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 Conference Paper %B IEEE Conference on Decision and Control (CDC2014) %D 2014 %T On Time-Optimal Trajectories for Differential Drive Vehicles with Field-Of-View Constraints %A Cristofaro, A. %A Salaris, P. %A L. Pallottino %A Giannoni, F. %A A. Bicchi %K Embedded Control %K Robotics %XThis paper presents the first step toward the study of minimum time trajectories for a differential drive robot, which is equipped with a fixed and limited Field-Of-View (FOV) camera, towards a desired configuration while keeping a given landmark in sight during maneuvers. While several previous works have provided a complete synthesis of shortest paths in case of both nonholonomic and FOV constraints, to the best of our knowledge, this paper represents the first analysis of minimum time trajectories with the two constraints. After showing the extremals of the problem at hand, i.e. straight lines, rotations on the spot, logarithmic spirals and involute of circles, we provide the optimal control laws that steer the vehicle along the path and the cost in terms of time along each extremal. Moreover, we compare some concatenations of extremals in order to reduce the complexity of the problem toward the definition of a sufficient finite set of optimal maneuvers.

%B IEEE Conference on Decision and Control (CDC2014) %I IEEE %C Los Angeles, USA, December 15-17 %P 2191 - 2197 %@ 978-1-4799-7746-8 %U http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7039723 %R 10.1109/CDC.2014.7039723