Research Papers

On the Construction of Minimum-Time Tours for a Dubins Vehicle in the Presence of Uncertainties

[+] Author and Article Information
Ross P. Anderson

Mechanical and Aerospace Engineering,
New York University Polytechnic
School of Engineering,
Brooklyn, NY 11201
e-mail: ross.anderson@nyu.edu

Dejan Milutinović

Associate Professor
Computer Engineering Department,
University of California,
Santa Cruz,
Santa Cruz, CA 95064
e-mail: dejan@soe.ucsc.edu

A receding horizon extension to the Dubins TSP appears in Ref. [25], and other strategies for re-optimizing the TSP solution are developed in Ref. [26].

A change is counted when the Hamming distance between the original sequence and an updated sequence is at least one.

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received January 28, 2014; final manuscript received July 4, 2014; published online October 21, 2014. Assoc. Editor: Jongeun Choi.

J. Dyn. Sys., Meas., Control 137(3), 031001 (Oct 21, 2014) (8 pages) Paper No: DS-14-1035; doi: 10.1115/1.4028552 History: Received January 28, 2014; Revised July 04, 2014

We propose an approach to the problem of computing a minimum-time tour through a series of waypoints for a Dubins vehicle in the presence of stochasticity. In this paper, we explicitly account for kinematic nonlinearities, the stochastic drift of the vehicle, the stochastic motion of the targets, and the possibility for the vehicle to service each of the targets or waypoints, leading to a new version of the Dubins vehicle traveling salesperson problem (TSP). Based on the Hamilton–Jacobi–Bellman (HJB) equation, we first compute the minimum expected time feedback control to reach one waypoint. Next, minimum expected times associated with the feedback control are used to construct and solve a TSP. We provide numerical results illustrating our solution, analyze how the stochasticity affects the solution, and consider the possibility for on-line recomputation of the waypoint ordering in a receding-horizon manner.

Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.


