Skip to main content

Safe Smooth Paths Between Straight Line Obstacles

  • Chapter
  • First Online:
Logics and Type Systems in Theory and Practice

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14560))

  • 218 Accesses

Abstract

We describe a collections of algorithm to compute smooth trajectories between obstacles, with the objective that the obtained trajectories should be smooth. In particular, we use a vertical cell decomposition algorithm to avoid the obstacles, a best first search algorithm to obtain trajectory sketches and Bézier curves to implement smoothness. We also provide some insights into the correctness arguments for these algorithms. These correctness arguments are intended for use in a formal proof. While we have running implementations of the full program, the formal proofs of correctness are still incomplete.

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

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Basu, S., Pollack, R., Roy M.F.: Algorithms in Real Algebraic Geometry, volume 10 of Algorithms and Computation in Mathematics, 2nd edn. Springer, Heidelberg (2006). https://doi.org/10.1007/3-540-33099-2

  2. Bertot, Y.: Formal Verification of a geometry algorithm: a quest for abstract views and symmetry in coq proofs. In: Fischer, B., Uustalu, T. (eds.) ICTAC 2018, vol. 11187, pp. 3–10. Springer, Heidelberg (2018). https://doi.org/10.1007/978-3-030-02508-3_1

    Chapter  Google Scholar 

  3. Bertot, Y., Guilhot, F., Mahboubi, A.: A formal study of Bernstein coefficients and polynomials. Math. Struct. Comput. Sci. 21(04), 731–761 (2011)

    Article  MathSciNet  Google Scholar 

  4. Cohen, C., Mahboubi, A.: Formal proofs in real algebraic geometry: from ordered fields to quantifier elimination. Logical Methods Comput. Sci. 8(102), 1–40 (2012)

    MathSciNet  Google Scholar 

  5. Dijkstra, E.W.: A note on two problems in connexion with graphs. Numer. Math. 1(1), 269–271 (1959)

    Article  MathSciNet  Google Scholar 

  6. Dufourd, J.F., Bertot, Y.: Formal study of plane Delaunay triangulation. In: Paulson, L., Kaufmann, M. (eds.) ITP 2010, vol. 6172, pp. 211–226. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14052-5_16

    Chapter  Google Scholar 

  7. Geuvers, H., Wiedijk, F., Zwanenburg, J.: A constructive proof of the fundamental theorem of algebra without using the rationals. In: Callaghan, P., Luo, Z., McKinna, J., Pollack, R. (eds.) TYPES 2000. LNCS, vol. 2277, pp. 96–111. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-45842-5_7

    Chapter  Google Scholar 

  8. Knuth, D.: Axioms and Hulls. Number 606 in Lecture Notes in Computer Science. Springer-Verlag, Heidelberg (1991). DOI: https://doi.org/10.1007/3-540-55611-7

  9. Latombe, J.-C.: Robot Motion Planning. Kluwer Academic Publishers, Norwell (1991)

    Book  Google Scholar 

  10. Pichardie, D., Bertot, Y.: Formalizing convex hull algorithms. In: Boulton, R.J., Jackson, P.B. (eds.) TPHOLs 2001. LNCS, vol. 2152, pp. 346–361. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44755-5_24

    Chapter  Google Scholar 

  11. Rizaldi, A., Immler, F., Schürmann, B., Althoff, M.: A Formally Verified Motion Planner for Autonomous Vehicles. In: Lahiri, S.K., Wang, C. (eds.) ATVA 2018. LNCS, vol. 11138, pp. 75–90. Springer, Heidelberg (2018). https://doi.org/10.1007/978-3-030-01090-4_5

    Chapter  Google Scholar 

  12. Zsidó, J.: Theorem of Three Circles in Coq. J. Autom. Reason. 53(2), 105–127 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The initial work on the vertical cell decomposition algorithms was done by Thomas Portet. Studies of potential collisions between Bézier curves and straight line segments were done by Quentin Vermande. Laurent Théry added the possibility to visualize the results on a web page.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yves Bertot .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Bertot, Y. (2024). Safe Smooth Paths Between Straight Line Obstacles. In: Capretta, V., Krebbers, R., Wiedijk, F. (eds) Logics and Type Systems in Theory and Practice. Lecture Notes in Computer Science, vol 14560. Springer, Cham. https://doi.org/10.1007/978-3-031-61716-4_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-61716-4_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-61715-7

  • Online ISBN: 978-3-031-61716-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics