the time-cost along each of them. Moreover, this being a crucial step toward numerical implementation, optimal trajectories are proved to be characterized by a finite number of switching points between different extremal arcs, i.e., the concatenations of extremal arcs with infinitely many junction times are shown to violate the optimality conditions.

%B Journal of Optimization Theory and Applications %P 1-27 %8 04/2017 %G eng %U http://download.springer.com/static/pdf/641/art%253A10.1007%252Fs10957-017-1110-7.pdf?originUrl=http%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%2Fs10957-017-1110-7&token2=exp=1492507070~acl=%2Fstatic%2Fpdf%2F641%2Fart%25253A10.1007%25252Fs10957-017-111 %R 10.1007/s10957-017-1110-7 %0 Journal Article %J IEEE Transactions on Robotics (T-RO) %D 2015 %T Epsilon-Optimal Synthesis for Unicycle-like Vehicles with Limited Field-Of-View Sensors %A P Salaris %A A. Cristofaro %A L. Pallottino %K Embedded Control %K Robotics %XIn 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 obtaining, for a realistic image plane modeled as a rectangle, a finite alphabet of extremal arcs and the overall synthesis. As shown, this objective can not 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 e–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 e–optimal path from each point in thatregion is univocally determined.

%B IEEE Transactions on Robotics (T-RO) %V 31 %P 1404 - 1418 %G eng %N 6 %R 10.1109/TRO.2015.2492878 %0 Conference Paper %B IEEE Conference on Decision and Control %D 2013 %T Shortest paths for wheeled robots with limited Field-Of-View: introducing the vertical constraint %A P Salaris %A A. Cristofaro %A L. Pallottino %A A. Bicchi %K Robotics %B IEEE Conference on Decision and Control %C Florence, Italy %P 5143 - 5149