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June  2017, 7(2): 259-288. doi: 10.3934/mcrf.2017009

## Minimal time synthesis for a kinematic drone model

 1 Université de Toulon, CNRS, LSIS, UMR 7296, F-83957 La Garde, France 2 Univ. Lyon, Université Claude Bernard Lyon 1, CNRS, LAGEP, UMR 5007, 43 bd du 11 novembre 1918, F-69100 Villeurbanne, France

* Corresponding author

Received  April 2016 Revised  October 2016 Published  April 2017

Fund Project: The authors were partially supported by the Grant ANR-12-BS03-0005 LIMICOS of the ANR.

In this paper, we consider a (rough) kinematic model for a UAV flying at constant altitude moving forward with positive lower and upper bounded linear velocities and positive minimum turning radius. For this model, we consider the problem of minimizing the time travelled by the UAV starting from a general configuration to connect a specified target being a fixed circle of minimum turning radius. The time-optimal synthesis is presented as a partition of the state space which defines a unique optimal path such that the target can be reached optimally.

Citation: Marc-Auréle Lagache, Ulysse Serres, Vincent Andrieu. Minimal time synthesis for a kinematic drone model. Mathematical Control & Related Fields, 2017, 7 (2) : 259-288. doi: 10.3934/mcrf.2017009
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##### References:
Candidate extremal trajectories of problem $\bf{(}{{\bf{P}}_{\bf{2}}}\bf{)}$ issued from ${\mathit{\tilde X}_{\rm{0}}}$
Extremal trajectories starting from ${\mathit{\tilde X}_{\rm{0}}}$ and having one switching
Time-optimal synthesis for the problem $\bf{(}{{\bf{P}}_{\bf{1}}}\bf{)}$
Domains on which at most one switching is possible
Bang-bang-singular-bang optimal trajectories. Two optimal trajectories solutions of problem $\bf{(}{{\bf{P}}_{\bf{1}}}\bf{)}$ starting from the same point in the cut locus (left) and the corresponding optimal trajectories solutions of problem $\bf{(}{{\bf{P}}_{\bf{0}}}\bf{)}$ (right)
Bang-bang-bang-bang-bang optimal trajectories and a bang-bang-bang-bang-bang-bang optimal trajectory. Three optimal trajectories solutions of problem $\bf{(}{{\bf{P}}_{\bf{1}}}\bf{)}$ starting from the same point in the cut locus (left) and the corresponding optimal trajectories solutions of problem $\bf{(}{{\bf{P}}_{\bf{0}}}\bf{)}$ (right)
The abnormal trajectory of problem $\bf{(}{{\bf{P}}_{\bf{1}}}\bf{)}$ and the corresponding trajectory for problem $\bf{(}{{\bf{P}}_{\bf{0}}}\bf{)}$ (right)
Intersection of $\gamma^{MPp}$ with $\gamma^{Mm}$ and $\gamma^{MmM}$ (left) and the corresponding cut locus (right)
Intersection of $\gamma^{MPpPMm}$ with $\gamma^{MmMP}$, $\gamma^{MmMPp}$ and $\gamma^{MPpPM}$
Notation of the five possible optimal controls
 Control Notation $(-1,1)$ $\mathit{m}$ $(1,1)$ $\mathit{p}$ $(-1,\eta)$ $\mathit{M}$ $(1,\eta)$ $\mathit{P}$ $u$-singular $\mathit{s}$
 Control Notation $(-1,1)$ $\mathit{m}$ $(1,1)$ $\mathit{p}$ $(-1,\eta)$ $\mathit{M}$ $(1,\eta)$ $\mathit{P}$ $u$-singular $\mathit{s}$
Color convention of the optimal synthesis
 $(-1,1)$-bang arc Blue $(1,1)$-bang arc Orange $(-1,\eta)$-bang arc Purple $(1,\eta)$-bang arc Red $u$-singular arc Magenta $u$-switching curves Dashed black $v$-switching curves Gray Cut Locus Green Abnormal Cut Locus Cyan
 $(-1,1)$-bang arc Blue $(1,1)$-bang arc Orange $(-1,\eta)$-bang arc Purple $(1,\eta)$-bang arc Red $u$-singular arc Magenta $u$-switching curves Dashed black $v$-switching curves Gray Cut Locus Green Abnormal Cut Locus Cyan
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