Skip to main content
SpringerLink
Log in

Lyapunov and Minimum-Time Path Planning for Drones

  • Published:
Journal of Dynamical and Control Systems Aims and scope Submit manuscript

Abstract

In this paper, we study the problem of controlling an unmanned aerial vehicle (UAV) to provide a target supervision and/or to provide convoy protection to ground vehicles. We first present a control strategy based upon a Lyapunov-LaSalle stabilization method to provide supervision of a stationary target. The UAV is expected to join a predesigned admissible circular trajectory around the target which is itself a fixed point in the space. Our strategy is presented for both high altitude long endurance (HALE) and medium altitude long endurance (MALE) types of UAVs. A UAV flying at a constant altitude (HALE type) is modeled as a Dubins vehicle (i.e., a planar vehicle with constrained turning radiusand constant forward velocity). For a UAV that might change its altitude (MALE type), we use the general kinematic model of a rigid body evolving in \(\mathbb {R}^{3}\). Both control strategies presented are smooth, and unlike what is usually proposed in the literature, these strategies asymptotically track a circular trajectory of exact minimum turning radius. We also present the time-optimal control synthesis for tracking a circle by a Dubins vehicle. This optimal strategy is very rich, although much simpler than the point-to-point time-optimal strategy studied in the 1990s. Finally, we propose control strategies to provide supervision of a moving target, which are based upon theprevious ones.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25

Similar content being viewed by others

Notes

  1. 1 According to [13], the set \(\{\tilde y=0\}\) is a turnpike (i.e., a trajectory that is locally time-optimal)

References

  1. Agrachev A A, Sachkov YL. Control theory from the geometric viewpoint, Vol. 87. Encyclopaedia of mathematical sciences. Control theory and optimization, II. Berlin: Springer; 2004.

  2. Agrachev A A, Sigalotti M. On the local structure of optimal trajectories in R 3. SIAM J Control Optim. 2003;42(2):513–531. electronic.

    Article  MATH  MathSciNet  Google Scholar 

  3. Ajami A, Maillot T, Boizot N, Balmat J-F, Gauthier J-P. Simulation of a uav ground control station. In: Proceedings of the 9th international conference of modeling and simulation, MOSIM’12, 2012. Bordeaux, France, 6–8 June.

  4. Ariyur K, Fregene K. Autonomous tracking of a ground vehicle by a UAV. In: American control conference; 2008. p. 669 –671.

  5. Bertuccelli L, Wu A, How J. Robust adaptive Markov decision processes: planning with model uncertainty. IEEE Control Syst. 2012;32(5):96–109.

    Article  MathSciNet  Google Scholar 

  6. Besançon G (ed). Nonlinear observers and applications, Vol. 363. Lecture notes in control and information sciences. Papers from the 28th international summer school on control held in Grenoble, September 2007. Berlin: Springer; 2007.

  7. Bhatia A, Graziano M, Karaman S, Naldi R, Frazzoli E. Dubins trajectory tracking using commercial off-the-shelf autopilots. In: AIAA guidance, navigation, and control conference. Honolulu, Hawaii; 2008.

  8. Boissonnat J-D, Cerezo A, Leblond J. Shortest paths of bounded curvature in the plane. In: Proceedings 1992 IEEE international conference on robotics and automation; 1992. vol. 3, p. 2315–2320.

  9. Boizot N, Gauthier J. Motion planning for kinematic systems. IEEE Trans Autom Control. 2013;58(6):1430–1442.

    Article  MathSciNet  Google Scholar 

  10. Bonnard B, Jurdjevic V, Kupka I, Sallet G. Transitivity of families of invariant vector fields on the semidirect products of Lie groups. Trans Amer Math Soc. 1982;271(2):525–535.

    Article  MATH  MathSciNet  Google Scholar 

  11. Boscain U, Chitour Y. Time-optimal synthesis for left-invariant control systems on S O(3). SIAM J Control Optim. 2005;44(1):111–139 . (electronic).

    Article  MATH  MathSciNet  Google Scholar 

  12. Boscain U, Piccoli B. Extremal synthesis for generic planar systems. J Dyn Control Syst. 2001;7(2):209–258.

    Article  MATH  MathSciNet  Google Scholar 

  13. Boscain U, Piccoli B. Optimal syntheses for control systems on 2-D manifolds, Vol. 43. Mathématiques and applications. Berlin: Springer; 2004.

  14. Bressan A, Piccoli B. A generic classification of time-optimal planar stabilizing feedbacks. SIAM J Control Optim. 1998;36(1):12–32. (electronic).

    Article  MATH  MathSciNet  Google Scholar 

  15. Bui X-N, Boissonnat J-d, Soueres P, Laumond J-P. Shortest path synthesis for Dubins non-holonomic robot. In: 1994 IEEE international conference on robotics and automation, Proceedings;1994. vol. 1 p. 2–7.

