Skip to main content
SpringerLink
Log in

Tightly Bounding the Shortest Dubins Paths Through a Sequence of Points

  • Published:
Journal of Intelligent & Robotic Systems Aims and scope Submit manuscript

Abstract

This article addresses an important path planning problem for robots and Unmanned Aerial Vehicles (UAVs), which is to find the shortest path of bounded curvature passing through a given sequence of target points on a ground plane. Currently, no algorithm exists that can compute an optimal solution to this problem. Therefore, tight lower bounds are vital in determining the quality of any feasible solution to this problem. Novel tight lower bounding algorithms are presented in this article by relaxing some of the heading angle constraints at the target points. The proposed approach requires us to solve variants of an optimization problem called the Dubins interval problem between two points where the heading angles at the points are constrained to be within a specified interval. These variants are solved using tools from optimal control theory. Using these approaches, two lower bounding algorithms are presented and these bounds are then compared with existing results in the literature. Computational results are presented to corroborate the performance of the proposed algorithms; the average reduction in the difference between upper bounds and lower bounds is 80 % to 85 % with respect to the trivial Euclidean lower bounds.

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

Similar content being viewed by others

References

  1. Agarwal, P.K., Raghavan, P., Tamaki, H.: Motion planning for a steering-constrained robot through moderate obstacles. In: Symposium on Theory of Computing, pp 343–352. ACM (1995)

  2. Agarwal, P.K., Wang, H.: Approximation algorithms for curvature-constrained shortest paths. SIAM J. Comput. 30(6), 1739–1772 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Boissonnat, J.D., Cérézo, A., Leblond, J.: Shortest paths of bounded curvature in the plane. J. Intell. Robot. Syst. 11(1-2), 5–20 (1994). doi:10.1007/BF01258291

    Article  MATH  Google Scholar 

  4. Boissonnat, J.D., Lazard, S.: A polynomial-time algorithm for computing shortest paths of bounded curvature amidst moderate obstacles. Int. J. Comput. Geom. Appl. 13(03), 189–229 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bui, X.N., Boissonnat, J.D., Soueres, P., Laumond, J.P.: Shortest path synthesis for Dubins non-holonomic robot. In: IEEE International Conference on Robotics and Automation, 1994. Proceedings, 1994. doi:10.1109/ROBOT.1994.351019, vol. 1, pp 2–7 (1994)

  6. Epstein, C., Cohen, I., Shima, T.: On the discretized dubins traveling salesman problem. Technical Report (2014)

  7. Goaoc, X., Kim, H.S., Lazard, S.: Bounded-curvature shortest paths through a sequence of points using convex optimization. SIAM J. Comput. 42(2), 662–684 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kenefic, R.J.: Finding good dubins tours for uavs using particle swarm optimization. J. Aerosp. Comput. Inf. Commun. 5(2), 47–56 (2008)

    Article  Google Scholar 

  9. Le Ny, J., Feron, E., Frazzoli, E.: On the dubins traveling salesman problem. IEEE Trans. Autom. Control 57(1), 265–270 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dubins, L.E.: On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents. Am. J. Math. 79(3), 487–516 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lee, J.H., Cheong, O., Kwon, W.C., Shin, S., Chwa, K.Y.: Approximation of curvature-constrained shortest paths through a sequence of points. In: Paterson, M. (ed.) Algorithms - ESA 2000, Lecture Notes in Computer Science, vol. 1879, pp 314–325. Springer, Berlin, Heidelberg (2000)

  12. Ma, X., Castañón, D., et al.: Receding horizon planning for Dubins traveling salesman problems. In: Conference on Decision and Control, pp 5453–5458. IEEE (2006)

  13. Macharet, D., Campos, M.: An orientation assignment heuristic to the Dubins traveling salesman problem. In: Bazzan, A.L., Pichara, K. (eds.) Advances in Artificial Intelligence – IBERAMIA 2014, Lecture Notes in Computer Science, vol. 8864, pp 457–468. Springer International Publishing (2014)

  14. Macharet, D.G., Alves Neto, A., da Camara Neto, V.F., Campos, M.F.: Efficient target visiting path planning for multiple vehicles with bounded curvature. In: IEEE International Conference on Intelligent Robots and Systems (IROS), pp 3830–3836 (2013)

  15. Macharet, D.G., Neto, A.A., da Camara Neto, V.F., Campos, M.F.: Nonholonomic path planning optimization for Dubins’ vehicles. In: IEEE International Conference on Robotics and Automation (Talk), pp 4208–4213 (2011)

