Abstract
In this paper, we address the shortest path problem of point-to-point maneuver for a new nonholonomic wheeled vehicle. The vehicle performs forward-only motion which is adopted by Dubins car, while the curvature radius of its path is restricted to a finitely bounded interval which is different from that of Dubins car with a lower-bounded curvature radius. We first provide the infimum of the path length of the vehicle of point-to-point maneuver without a start or final orientation constraint. Then we study the infimum of the path length of the vehicle for point-to-point maneuver with a start and final orientation constraints. Next, we derive the explicit expressions for the candidate infimums. The infimum of the length path is the minimum of the candidates. Moreover, several examples are provided to verify the main results. The results in this paper extend the results on the existence of the shortest path of Dubins car to the new type of nonholonomic vehicles.
Similar content being viewed by others
References
Dubins, L.E.: On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents. Am. J. Math. 79, 497–516 (1957)
Chitsaz, H., LaValle, S.M., Balkcom, D.J., et al.: Minimum wheel-rotation paths for differential-drive mobile robots. Int. J. Robot. Res. 28, 66–80 (2009)
Agarwal, P.K., Biedl, T., Lazard, S., et al.: Curvature-constrained shortest paths in a convex polygon. SIAM J. Comput. 31, 1814–1851 (2002)
Esquivel, W.D., Chiang, L.E.: Nonholonomic path planning among obstacles subject to curvature restrictions. Robotica 20, 49–58 (2002)
Lee, J., Cheong, O., Kwon, W., et al.: Approximation of curvature-constrained shortest paths through a sequence of points. In: Lecture Notes in Computer Science vol. 1879. Springer, Berlin/Heidelberg (2000)
Janiak, M., Tchon, K.: Constrained motion planning of nonholonomic systems. Syst. Control Lett. 60, 625–631 (2011)
Ding, X.C., Rahmani, A., Egerstedt, M.: Multi-uav convoy protection: an optimal approach to path planning and coordination. IEEE Trans. Robot. 26, 256–268 (2010)
Ny, J.L., Frazzoli, E., Feron, E.: The curvature-constrained traveling salesman problem for high point densities. In: Proceedings of IEEE International Conference Decision and Control, pp 5985–5990. New Orleans (2007)
Chang, A., Brazil, M., Rubinstein, J., et al.: Curvature-constrained directional-cost paths in the plane. J. Global. Optim. 53, 663–681 (2012)
Chang, A., Brazil, M., Thomas, D., et al.: Optimal curvature-constrained paths with anisotropic costs in the plane. In: Proceedings of the Australian Control Conference, pp 112–117. Melbourne (2011)
Savla, K., Frazzoli, E., Bullo, F.: On the point-to-point and traveling salesperson problems for dubins vehicle. In: Proceedings of the IEEE American Control Conference, pp 786–791. Portland (2005)
Backer, J., Kirkpatrick, D.: Finding curvature-constrained paths that avoid polygonal obstacles. In: Proceedings of the Symposium on the Computational Geometry, pp 66–73. New York (2007)
Wang, H., Chen, Y., Soueres, P.: A geometric algorithm to compute time-optimal trajectories for a bidirectional steered robot. IEEE Trans. Robot. 25, 399–413 (2009)
Lamiraux, F., Lammond, J.P.: Smooth motion planning for car-like vehicles. IEEE Trans. Robot. Autom. 17, 498–501 (2001)
Wolek, A., Woolsey, C.: Disturbance rejection in dubins path planning. In: Proceedings of the IEEE American Control Conference, pp 4873–4878. Montreal (2012)
Hota, S., Ghose, D.: A modified dubins method for optimal path planning of a miniature air vehicle converging to a straight line path. In: Proceedings of the IEEE American Control Conference, pp 2397–2402. Missouri (2009)
Choi, H., Atkins, E.: Smooth transitions for a turning dubins vehicle. In: Proceedings of the AIAA Guidance, Navigation, and Control Conference. Toronto (2010)
Filippis, L., Guglieri, G., Quagliotti, F.: Path planning strategies for uavs in 3d environments. J. Intell. Robot. Syst. 65, 247–264 (2012)
Hwangbo, M., Kuffner, J., Kanade, T.: Efficient two-phase 3d motion planning for small fixed-wing uavs. In: Proceedings of the IEEE International Conference Robotics and Automation, pp 1035–1041. Roma (2007)
Mahmoudian, N.: Efficient Motion Planning and Control for Underwater Gliders. PhD. Thesis. Virginia Polytechnic Institute and State University, Virginia (2009)
Mahmoudian, N., Woolsey, C.: Underwater glider motion control. In: Proceedings of the IEEE International Conference Decision and Control, pp 552–557. Cancun (2008)
Mahmoudian, N., Geisbert, J., Woolsey, C.: Approximate analytical turning conditions for underwater gliders: implications for motion control and path planning. IEEE J. Ocean. Eng. 35, 131–143 (2010)
Ito, S., Takeuchi, S., Sasaki, M.: Motion measurement of a two-wheeled skateboard and its dynamical simulation. Appl. Math. Model. 36, 2178–2191 (2012)
Shammas, E.A., Choset, H., Rizzi, A.A.: Towards a unified approach to motion planning for dynamic underactuated mechanical systems with nonholonomic constraints. Int. J. Robot. Res. 26, 1075–1124 (2007)
Agarwal, P., Wang, H.: Approximation algorithms for curvature-constrained shortest paths. SIAM J. Sci. Comput. 30, 1739–1772 (2001)
Wang, H., Chen, Y., Soueres, P.: A geometric algorithm to compute time-optimal trajectories for a bidirectional steered robot. IEEE Trans. Robot. 25, 399–413 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Su, B., Wang, T., Wu, R. et al. Infimum of Path Length of Nonholonomic Vehicle with Finitely Bounded Curvature Radius. J Intell Robot Syst 79, 197–210 (2015). https://doi.org/10.1007/s10846-014-0053-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10846-014-0053-6