  16. Bui X-N, Soueres P, Boissonnat J-D, Laumond J-P. The shortest path synthesis for non-holonomic robots moving forwards. Rapport de recherche RR-2153. Nice: INRIA; 1994.

    Google Scholar 

  17. Bullo F, Lewis A D. Geometric control of mechanical systems, Vol. 49. Texts in Applied Mathematics. Modeling, analysis, and design for simple mechanical control systems. New York: Springer; 2005.

  18. Campolo D, Schenato L, Guglielmelli E, Sastry S. A Lyapunov-based approach for the control of biomimetic robotic systems with periodic forcing inputs. In: Proceedings of 16th IFAC World congress on automatic control (IFAC05); 2005.

  19. Chen H, Chang K, Agate CS. Tracking with UAV using tangent-plus-Lyapunov vector field guidance. In: 12th International conference on information fusion. FUSION ’09; 2009. p. 363–372.

  20. Chitour Y, Sigalotti M. Dubins’ problem on surfaces. I. Nonnegative curvature. J Geom Anal. 2005;15(4):565–587.

    Article  MATH  MathSciNet  Google Scholar 

  21. Chitsaz H, LaValle S. Time-optimal paths for a Dubins airplane. In: 46th IEEE conference on decision and control; 2007. p. 2379–2384.

  22. Cichella V, Kaminer I, Dobrokhodov V, Xargay E, Hovakimyan N, Pascoal A. Geometric 3D path-following control for a fixed-wing UAV on SO(3). In: AIAA guidance, navigation, and control conference. American Institute of Aeronautics and Astronautics, 2013/05/02; 2011.

  23. Dimarogonas D V, Loizou S G, Kyriakopoulos K J, Zavlanos M M. A feedback stabilization and collision avoidance scheme for multiple independent non-point agents. Automatica 2006;42(2):229–243.

    Article  MATH  MathSciNet  Google Scholar 

  24. Ding X, Rahmani A, Egerstedt M. Multi-UAV convoy protection: an optimal approach to path planning and coordination. IEEE Trans Robot. 2010;26(2):256–268.

    Article  Google Scholar 

  25. Dubins L E. On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents. Amer J Math. 1957; 79:497–516.

    Article  MATH  MathSciNet  Google Scholar 

  26. Frew E, Lawrence D, Dixon C, Elston J, Pisano W. Lyapunov guidance vector fields for unmanned aircraft applications. In: American Control Conference. ACC’07; 2007. p. 371–376.

  27. Frew E, McGee T, Kim Z, Xiao X, Jackson S, Morimoto M, Rathinam S, Padial J, Sengupta R. Vision-based road-following using a small autonomous aircraft. In: Aerospace conference. Proceedings. 2004 IEEE; 2004. vol. 5, p. 3006–3015.

  28. Frew E W, Lawrence DA, Morris S. Coordinated standoff tracking of moving targets using Lyapunov guidance vector fields. J Guid Control Dyn. 2008;31(2):290–306. 2013/02/21.

    Article  Google Scholar 

  29. Gauthier J-P, Kupka I. Deterministic observation theory and applications. Cambridge: Cambridge University Press; 2001.

    Book  MATH  Google Scholar 

  30. Gauthier J-P, Zakalyukin V. On the one-step-bracket-generating motion planning problem. J Dyn Control Syst. 2005;11(2):215–235.

    Article  MATH  MathSciNet  Google Scholar 

  31. Goerzen C, Kong Z, Mettler B. A survey of motion planning algorithms from the perspective of autonomous UAV guidance. J Intell Robot Syst. 2010;57(1–4):65–100.

    Article  MATH  Google Scholar 

  32. Hua M-D, Hamel T, Morin P, Samson C. A control approach for thrust-propelled underactuated vehicles and its application to VTOL drones. IEEE Trans Autom Control 2009;54(8):1837–1853.

    Article  MathSciNet  Google Scholar 

  33. Kokkeby K L, Lutter R P, Munoz M L, Cathey F W, Hilliard J D, Olson T L. System and methods for autonomous tracking and surveillance, Vol. 06;2009.

  34. LaSalle J. Stability theory for ordinary differential equations. J Diff Equ. 1968;4(1):57–65.

    Article  MATH  MathSciNet  Google Scholar 

  35. Laumond J-P (ed). Robot motion planning and control, Vol. 229. Lecture notes in control and information sciences. London: Springer; 1998. http://www.laas.fr/~jpl/book.html.

  36. Lawrence D, Frew E, Pisano W. Lyapunov vector fields for autonomous UAV flight control. In: AIAA guidance, navigation and control conference and exhibit. American Institute of Aeronautics and Astronautics, 2013/02/21;2007.

  37. Lee J, Huang R, Vaughn A, Xiao X, Hedrick J K, Zennaro M, Sengupta R. Strategies of path-planning for a UAV to track a ground vehicle. AINS Conference; 2003.

