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.

References

References
1.
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
.10.1109/TRA.2002.805653
2.
Tang
,
Z.
, and
Ozguner
,
U.
,
2005
, “
Motion Planning for Multitarget Surveillance With Mobile Sensor Agents
,”
IEEE Trans. Rob.
,
21
(
5
), pp.
898
908
.10.1109/TRO.2005.847567
3.
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
.10.1109/JPROC.2011.2158181
4.
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
.10.1109/IROS.2012.6385678
5.
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-MOVIC2012-8720.10.1115/DSCC2012-MOVIC2012-8720
6.
Papadimitriou
,
C.
, and
Steiglitz
,
K.
,
2007
,
Combinatorial Optimization: Algorithms and Complexity
,
Dover Publications
,
Mineola, NY
.
7.
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
.10.2307/2372560
8.
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
.10.1109/ACC.2005.1470055
9.
Savla
,
K.
,
Frazzoli
,
E.
, and
Bullo
,
F.
,
2008
, “
Traveling Salesperson Problems for the Dubins Vehicle
,”
IEEE Trans. Automa. Control
,
53
(
6
), pp.
1378
1391
.10.1109/TAC.2008.925814
10.
Le Ny
,
J.
,
Feron
,
E.
, and
Frazzoli
,
E.
,
2012
, “
On the Dubins Traveling Salesman Problem
,”
IEEE Trans. Automa. Control
,
57
(
1
), pp.
265
270
.10.1109/TAC.2011.2166311
11.
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
.10.1109/CDC.2004.1429220
12.
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
.10.2514/1.48949
13.
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
.10.1109/CDC.2006.377339
14.
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
.
15.
Leipälä
,
T.
,
1978
, “
On the Solutions of Stochastic Traveling Salesman Problems
,”
Euro. J. Oper. Res.
,
2
(
4
), pp.
291
297
.10.1016/0377-2217(78)90044-9
16.
Berman
,
O.
, and
Simchi-Levi
,
D.
,
1989
, “
The Traveling Salesman Location Problem on Stochastic Networks
,”
Transp. Sci.
,
23
(
1
), pp.
54
57
.10.1287/trsc.23.1.54
17.
Kenyon
,
A. S.
, and
Morton
,
D. P.
,
2003
, “
Stochastic Vehicle Routing With Random Travel Times
,”
Transp. Sci.
,
37
(
1
), pp.
69
82
.10.1287/trsc.37.1.69.12820
18.
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
.10.1287/opre.39.4.601
19.
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
.10.1287/moor.16.1.72
20.
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.
21.
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
.10.1109/ACC.2007.4282819
22.
Laporte
,
G.
,
Louveaux
,
F.
, and
Mercure
,
H.
,
1994
, “
A Priori Optimization of the Traveling Salesman Problem
,”
Oper. Res.
,
42
(
3
), pp.
543
549
.10.1287/opre.42.3.543
23.
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
.
24.
Choubey
,
N. S.
,
2013
, “
Moving Target Travelling Salesman Problem Using Genetic Algorithm
,”
Int. J. Comput. Appl. Technol.
70
(
2
), pp.
30
34
.
25.
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
.10.1109/CDC.2006.376928
26.
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
.
27.
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
.10.2514/1.59512
28.
Anderson
,
R. P.
, and
Milutinović
,
D.
,
2013
, “
The Dubins Traveling Salesperson Problem With Stochastic Dynamics
,”
ASME
Paper No. DSCC2013-3846.10.1115/DSCC2013-3846
29.
Bertsimas
,
D.
,
Chervi
,
P.
, and
Peterson
,
M.
,
1995
, “
Computational Approaches to Stochastic Vehicle Routing Problems
,”
Transp. Sci.
,
29
(
4
), pp.
342
352
.10.1287/trsc.29.4.342
30.
Gardiner
,
C.
,
2009
,
Sciences
,
4th ed.
,
Springer
, Berlin, Germany.
31.
Kushner
,
H. J.
, and
Dupuis
,
P.
,
2001
,
Numerical Methods for Stochastic Control Problems in Continuous Time
,
2nd ed.
,
Springer
,
New York
.
32.
Oksendal
,
B.
,
2003
,
Stochastic Differential Equations: An Introduction With Applications
,
6th ed.
,
Springer-Verlag
,
Berlin
, Germany.
33.
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
.10.1057/jors.1996.190
34.
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.
35.
Helsgaun
,
K.
,
2000
, “
An Effective Implementation of the Lin-Kernighan Traveling Salesman Heuristic
,”
Eur. J. Oper. Res.
,
126
(
1
), pp.
106
130
.10.1016/S0377-2217(99)00284-2
You do not currently have access to this content.