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Research Papers: Design Automation

A Novel Sampling Technique for Probabilistic Static Coverage Problems

[+] Author and Article Information
Binbin Zhang

DART Lab,
Department of Mechanical and
Aerospace Engineering,
University at Buffalo,
Buffalo, NY 14260

Nagavenkat Adurthi, Puneet Singla

LAIRS Lab,
Department of Mechanical and
Aerospace Engineering,
University at Buffalo,
Buffalo, NY 14260

Rahul Rai

DART Lab,
Department of Mechanical and
Aerospace Engineering,
University at Buffalo,
Buffalo, NY 14260
e-mail: rahulrai@buffalo.edu

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received March 3, 2015; final manuscript received December 17, 2015; published online January 20, 2016. Assoc. Editor: Gary Wang.

J. Mech. Des 138(3), 031403 (Jan 20, 2016) (9 pages) Paper No: MD-15-1181; doi: 10.1115/1.4032395 History: Received March 03, 2015; Revised December 17, 2015

Resource allocation in the presence of constraints is an important activity in many systems engineering problems such as surveillance, infrastructure planning, environmental monitoring, and cooperative task performance. The resources in many important problems are agents such as a person, machine, unmanned aerial vehicles (UAVs), infrastructures, and software. Effective execution of a given task is highly correlated with effective allocation of resources to execute the task. An important class of resource allocation problem in the presence of limited resources is static coverage problem. In static coverage problems, it is necessary to allocate resources (stationary configuration of agents) to cover an area of interest so that an event or spatial property of the area can be detected or monitored with high probability. In this paper, we outline a novel sampling algorithm for the static coverage problem in presence of probabilistic resource intensity allocation maps (RIAMs). The key intuition behind our sampling approach is to use the finite number of samples to generate an accurate representation of RIAM. The outlined sampling technique is based on an optimization framework that approximates the RIAM with piecewise linear surfaces on the Delaunay triangles and optimizes the sample placement locations to decrease the difference between the probability distribution and Delaunay triangle surface. Numerical results demonstrate that the algorithm is robust to the initial sample point locations and has superior performance in a wide range of theoretical problems and real-life applications. In a real-life application setting, we demonstrate the efficacy of the proposed algorithm to predict the position of wind stations for monitoring wind speeds across the U.S. The algorithm is also used to give recommendations on the placement of police cars in San Francisco and weather buoys in Pacific Ocean.

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Topics: Algorithms
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Figures

Grahic Jump Location
Fig. 1

Static SMC configuration (blue circles) obtained for two multivariate normal distribution functions f with different initial sample positions (red diamonds): (a) corner, (b) whole range, (c) center, and (d) center,

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Fig. 2

(a) Delaunay triangulation and Voronoi diagram of vertices (green dots) and (b) projection of Kriging surface SKrig or true surface STrue and Delaunay triangle surface SDeT in the Voronoi cell where sk located (the bold blue line is the boundary of SKrig or STrue and SDeT in Eq. (5))

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Fig. 4

Configuration of agents (red dots) with different K number after 50 iterations: (a) K = 10 and (b) K = 20

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Fig. 5

Two-norm relative difference plot and convergence rate plot for different number K of agents in 50 iterations: (a) relative difference plot for K = 10 agents, (b) relative difference plot for K = 20 agents, (c) convergence rate plot for K = 10 agents, and (d) convergence rate plot for K = 20 agents

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Fig. 6

Configuration of K = 10 agents (red dots) with different initial positions (blue crosses) after 50 iterations: (a) right corner, (b) left corner, (c) center, and (d) whole range

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Fig. 7

Variance of δ with different initial positions. The (a)–(d) are corresponding to initial positions in Figs. 6(a)6(d). Each box stands for the variance of ten relative difference values. The red lines show median relative difference.

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Fig. 8

True surface plot of complex surface Eq. (9)

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Fig. 9

Configuration (red dots) of different number K of agents after 20 iterations: (a) K = 10, (b) K = 20, (c) K = 30, and (d) K = 40

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Fig. 10

(a) U direction wind speed data map of U.S. [30]. (b) Surrogate Kriging Surface of wind speed data (in U direction). The surface is corresponding to the contour (a).

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Fig. 11

Configuration (red dots) of different number K of agents after 20 iterations: (a) K = 10, (b) K = 30, (c) K = 50, and (d) K = 70

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Fig. 12

Variance of relative difference for different number K of agents. The (a)–(d) are corresponding to K=10,30,50,and 70. Each box stands for the variance of ten relative difference values. The red lines show median relative difference.

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Fig. 13

(a) Probability contour map of crime incidence in San Francisco. The crime incidents data are cited from Ref. [31]. (b) Probability contour map and recommended configuration of police cars (red dots) with K = 100.

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Fig. 14

Two-norm relative distance (between true surface and Delaunay triangle surface) versus iterations plot for K = 50 agents. True surface is corresponding to Eq. (10).

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