On the Length of Dubins Path with Any Initial and Terminal Configurations

Li Dai; Zheng Xie

doi:doi:10.11648/j.pamj.20150406.14

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On the Length of Dubins Path with Any Initial and Terminal Configurations

Li Dai, Zheng Xie

Published in Pure and Applied Mathematics Journal (Volume 4, Issue 6)

Received: 27 September 2015 Accepted: 8 October 2015 Published: 21 October 2015

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Abstract

Dubins has proved in 1957 that the minimum length path between an initial and a terminal configuration can be found among the six paths {LSL, RSR, LSR, RSL, RLR, LRL}. Skel and Lumelsky have studied the length of Dubins path with the initial configuration (0, 0; α) and the terminal configuration (d, 0; β) and the minimal turning radius ρ=1 in 2001. We extended the Skel and Lumelsky’s results to the case that the initial and terminal configuration is(x₀, y₀, α), (x₁, y₁, β), respectively (where x₀, y₀, x₁, y₁ϵℝ), and the minimal turning radius is ρ>0.

Published in	Pure and Applied Mathematics Journal (Volume 4, Issue 6)
DOI	10.11648/j.pamj.20150406.14
Page(s)	248-254
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Dubins Set, Dubins Path, Dubins Traveling Salesman Problem, Computational Geometry

References

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[4]	J. Barraqu and, J. C. Latombe. On nonholonomic mobile robots and optimal maneuvering. Proceedings of the Fourth International Symposium on Intelligent Control, Albany, NY, 1989.
[5]	J. T. Isaacs and J. P. Hespanha. Dubins traveling salesman problem with neighborhoods: a graph-based approach. Algorithms, 6(2013): 84-99.
[6]	B. Yuan, M. Orlowska and S. Sadiq. On the optimal robot routing problem in wireless sendor networks. IEEE Trans. Data Eng. 19(2007):1252-1261.
[7]	K. J. Obermeyer. Path planning for a UAV performing reconnaissance of static ground targets in terrain. Proceedings of the AIAA Conference on Guidance, Navigation, and Control, Chicago, Illinois, USA, August 2009: 10-13.
[8]	L. E. Dubins. On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and Tangents. American Journal of Mathematics, 79 (1957) 497-516.
[9]	A. M. Shkel and V. Lumelsky. Classification of the Dubins set. Robotics and Autonomous Systems, 34(2001)179-202.
[10]	J. L. Ny, E. Frazzoli and E. Feron. The Curvature-constrained Traveling Salesman Problem for High Point Densities. Proceedings of the IEEE Conference on Decision and Control, New Orleans, Louisiana, USA, 12-14 December 2007:5985-5990.
[11]	K. Savla, E. Frazzoli and F. Bullo. Traveling salesperson problems for the Dubins vehicle. IEEETrans. Autom. Control 53(2008) 1378-1391.
[12]	X. Ma and D. Castanon. Receding Horizon Planning for Dubins Traveling Salesman Problems. Proceedings of the IEEE Conference on Decision and Control, San Diego, California, USA, 13–15December 2006; pp. 5453-5458.
[13]	S. Rathinam, R. Sengupta and S. Darbha. A resource allocation algorithm for multiple vehiclesystems with non-holnomic constraints. IEEE Trans. Autom. Sci. Eng. 4 (2007) 98–104.
[14]	L. N. Jerone, J. Eric and F. Emilio. On the Dubins Traveling Salesman Problem. IEEE Transactions on Automatic Control, 57(2012):265-270.
[15]	I. Pantelis and S. Tal. Motion planning algorithms for the Dubins Traveling Salesperson Problem. Automatica, 53(2015)247-255.
[16]	E. B. Sylvester, K. David and P. Valentin. Discrete Dubins Paths. Ar Xiv: 1211.2365v1 [cs.DM] 11 Nov 2012.

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APA Style

Li Dai, Zheng Xie. (2015). On the Length of Dubins Path with Any Initial and Terminal Configurations. Pure and Applied Mathematics Journal, 4(6), 248-254. https://doi.org/10.11648/j.pamj.20150406.14

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ACS Style

Li Dai; Zheng Xie. On the Length of Dubins Path with Any Initial and Terminal Configurations. Pure Appl. Math. J. 2015, 4(6), 248-254. doi: 10.11648/j.pamj.20150406.14

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AMA Style

Li Dai, Zheng Xie. On the Length of Dubins Path with Any Initial and Terminal Configurations. Pure Appl Math J. 2015;4(6):248-254. doi: 10.11648/j.pamj.20150406.14

