IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v73y2019i2d10.1007_s10589-019-00076-y.html
   My bibliography  Save this article

Markov–Dubins interpolating curves

Author

Listed:
  • C. Yalçın Kaya

    (University of South Australia)

Abstract

A realistic generalization of the Markov–Dubins problem, which is concerned with finding the shortest planar curve of constrained curvature joining two points with prescribed tangents, is the requirement that the curve passes through a number of prescribed intermediate points/nodes. We refer to this generalization as the Markov–Dubins interpolation problem. We formulate this interpolation problem as an optimal control problem and obtain results about the structure of its solution using optimal control theory. The Markov–Dubins interpolants consist of a concatenation of circular (C) and straight-line (S) segments. Abnormal interpolating curves are shown to exist and characterized; however, if the interpolating curve contains a straight-line segment then it cannot be abnormal. We derive results about the stationarity, or criticality, of the feasible solutions of certain structure. In particular, any feasible interpolant with arc types of CSC in each stage is proved to be stationary, i.e., critical. We propose a numerical method for computing Markov–Dubins interpolating paths. We illustrate the theory and the numerical approach by four qualitatively different examples.

Suggested Citation

  • C. Yalçın Kaya, 2019. "Markov–Dubins interpolating curves," Computational Optimization and Applications, Springer, vol. 73(2), pages 647-677, June.
    • Handle: RePEc:spr:coopap:v:73:y:2019:i:2:d:10.1007_s10589-019-00076-y
    DOI: 10.1007/s10589-019-00076-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-019-00076-y
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10589-019-00076-y?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. C. Kaya & Helmut Maurer, 2014. "A numerical method for nonconvex multi-objective optimal control problems," Computational Optimization and Applications, Springer, vol. 57(3), pages 685-702, April.
    2. C. Yalçın Kaya, 2017. "Markov–Dubins path via optimal control theory," Computational Optimization and Applications, Springer, vol. 68(3), pages 719-747, December.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Nguyen Thi Toan & Le Quang Thuy, 2023. "S-Derivative of the Extremum Multifunction to a Multi-objective Parametric Discrete Optimal Control Problem," Journal of Optimization Theory and Applications, Springer, vol. 196(1), pages 240-265, January.
    2. Bui Trong Kien & Trinh Duy Binh, 2023. "On the second-order optimality conditions for multi-objective optimal control problems with mixed pointwise constraints," Journal of Global Optimization, Springer, vol. 85(1), pages 155-183, January.
    3. Owen Davis & Siavash Radpour, 2022. "Older Workers’ Wages Are Growing—But Not Fast Enough," SCEPA publication series. 2022-02, Schwartz Center for Economic Policy Analysis (SCEPA), The New School.
    4. Stephan Dempe & Gabriele Eichfelder & Jörg Fliege, 2015. "On the effects of combining objectives in multi-objective optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 82(1), pages 1-18, August.
    5. C. Yalçın Kaya, 2020. "Optimal Control of the Double Integrator with Minimum Total Variation," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 966-981, June.
    6. Cui, Yunfei & Geng, Zhiqiang & Zhu, Qunxiong & Han, Yongming, 2017. "Review: Multi-objective optimization methods and application in energy saving," Energy, Elsevier, vol. 125(C), pages 681-704.
    7. Regina S. Burachik & Alexander C. Kalloniatis & C. Yalçın Kaya, 2021. "Sparse Network Optimization for Synchronization," Journal of Optimization Theory and Applications, Springer, vol. 191(1), pages 229-251, October.
    8. C. Yalçın Kaya, 2017. "Markov–Dubins path via optimal control theory," Computational Optimization and Applications, Springer, vol. 68(3), pages 719-747, December.
    9. Andreas Lichtenberger & Joao Paulo Braga & Willi Semmler, 2022. "Green Bonds for the Transition to a Low-Carbon Economy," Econometrics, MDPI, vol. 10(1), pages 1-31, March.
    10. Andreas Lichtenberger & Joao Paulo Braga & Willi Semmler, 2022. "Green Bonds for the Transition to a Low-Carbon Economy," SCEPA working paper series. 2022-02, Schwartz Center for Economic Policy Analysis (SCEPA), The New School.
    11. Nguyen Thi Toan & Le Quang Thuy & Nguyen Tuyen & Yi-Bin Xiao, 2021. "Second-order KKT optimality conditions for multiobjective discrete optimal control problems," Journal of Global Optimization, Springer, vol. 79(1), pages 203-231, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:coopap:v:73:y:2019:i:2:d:10.1007_s10589-019-00076-y. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.