Abstract
We study the motion-planning problem for a car-like robot whose turning radius is bounded from below by one and which is allowed to move in the forward direction only (Dubins car). For two robot configurations σ, σ′, let ℓ(σ, σ′) be the length of a shortest bounded-curvature path from σ to σ′ without obstacles. For d ≥ 0, let ℓ(d) be the supremum of ℓ(σ, σ′), over all pairs (σ, σ′) that are at Euclidean distance d. We study the function dub(d) = ℓ(d) − d, which expresses the difference between the bounded-curvature path length and the Euclidean distance of its endpoints. We show that dub(d) decreases monotonically from dub(0) = 7π/3 to dub(d ∗ ) = 2π, and is constant for d ≥ d ∗ . Here d ∗ ≈ 1.5874. We describe pairs of configurations that exhibit the worst-case of dub(d) for every distance d.
This research is supported in part by Mid-career Researcher Program through NRF grant (No. R01-2008-000-11607-0) funded by the MEST and in part by SRC-GAIA through NRF grant (No. 2011-0030044) funded by the Government of Korea.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Dubins, L.E.: On curves of minimal length with a constraint on average curvature, and prescribed initial and terminal positions and tangents. American Journal of Mathematics 79(3), 497–516 (1957)
Goaoc, X., Kim, H.-S., Lazard, S.: Bounded-curvature shortest paths through a sequence of points. Technical Report inria-00539957, INRIA (2010), http://hal.inria.fr/inria-00539957/en
Kim, H.-S., Cheong, O.: The cost of bounded curvature (2011), http://arxiv.org/abs/1106.6214
Kim, J.-H.: The upper bound of bounded curvature path. Master’s thesis, KAIST (2008)
LaValle, S.M.: Planning Algorithms. Cambridge University Press (2006)
Lee, J.-H., Cheong, O., Kwon, W.-C., Shin, S.-Y., Chwa, K.-Y.: Approximation of Curvature-Constrained Shortest Paths through a Sequence of Points. In: Paterson, M. (ed.) ESA 2000. LNCS, vol. 1879, pp. 314–325. Springer, Heidelberg (2000)
Ma, X., Castañón, D.A.: Receding horizon planning for Dubins traveling salesman problems. In: 45th IEEE Conference on Decision and Control, pp. 5453–5458 (December 2006)
Le Ny, J., Feron, E., Frazzoli, E.: The curvature-constrained traveling salesman problem for high point densities. In: Proceedings of the 46th IEEE Conference on Decision and Control, pp. 5985–5990 (2007)
Savla, K., Frazzoli, E., Bullo, F.: Traveling salesperson problems for the Dubins vehicle. IEEE Transactions on Automatic Control 53, 1378–1391 (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kim, HS., Cheong, O. (2012). The Cost of Bounded Curvature. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds) Computing and Combinatorics. COCOON 2012. Lecture Notes in Computer Science, vol 7434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32241-9_22
Download citation
DOI: https://doi.org/10.1007/978-3-642-32241-9_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32240-2
Online ISBN: 978-3-642-32241-9
eBook Packages: Computer ScienceComputer Science (R0)