Beard, R., McLain, T., and Goodrich, M., 2002, “Coordinated Target Assignment and Intercept for Unmanned Air Vehicles,” Proceedings IEEE International Conference on Robotics and Automation, Washington, DC, May 11–15, pp. 2581–2586. [CrossRef]
Tang, Z., and Ozguner, U., 2005, “Motion Planning for Multitarget Surveillance With Mobile Sensor Agents,” IEEE Trans. Rob., 21(5), pp. 898–908. [CrossRef]
Bullo, F., Frazzoli, E., Pavone, M., Savla, K., and Smith, S. L., 2011, “Dynamic Vehicle Routing for Robotic Systems,” Proc. IEEE, 99(9), pp. 1482–1504. [CrossRef]
Kumar, S., and Chakravorty, S., 2012, “Multi-Agent Generalized Probabilistics RoadMaps: MAGPRM,” Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, IEEE, Vilamoura, Portugal, Oct. 7–12, pp. 3747–3753. [CrossRef]
Anderson, R. P., Dinolov, G., Milutinovic, D., and Moore, A., 2012, “Maximally-Informative Ocean Modeling System (ROMS) Navigation of an AUV in Uncertain Ocean Currents,” ASME Paper No. DSCC2012-MOVIC2012-8720. [CrossRef]
Papadimitriou, C., and Steiglitz, K., 2007, Combinatorial Optimization: Algorithms and Complexity, Dover Publications, Mineola, NY.
Dubins, L. E., 1957, “On Curves of Minimal Length With a Constraint on Average Curvature, and With Prescribed Initial and Terminal Positions and Tangents,” Am. J. Math., 79(3), pp. 497–516. [CrossRef]
Savla, K., Frazzoli, E., and Bullo, F., 2005, “On the Point-to-Point and Traveling Salesperson Problems for Dubins' Vehicle,” Proceedings of the American Control Conference, IEEE, Portland, OR, June 8–10, pp. 786–791. [CrossRef]
Savla, K., Frazzoli, E., and Bullo, F., 2008, “Traveling Salesperson Problems for the Dubins Vehicle,” IEEE Trans. Automa. Control, 53(6), pp. 1378–1391. [CrossRef]
Le Ny, J., Feron, E., and Frazzoli, E., 2012, “On the Dubins Traveling Salesman Problem,” IEEE Trans. Automa. Control, 57(1), pp. 265–270. [CrossRef]
Frazzoli, E., and Bullo, F., 2004, “Decentralized Algorithms for Vehicle Routing in a Stochastic Time-Varying Environment,” Proceedings of the 43rd IEEE Conference on Decision and Control, Nassau, Bahamas, Dec. 14–17, pp. 3357–3364. [CrossRef]
Obermeyer, K. J., Oberlin, P., and Darbha, S., 2012, “Sampling-Based Path Planning for a Visual Reconnaissance UAV,” AIAA J. Guid. Control Dyn., 35(2), pp. 619–631. [CrossRef]
Enright, J. J., and Frazzoli, E., 2006, “The Traveling Salesman Problem for the Reeds-Shepp Car and the Differential Drive Robot,” Proceedings of the 45th IEEE Conference on Decision and Control, IEEE, San Diego, CA, Dec. 13–15, pp. 3058–3064. [CrossRef]
Itani, S., Frazzoli, E., and Dahleh, M. A., 2008, “Travelling Salesperson Problem for Dynamic Systems,” Proceedings of the 17th IFAC World Congress, IFAC, Seoul, Korea, July 6–11, pp. 13318–13323.
Leipälä, T., 1978, “On the Solutions of Stochastic Traveling Salesman Problems,” Euro. J. Oper. Res., 2(4), pp. 291–297. [CrossRef]
Berman, O., and Simchi-Levi, D., 1989, “The Traveling Salesman Location Problem on Stochastic Networks,” Transp. Sci., 23(1), pp. 54–57. [CrossRef]
Kenyon, A. S., and Morton, D. P., 2003, “Stochastic Vehicle Routing With Random Travel Times,” Transp. Sci., 37(1), pp. 69–82. [CrossRef]
Bertsimas, D. J., and G.van Ryzin, 1991, “A Stochastic and Dynamic Vehicle Routing Problem in the Euclidean Plane,” Oper. Res., 39(4), pp. 601–615. [CrossRef]
Goemans, M. X., and Bertsimas, D. J., 1991, “Probabilistic Analysis of the Held and Karp Lower Bound for the Euclidean Traveling Salesman Problem,” Math. Oper. Res., 16(1), pp. 72–89. [CrossRef]
Enright, J. J., and Frazzoli, E., 2005, “UAV Routing in a Stochastic, Time-Varying Environment,” Proceedings of the 16th IFAC World Congress, Vol. 16, IFAC, Prague, Czech Republic, July 3–8, pp. 2009–2015.
Itani, S., and Dahleh, M. A., 2007, “On the Stochastic TSP for the Dubins Vehicle,” Proceedings of the American Control Conference, IEEE, New York, July 9–13, pp. 443–448. [CrossRef]
Laporte, G., Louveaux, F., and Mercure, H., 1994, “A Priori Optimization of the Traveling Salesman Problem,” Oper. Res., 42(3), pp. 543–549. [CrossRef]
Helvig, C., Robins, G., and Zelikovsky, A., 1998, “Moving-Target TSP and Related Problems,” LNCS: Proceedings of the 6th Annual European Symposium, Venice, Italy, Aug. 24–26, pp. 453–464.
Choubey, N. S., 2013, “Moving Target Travelling Salesman Problem Using Genetic Algorithm,” Int. J. Comput. Appl. Technol.70(2), pp. 30–34.
Ma, X., and Casta, D. A., 2006, “Receding Horizon Planning for Dubins Traveling Salesman Problems,” Proceedings of the 45th IEEE Conference on Decision and Control, IEEE, San Diego, CA, Dec. 13–15, pp. 5453–5458. [CrossRef]
Powell, W. B., Jaillet, P., and Odoni, A., 1995, “Stochastic and Dynamic Networks and Routing,” Handbook in Operations Research and Management Science, Vol. 8, Elsevier, Amsterdam, The Netherlands, pp. 141–195.
Anderson, R. P., Bakolas, E., Milutinović, D., and Tsiotras, P., 2013, “Optimal Feedback Guidance of a Small Aerial Vehicle in the Presence of Stochastic Wind,” AIAA J. Guid. Navig. Control, 36(4), pp. 975–985. [CrossRef]
Anderson, R. P., and Milutinović, D., 2013, “The Dubins Traveling Salesperson Problem With Stochastic Dynamics,” ASME Paper No. DSCC2013-3846. [CrossRef]
Bertsimas, D., Chervi, P., and Peterson, M., 1995, “Computational Approaches to Stochastic Vehicle Routing Problems,” Transp. Sci., 29(4), pp. 342–352. [CrossRef]
Gardiner, C., 2009, Sciences, 4th ed., Springer, Berlin, Germany.
Kushner, H. J., and Dupuis, P., 2001, Numerical Methods for Stochastic Control Problems in Continuous Time, 2nd ed., Springer, New York.
Oksendal, B., 2003, Stochastic Differential Equations: An Introduction With Applications, 6th ed., Springer-Verlag, Berlin, Germany.
Laporte, G., Asef-Vaziri, A., and Sriskandarajah, C., 1996, “Some Applications of the Generalized Travelling Salesman Problem,” J. Oper. Res. Soc., 47(12), pp. 1461–1467. [CrossRef]
Behzad, A., and Modarres, M., 2002, “A New Efficient Transformation of the Generalized Traveling Salesman Problem Into Traveling Salesman Problem,” Proceedings of the 15th International Conference of Systems Engineering, Las Vegas, NV, Aug. 6–8.
Helsgaun, K., 2000, “An Effective Implementation of the Lin-Kernighan Traveling Salesman Heuristic,” Eur. J. Oper. Res., 126(1), pp. 106–130. [CrossRef]