  38. Park S, Deyst J, How JP. A new nonlinear guidance logic for trajectory tracking. In: AIAA guidance, navigation, and control conference and exhibit. American Institute of Aeronautics and Astronautics; 2004.

  39. Piccoli B. Classification of generic singularities for the planar time-optimal synthesis. SIAM J Control Optim. 1996;34(6):1914–1946.

    Article  MATH  MathSciNet  Google Scholar 

  40. Piccoli B, Sussmann H J. Regular synthesis and sufficiency conditions for optimality. SIAM J Control Optim. 2000;39(2):359–410. (electronic).

    Article  MATH  MathSciNet  Google Scholar 

  41. Prévost C G, Desbiens A, Gagnon D, Hodouin E. UAV optimal cooperative target tracking and collision avoidance of moving objects, Vol. 17. In: Chung M, Jin, Misra, Pradeep, editors. The International Federation of Automatic Control; 2008, pp. 5724–5729.

  42. Rafi F, Khan S, Shafiq K, Shah M. Autonomous target following by unmanned aerial vehicles. Unmanned Syst Tech VIII. 2006;6230(1):623010.

    Article  Google Scholar 

  43. Sachkov Y. Control theory on lie groups. J Math Sci. 2009;156:381–439.

    Article  MATH  MathSciNet  Google Scholar 

  44. Schättler H. On the local structure of time-optimal bang-bang trajectories in R 3. SIAM J Control Optim. 1988;26(1):186–204.

    Article  MATH  MathSciNet  Google Scholar 

  45. Sebesta K, Boizot N. A real-time adaptive high-gain EKF, applied to a quadcopter inertial navigation system. IEEE Trans Indus Elec. 2014;61(1):495–503.

    Article  Google Scholar 

  46. Sigalotti M, Chitour Y. Dubins’ problem on surfaces. II. Nonpositive curvature. SIAM J Control Optims 2006;45(2):457–482. (electronic).

    Article  MathSciNet  Google Scholar 

  47. Souères P, Balluchi A, Bicchi A. Optimal feedback control for route tracking with a bounded-curvature vehicle. Int J Control. 2001;74(10):1009–1019.

    Article  MATH  Google Scholar 

  48. Sussmann H J. Regular synthesis for time-optimal control of single-input real analytic systems in the plane. SIAM J Control Optim. 1987;25(5):1145–1162.

    Article  MathSciNet  Google Scholar 

  49. Theodorakopoulos P, Lacroix S. A strategy for tracking a ground target with a UAV. In: IEEE/RSJ international conference on intelligent robots and systems. IROS; 2008. p. 1254–1259.

  50. Walsh G, Montgomery R, Sastry S. Optimal path planning on matrix lie groups. In: Proceedings of the 33rd IEEE conference on decision and control 1994. vol. 2, p. 1258–1263.

  51. Xargay E, Dobrokhodov V, Kaminer I, Pascoal A, Hovakimyan N, Cao C. Time-critical cooperative control of multiple autonomous vehicles: robust distributed strategies for path-following control and time-coordination over dynamic communications networks. IEEE Control Syst. 2012;32(5):49–73.

    Article  MathSciNet  Google Scholar 

  52. Zhu S, Wang D, Low C. Ground target tracking using UAV with input constraints. J Intell Robot Syst. 2013;69:417–429.

    Article  Google Scholar 

Download references

Acknowledgments

This research has been supported by the French FUI SHARE project (see [3]), supported by a consortium of companies and research labs (Opéra Ergonomie, ONERA, Thales Alénia Space, Eurocopter, Adetel group) and by the European Research Council, ERC StG 2009 “GeCoMethods”, contract number 239748.

Author information

Authors and Affiliations

  1. Université de Toulon, LSIS, UMR CNRS 7296, B.P 20132, 83957, La Garde Cedex, France

    Thibault Maillot & Jean-Paul Gauthier

  2. CNRS, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), UMR CNRS 7641, Route de Saclay, 91128, Palaiseau Cedex, France

    Ugo Boscain

  3. INRIA GECO Project, La Garde Cedex, France

    Ugo Boscain & Jean-Paul Gauthier

  4. Université Claude Bernard Lyon 1, LAGEP, UMR CNRS 5007, 43 bd du 11 novembre 1918, 69100, Villeurbanne, France

    Ulysse Serres

Authors
  1. Thibault Maillot
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ugo Boscain
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jean-Paul Gauthier
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ulysse Serres
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ulysse Serres.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Maillot, T., Boscain, U., Gauthier, JP. et al. Lyapunov and Minimum-Time Path Planning for Drones. J Dyn Control Syst 21, 47–80 (2015). https://doi.org/10.1007/s10883-014-9222-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10883-014-9222-y

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation