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 con- sidered 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 orien- tations 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.

On the Minimum-Time Control Problem for Differential Drive Robots with Bearing Constraints / Cristofaro, Andrea; Salaris, Paolo; Pallottino, Lucia; Giannoni, Fabio; Bicchi, Antonio. - In: JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS. - ISSN 0022-3239. - 173:3(2017), pp. 967-993. [10.1007/s10957-017-1110-7]

### On the Minimum-Time Control Problem for Differential Drive Robots with Bearing Constraints

#### Abstract

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 con- sidered 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 orien- tations 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.
##### Scheda breve Scheda completa
2017
Time-optimal paths; Nonholonomic dynamical system; Bearing constraints; Differential drive vehicles
01 Pubblicazione su rivista::01a Articolo in rivista
On the Minimum-Time Control Problem for Differential Drive Robots with Bearing Constraints / Cristofaro, Andrea; Salaris, Paolo; Pallottino, Lucia; Giannoni, Fabio; Bicchi, Antonio. - In: JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS. - ISSN 0022-3239. - 173:3(2017), pp. 967-993. [10.1007/s10957-017-1110-7]
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11573/1329697`