Grahic Jump Location
Fig. 1

(a) Diagram of a Dubins vehicle at heading angle θ reaching a sequence of targets μ(1), μ(2),… after which it should return to its initial condition. (b) A simulated example path through five waypoints using the optimal minimum-time control to reach each without regard to the incident heading angle (return path and stochastic effects not shown).

Grahic Jump Location
Fig. 2

(a) Expected time for a Dubins vehicle to reach a waypoint at −180 deg when current θ = −144 deg. (b) Expected time for a Dubins vehicle to reach a waypoint with free incident heading when θ = −144 deg. (c) and (d) The corresponding optimal turning rates (white for umax and black for -umax) for fixed incident heading and free incident heading, respectively.

Grahic Jump Location
Fig. 3

Expected time T¯(Δxi,Δyi,θ,t|θi0) for a Dubins vehicle to first service the previous waypoint for a duration t¯ before traveling to and reaching the next waypoint at θi0 = -180 deg when current θ = −144 deg. From left to right, top to bottom, the time t increases from 0 to 5.6 in 0.7 s increments, and the time t = t¯ = 6.3 s (the boundary condition used to compute these time-to-go plots) can be found in Fig. 2(b).

Grahic Jump Location
Fig. 4

(a) Distribution of difference in intended and actual incident heading angles with σ = 0.1, σT = 0 (mean = −0.08, var = 0.03). (b) Histogram of time to reach a single waypoint, where multiple peaks indicate the occurrence of missing the waypoint. (c) Same as (a) but σ = 0.1, σT = 0.1 (mean = 0.08, var = 0.04). (d) Same as (b) but σ = 0.1, σT = 0.1.

Grahic Jump Location
Fig. 5

Minimum-time path for the Dubins vehicle through five waypoints. The gray background represents the probability density of the spatial locations traveled by the Dubins vehicle over 1000 simulations. In the case of moving targets, the contours represent the density of the target locations. A single sample trajectory is shown in each case for the Dubins vehicle as a dashed line, and, in the case of moving targets, the paths of the targets are solid. If present, arrows indicate the direction of the fixed incident heading angle chosen by the TSP; otherwise, the incident heading angle was free. For receding-horizon cases, trajectories while servicing target not shown. Depicted in (a) in white is the realization without stochasticity (σ = σT = 0).

Grahic Jump Location
Fig. 6

Distribution of times from 1000 simulations. Panels (a)–(f) correspond to the same conditions of Figs. 5(a)5(f). Times required to service each target in panels (e) and (f) are not included. The vertical dotted line is the minimum TSP travel time (12.33 s) without nonholonomic vehicle constraints, i.e., a regular TSP.

Grahic Jump Location
Fig. 7

Minimum-time paths using the substitute control against moving targets σ = 0.1, σT = 0.1, (inset) Distribution of times, (mean(Ttot), std(Ttot)) = (27.7, 5.4).

Grahic Jump Location
Fig. 8

Percent of 1000 simulations from Figs. 5(e) and 5(f) whose waypoint orderings differ after TSP recomputation, for free incident heading simulations (○) and fixed incident headings (∇). The percentage of simulations that change the fixed incident heading angle θi0 after recomputation are Δ.




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In