## “Continuous-Curvature Path Generation Using Fermat's Spiral”Authors: Anastasios M. Lekkas, Andreas Reason Dahl, Morten Breivik and Thor I. Fossen,
Affiliation: NTNU, Centre for Ships and Ocean Structures, NTNU, Department of Marine Technology, NTNU, Department of Engineering Cybernetics and NTNU, Centre for Autonomous Marine Operations and Systems
Reference: 2013, Vol 34, No 4, pp. 183-198. |

**Keywords:**Path planning, Fermat's spiral, continuous curvature, parametric curve, path tracking

**Abstract:**This paper proposes a novel methodology, based on Fermat's spiral (FS), for constructing curvature-continuous parametric paths in a plane. FS has a zero curvature at its origin, a property that allows it to be connected with a straight line smoothly, that is, without the curvature discontinuity which occurs at the transition point between a line and a circular arc when constructing Dubins paths. Furthermore, contrary to the computationally expensive clothoids, FS is described by very simple parametric equations that are trivial to compute. On the downside, computing the length of an FS arc involves a Gaussian hypergeometric function. However, this function is absolutely convergent and it is also shown that it poses no restrictions to the domain within which the length can be calculated. In addition, we present an alternative parametrization of FS which eliminates the parametric speed singularity at the origin, hence making the spiral suitable for path-tracking applications. A detailed description of how to construct curvature-continuous paths with FS is given.

PDF (889 Kb) DOI: 10.4173/mic.2013.4.3

**DOI forward links to this article:**

[1] Daniel de A. Fernandes, Asgeir J. Sørensen and Decio C. Donha (2015), doi:10.4173/mic.2015.2.2 | |

[2] Anastasios M. Lekkas and Thor I. Fossen (2014), doi:10.1109/TCST.2014.2306774 | |

[3] Jinhong Noh, Jaehyung Park, Haiyun Wang, Jonghun Park, Seongryong Chang and Ukyoul Huh (2015), doi:10.1109/ICIT.2015.7125154 | |

[4] Anastasios M. Lekkas, Ann L. Roald and Morten Breivik (2016), doi:10.1016/j.ifacol.2016.10.313 | |

[5] Ibrahim A. Hameed (2017), doi:10.1007/978-3-319-48308-5_86 | |

[6] Mauro Candeloro, Anastasios M. Lekkas and Asgeir J. Sørensen (2017), doi:10.1016/j.conengprac.2017.01.007 | |

[7] Artur Wolek and Craig A. Woolsey (2017), doi:10.1007/978-3-319-55372-6_9 |

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**BibTeX:**

@article{MIC-2013-4-3,

title={{Continuous-Curvature Path Generation Using Fermat's Spiral}},

author={Lekkas, Anastasios M. and Dahl, Andreas Reason and Breivik, Morten and Fossen, Thor I.},

journal={Modeling, Identification and Control},

volume={34},

number={4},

pages={183--198},

year={2013},

doi={10.4173/mic.2013.4.3},

publisher={Norwegian Society of Automatic Control}

};