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@article{10.11648/j.pamj.20150406.14,
  author = {Li Dai and Zheng Xie},
  title = {On the Length of Dubins Path with Any Initial and Terminal Configurations},
  journal = {Pure and Applied Mathematics Journal},
  volume = {4},
  number = {6},
  pages = {248-254},
  doi = {10.11648/j.pamj.20150406.14},
  url = {https://doi.org/10.11648/j.pamj.20150406.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20150406.14},
  abstract = {Dubins has proved in 1957 that the minimum length path between an initial and a terminal configuration can be found among the six paths {LSL, RSR, LSR, RSL, RLR, LRL}. Skel and Lumelsky have studied the length of Dubins path with the initial configuration (0, 0; α) and the terminal configuration (d, 0; β) and the minimal turning radius ρ=1 in 2001. We extended the Skel and Lumelsky’s results to the case that the initial and terminal configuration is(x0, y0, α), (x1, y1, β), respectively (where x0, y0, x1, y1ϵℝ), and the minimal turning radius is ρ>0.},
 year = {2015}
}

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TY  - JOUR
T1  - On the Length of Dubins Path with Any Initial and Terminal Configurations
AU  - Li Dai
AU  - Zheng Xie
Y1  - 2015/10/21
PY  - 2015
N1  - https://doi.org/10.11648/j.pamj.20150406.14
DO  - 10.11648/j.pamj.20150406.14
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 248
EP  - 254
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20150406.14
AB  - Dubins has proved in 1957 that the minimum length path between an initial and a terminal configuration can be found among the six paths {LSL, RSR, LSR, RSL, RLR, LRL}. Skel and Lumelsky have studied the length of Dubins path with the initial configuration (0, 0; α) and the terminal configuration (d, 0; β) and the minimal turning radius ρ=1 in 2001. We extended the Skel and Lumelsky’s results to the case that the initial and terminal configuration is(x0, y0, α), (x1, y1, β), respectively (where x0, y0, x1, y1ϵℝ), and the minimal turning radius is ρ>0.
VL  - 4
IS  - 6
ER  -

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Author Information

Li Dai

College of Science, National University of Defense Technology, Changsha, Hunan, China
Zheng Xie

College of Science, National University of Defense Technology, Changsha, Hunan, China

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APA Style

Li Dai, Zheng Xie. (2015). On the Length of Dubins Path with Any Initial and Terminal Configurations. Pure and Applied Mathematics Journal, 4(6), 248-254. https://doi.org/10.11648/j.pamj.20150406.14

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ACS Style

Li Dai; Zheng Xie. On the Length of Dubins Path with Any Initial and Terminal Configurations. Pure Appl. Math. J. 2015, 4(6), 248-254. doi: 10.11648/j.pamj.20150406.14

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AMA Style

Li Dai, Zheng Xie. On the Length of Dubins Path with Any Initial and Terminal Configurations. Pure Appl Math J. 2015;4(6):248-254. doi: 10.11648/j.pamj.20150406.14

Copy | Download

@article{10.11648/j.pamj.20150406.14,
  author = {Li Dai and Zheng Xie},
  title = {On the Length of Dubins Path with Any Initial and Terminal Configurations},
  journal = {Pure and Applied Mathematics Journal},
  volume = {4},
  number = {6},
  pages = {248-254},
  doi = {10.11648/j.pamj.20150406.14},
  url = {https://doi.org/10.11648/j.pamj.20150406.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20150406.14},
  abstract = {Dubins has proved in 1957 that the minimum length path between an initial and a terminal configuration can be found among the six paths {LSL, RSR, LSR, RSL, RLR, LRL}. Skel and Lumelsky have studied the length of Dubins path with the initial configuration (0, 0; α) and the terminal configuration (d, 0; β) and the minimal turning radius ρ=1 in 2001. We extended the Skel and Lumelsky’s results to the case that the initial and terminal configuration is(x0, y0, α), (x1, y1, β), respectively (where x0, y0, x1, y1ϵℝ), and the minimal turning radius is ρ>0.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - On the Length of Dubins Path with Any Initial and Terminal Configurations
AU  - Li Dai
AU  - Zheng Xie
Y1  - 2015/10/21
PY  - 2015
N1  - https://doi.org/10.11648/j.pamj.20150406.14
DO  - 10.11648/j.pamj.20150406.14
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 248
EP  - 254
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20150406.14
AB  - Dubins has proved in 1957 that the minimum length path between an initial and a terminal configuration can be found among the six paths {LSL, RSR, LSR, RSL, RLR, LRL}. Skel and Lumelsky have studied the length of Dubins path with the initial configuration (0, 0; α) and the terminal configuration (d, 0; β) and the minimal turning radius ρ=1 in 2001. We extended the Skel and Lumelsky’s results to the case that the initial and terminal configuration is(x0, y0, α), (x1, y1, β), respectively (where x0, y0, x1, y1ϵℝ), and the minimal turning radius is ρ>0.
VL  - 4
IS  - 6
ER  -

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