  16. Macharet, D.G., Neto, A.A., da Camara Neto, V.F., Campos, M.F.: Data gathering tour optimization for Dubins’ vehicles. In: Congress on Evolutionary Computation (CEC), pp 1–8 (2012)

  17. Manyam, S., Rathinam, S.: A tight lower bounding procedure for the dubins traveling salesman problem. Presented at the International Symposium on Mathematical Programming (2015). arXiv:1506.08752

  18. Manyam, S., Rathinam, S., Casbeer, D.: Dubins paths through a sequence of points: lower and upper bounds. In: 2016 International Conference on Unmanned Aircraft Systems (ICUAS), pp 284–291. IEEE (2016)

  19. Manyam, S.G., Rathinam, S., Darbha, S.: Computation of lower bounds for a multiple depot, multiple vehicle routing problem with motion constraints. J. Dyn. Syst. Meas. Control. 137(9), 094,501 (2015)

    Article  Google Scholar 

  20. Manyam, S.G., Rathinam, S., Swaroop, D., Obermeyer, K.: Lower bounds for a vehicle routing problem with motion constraints. Int. J. Robot. Autom. 30(3) (2015)

  21. Manyam, S.G., Sivakumar R., Swaroop D.: Computation of lower bounds for a multiple depot, multiple vehicle routing problem with motion constraints. In: 52nd IEEE Conference on Decision and Control, Firenze, pp. 2378-2383 (2013)

  22. Medeiros, A.C., Urrutia, S.: Discrete optimization methods to determine trajectories for dubins’ vehicles. Electron Notes Discrete Math. 36, 17–24 (2010). doi:10.1016/j.endm.2010.05.003. {ISCO} 2010 - International Symposium on Combinatorial Optimization http://www. sciencedirect.com/science/article/pii/S1571065310000041

    Article  MATH  Google Scholar 

  23. Oberlin, P., Rathinam, S., Darbha, S.: Today’s traveling salesman problem. IEEE Robot. Autom. Mag. 17(4), 70–77 (2010)

    Article  Google Scholar 

  24. Pontryagin, L.S., Boltianski, V.G., Gamkrelidze, R.V., Mishchenko, E.F., Brown, D.E.: The Mathematical Theory of Optimal Processes. A Pergamon Press (1964). http://opac.inria.fr/record=b1122221

  25. Rathinam, S., Sengupta, R., Darbha, S.: A resource allocation algorithm for multivehicle systems with nonholonomic constraints. IEEE Trans. Autom. Sci. Eng. 4, 98–104 (2007). doi:10.1109/TASE.2006.872110

    Article  Google Scholar 

  26. Rathinam, S., Pramod K.: An Approximation Algorithm for a Shortest Dubins Path Problem. arXiv:1604.05064 (2016)

  27. Sadeghi, A., Smith, S.L.: On efficient computation of shortest dubins paths through three consecutive points. arXiv:1609.06662 (2016)

  28. Sujit, P., Hudzietz, B., Saripalli, S.: Route planning for angle constrained terrain mapping using an unmanned aerial vehicle. J. Intell. Robot. Syst. 69(1-4), 273–283 (2013)

    Article  Google Scholar 

  29. Sussmann, H.J., Tang, G.: Shortest paths for the reeds-shepp car: a worked out example of the use of geometric techniques in nonlinear optimal control. Rutgers Center for Systems and Control Technical Report 10, 1–71 (1991)

    Google Scholar 

  30. Tang, Z., Ozguner, U.: Motion planning for multitarget surveillance with mobile sensor agents. IEEE Trans. Robot. 21(5), 898–908 (2005). doi:10.1109/TRO.2005.847567

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Control Science Center of Excellence, Air Force Research Laboratory, WPAFB, OH, USA

    Satyanarayana G. Manyam, David Casbeer & Eloy Garcia

  2. Department of Mechanical Engineering, Texas A & M University, College Station, TX, USA

    Sivakumar Rathinam

Authors
  1. Satyanarayana G. Manyam
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sivakumar Rathinam
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. David Casbeer
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Eloy Garcia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Satyanarayana G. Manyam.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Manyam, S.G., Rathinam, S., Casbeer, D. et al. Tightly Bounding the Shortest Dubins Paths Through a Sequence of Points. J Intell Robot Syst 88, 495–511 (2017). https://doi.org/10.1007/s10846-016-0459-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10846-016-0459-4

Keywords

Search

Navigation