Continuous-Curvature Path Generation Using Fermat's Spiral
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 discontinuit...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Norwegian Society of Automatic Control
2013-10-01
|
| Series: | Modeling, Identification and Control |
| Subjects: | |
| Online Access: | http://www.mic-journal.no/PDF/2013/MIC-2013-4-3.pdf |
| _version_ | 1818387861419851776 |
|---|---|
| author | Anastasios M. Lekkas Andreas Reason Dahl Morten Breivik Thor I. Fossen |
| author_facet | Anastasios M. Lekkas Andreas Reason Dahl Morten Breivik Thor I. Fossen |
| author_sort | Anastasios M. Lekkas |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-14T04:16:40Z |
| format | Article |
| id | doaj.art-bb0e1338a7fb470fb26ac473d6a51c47 |
| institution | Directory Open Access Journal |
| issn | 0332-7353 1890-1328 |
| language | English |
| last_indexed | 2024-12-14T04:16:40Z |
| publishDate | 2013-10-01 |
| publisher | Norwegian Society of Automatic Control |
| record_format | Article |
| series | Modeling, Identification and Control |
| spelling | doaj.art-bb0e1338a7fb470fb26ac473d6a51c472022-12-21T23:17:31ZengNorwegian Society of Automatic ControlModeling, Identification and Control0332-73531890-13282013-10-0134418319810.4173/mic.2013.4.3Continuous-Curvature Path Generation Using Fermat's SpiralAnastasios M. LekkasAndreas Reason DahlMorten BreivikThor I. FossenThis 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.http://www.mic-journal.no/PDF/2013/MIC-2013-4-3.pdfPath planningFermat's spiralcontinuous curvatureparametric curvepath tracking |
| spellingShingle | Anastasios M. Lekkas Andreas Reason Dahl Morten Breivik Thor I. Fossen Continuous-Curvature Path Generation Using Fermat's Spiral Modeling, Identification and Control Path planning Fermat's spiral continuous curvature parametric curve path tracking |
| title | Continuous-Curvature Path Generation Using Fermat's Spiral |
| title_full | Continuous-Curvature Path Generation Using Fermat's Spiral |
| title_fullStr | Continuous-Curvature Path Generation Using Fermat's Spiral |
| title_full_unstemmed | Continuous-Curvature Path Generation Using Fermat's Spiral |
| title_short | Continuous-Curvature Path Generation Using Fermat's Spiral |
| title_sort | continuous curvature path generation using fermat s spiral |
| topic | Path planning Fermat's spiral continuous curvature parametric curve path tracking |
| url | http://www.mic-journal.no/PDF/2013/MIC-2013-4-3.pdf |
| work_keys_str_mv | AT anastasiosmlekkas continuouscurvaturepathgenerationusingfermatsspiral AT andreasreasondahl continuouscurvaturepathgenerationusingfermatsspiral AT mortenbreivik continuouscurvaturepathgenerationusingfermatsspiral AT thorifossen continuouscurvaturepathgenerationusingfermatsspiral |