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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

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

Dejan Milutinović

Associate Professor
Mem. ASME
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.

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References

Figures

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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).

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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.

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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).

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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.

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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).

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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.

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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).

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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.

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