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Research Papers

On Tightly Bounding the Dubins Traveling Salesman's Optimum

[+] Author and Article Information
Satyanarayana G. Manyam

Infoscitex Corporation,
DCS Company,
Dayton, OH 45431
e-mail: msngupta@gmail.com

Sivakumar Rathinam

Department of Mechanical Engineering,
Texas A&M University,
College Station, TX 77843
e-mail: srathinam@tamu.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received February 10, 2016; final manuscript received January 22, 2018; published online March 7, 2018. Assoc. Editor: Azim Eskandarian.

J. Dyn. Sys., Meas., Control 140(7), 071013 (Mar 07, 2018) (12 pages) Paper No: DS-16-1081; doi: 10.1115/1.4039099 History: Received February 10, 2016; Revised January 22, 2018

The Dubins traveling salesman problem (DTSP) has generated significant interest over the last decade due to its occurrence in several civil and military surveillance applications. This problem requires finding a curvature constrained shortest path for a vehicle visiting a set of target locations. Currently, there is no algorithm that can find an optimal solution to the DTSP. In addition, relaxing the motion constraints and solving the resulting Euclidean traveling salesman problem (ETSP) provide the only lower bound available for the DTSP. However, in many problem instances, the lower bound computed by solving the ETSP is far below the cost of the feasible solutions obtained by some well-known algorithms for the DTSP. This paper addresses this fundamental issue and presents the first systematic procedure for developing tight lower bounds for the DTSP.

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References

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Figures

Grahic Jump Location
Fig. 1

There are four possible headings at each target. A feasible solution for the DTSP can be obtained by choosing a heading at each target and finding a corresponding optimal TSP path.

Grahic Jump Location
Fig. 2

A comparison between the cost of the feasible solution (upper bound) obtained by solving the one-in-a-set TSP with 32 discretizations and the optimal cost of the corresponding ETSP (lower bound) for 25 instances. There are 20 targets in each instance, and the location of each target is uniformly sampled from a 1000 × 1000 square. Also, the minimum turning radius of the vehicle is set to 100.

Grahic Jump Location
Fig. 5

Given θ1, the length of the RSL path varies monotonically with respect to θ2 wherever the path exists and none of its curved segments vanishes

Grahic Jump Location
Fig. 6

LRL path for θ1=0

Grahic Jump Location
Fig. 4

A feasible solution to the Dubins interval problem

Grahic Jump Location
Fig. 3

There are four intervals at each target. A lower bound for the DTSP can be obtained by choosing an interval and restricting both the arrival and the departure angles to be in the chosen interval at each target and then finding a corresponding optimal TSP path. The shaded interval at each target shows the chosen interval with the arrival and departure angles.

Grahic Jump Location
Fig. 7

Given θ1, the length of the LRL path reaches a maximum when θ2=(π/2)+α, as shown. This figure also shows the values of θ2 where the LRL path just ceases to exist.

Grahic Jump Location
Fig. 8

Given θ1, the LRL paths when the arc angle in the right turn is π. This figure shows the angles for θ2 when the LRL path does not exist.

Grahic Jump Location
Fig. 9

One can solve minθ1∈I1{LS1(θ1)} by considering the reflections of the points with respect to the x-axis and solving the corresponding minθ1∈I1{RS1(θ1)}

Grahic Jump Location
Fig. 12

Lower bounds computed with 4, 8, 16, and 32 intervals at each target for 25 instances: (a) 10 targets, (b) 15 targets, and (c) 20 targets

Grahic Jump Location
Fig. 13

Comparison between lower bounds and upper bounds for 32 discretizations, along with the optimal Euclidean TSP cost: (a) 10 targets, (b) 15 targets, and (c) 20 targets

Grahic Jump Location
Fig. 14

A feasible Dubins path for an instance with 20 targets and the path obtained from lower bound computation

Grahic Jump Location
Fig. 15

RS path: examples illustrating D(θ1) and θ2(RS,θ1): (a) x¯>2ρ, (b) x¯>2ρ, (c) x¯≤2ρ, and (d) x¯≤2ρ

Grahic Jump Location
Fig. 16

RL path: examples illustrating D(θ1) and θ2(RL,θ1) for the case when 0≤ϕ+θ2≤π: (a) x¯>2ρ, (b) x¯>2ρ, (c) x¯≤2ρ, and (d) x¯≤2ρ

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