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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
##### References:
 [1] A. A. Agrachev and Y. L. Sachkov, Control Theory From the Geometric Viewpoint, vol. 87 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-06404-7. [2] A. Balluchi, A. Bicchi, B. Piccoli and P. Souères, Stability and robustness of optimalsynthesis for route tracking by Dubins' Vehicules, in Decision and Control (CDC), 2000 IEEE 39th Annual Conference on, 2000,581-586. [3] U. Boscain, T. Chambrion and G. Charlot, Nonisotropic 3-level quantum systems: Complete solutions for minimum time and minimum energy, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 957-990 (electronic). doi: 10.3934/dcdsb.2005.5.957. [4] U. Boscain, F. Grönberg, R. Long and H. Rabitz, Minimal time trajectories for two-level quantum systems with two bounded controls J. Math. Phys. , 55 (2014), 062106, 25pp. doi: 10.1063/1.4882158. [5] U. Boscain and B. Piccoli, Extremal synthesis for generic planar systems, J. Dynam. Control Systems, 7 (2001), 209-258. doi: 10.1023/A:1013003204923. [6] U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, vol. 43 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 2004. [7] U. Boscain and B. Piccoli, Synthesis theory in optimal control, Encyclopedia of Systems and Control, Springer London, London, (2014), 1-11. doi: 10.1007/978-1-4471-5102-9_50-1. [8] A. Bressan and B. Piccoli, A generic classification of time-optimal planar stabilizing feedbacks, SIAM J. Control Optim., 36 (1998), 12-32 (electronic). doi: 10.1137/S0363012995291117. [9] L. E. Dubins, On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents, Amer. J. Math., 79 (1957), 497-516. doi: 10.2307/2372560. [10] M. A. Lagache, U. Serres and V. Andrieu, Time minimum synthesis for a kinematic drone model, in 2015 54th IEEE Conference on Decision and Control (CDC), 2015,4067-4072. doi: 10.1109/CDC.2015.7402852. [11] T. Maillot, U. Boscain, J.-P. Gauthier and U. Serres, Lyapunov and Minimum-Time Path Planning for Drones, J. Dyn. Control Syst., 21 (2015), 47-80. doi: 10.1007/s10883-014-9222-y. [12] B. Piccoli, Classification of generic singularities for the planar time-optimal synthesis, SIAM J. Control Optim., 34 (1996), 1914-1946. doi: 10.1137/S0363012993256149. [13] B. Piccoli and H. Sussmann, Regular synthesis and sufficiency conditions for optimality, SIAM J. Control Optim., 39 (2000), 359-410 (electronic). doi: 10.1137/S0363012999322031. [14] P. Souères, A. Balluchi and A. Bicchi, Optimal feedback control for route tracking with a bounded-curvature vehicle, Internat. J. Control, 74 (2001), 1009-1019. doi: 10.1080/00207170110052211. [15] H. Sussmann, Regular synthesis for time-optimal control of single-input real analytic systems in the plane, SIAM J. Control Optim., 25 (1987), 1145-1162. doi: 10.1137/0325062.

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##### References:
 [1] A. A. Agrachev and Y. L. Sachkov, Control Theory From the Geometric Viewpoint, vol. 87 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-06404-7. [2] A. Balluchi, A. Bicchi, B. Piccoli and P. Souères, Stability and robustness of optimalsynthesis for route tracking by Dubins' Vehicules, in Decision and Control (CDC), 2000 IEEE 39th Annual Conference on, 2000,581-586. [3] U. Boscain, T. Chambrion and G. Charlot, Nonisotropic 3-level quantum systems: Complete solutions for minimum time and minimum energy, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 957-990 (electronic). doi: 10.3934/dcdsb.2005.5.957. [4] U. Boscain, F. Grönberg, R. Long and H. Rabitz, Minimal time trajectories for two-level quantum systems with two bounded controls J. Math. Phys. , 55 (2014), 062106, 25pp. doi: 10.1063/1.4882158. [5] U. Boscain and B. Piccoli, Extremal synthesis for generic planar systems, J. Dynam. Control Systems, 7 (2001), 209-258. doi: 10.1023/A:1013003204923. [6] U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, vol. 43 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 2004. [7] U. Boscain and B. Piccoli, Synthesis theory in optimal control, Encyclopedia of Systems and Control, Springer London, London, (2014), 1-11. doi: 10.1007/978-1-4471-5102-9_50-1. [8] A. Bressan and B. Piccoli, A generic classification of time-optimal planar stabilizing feedbacks, SIAM J. Control Optim., 36 (1998), 12-32 (electronic). doi: 10.1137/S0363012995291117. [9] L. E. Dubins, On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents, Amer. J. Math., 79 (1957), 497-516. doi: 10.2307/2372560. [10] M. A. Lagache, U. Serres and V. Andrieu, Time minimum synthesis for a kinematic drone model, in 2015 54th IEEE Conference on Decision and Control (CDC), 2015,4067-4072. doi: 10.1109/CDC.2015.7402852. [11] T. Maillot, U. Boscain, J.-P. Gauthier and U. Serres, Lyapunov and Minimum-Time Path Planning for Drones, J. Dyn. Control Syst., 21 (2015), 47-80. doi: 10.1007/s10883-014-9222-y. [12] B. Piccoli, Classification of generic singularities for the planar time-optimal synthesis, SIAM J. Control Optim., 34 (1996), 1914-1946. doi: 10.1137/S0363012993256149. [13] B. Piccoli and H. Sussmann, Regular synthesis and sufficiency conditions for optimality, SIAM J. Control Optim., 39 (2000), 359-410 (electronic). doi: 10.1137/S0363012999322031. [14] P. Souères, A. Balluchi and A. Bicchi, Optimal feedback control for route tracking with a bounded-curvature vehicle, Internat. J. Control, 74 (2001), 1009-1019. doi: 10.1080/00207170110052211. [15] H. Sussmann, Regular synthesis for time-optimal control of single-input real analytic systems in the plane, SIAM J. Control Optim., 25 (1987), 1145-1162. doi: 10.1137/0325062.
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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