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Research Papers

A Single Forward-Velocity Control Signal for Stochastic Source Seeking With Multiple Nonholonomic Vehicles

[+] Author and Article Information
Paul Frihauf

Mechanical and Aerospace,
Engineering Department,
University of California,
San Diego, CA 92093
e-mail: pfrihauf@ucsd.edu

Shu-Jun Liu

Department of Mathematics,
Southeast University,
Nanjing 211189, China
e-mail: sjliu@seu.edu.cn

Miroslav Krstic

Mechanical and Aerospace,
Engineering Department,
University of California,
San Diego, CA 92093
e-mail: krstic@ucsd.edu

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received August 6, 2013; final manuscript received April 26, 2014; published online July 10, 2014. Assoc. Editor: John B. Ferris.

J. Dyn. Sys., Meas., Control 136(5), 051024 (Jul 10, 2014) (9 pages) Paper No: DS-13-1304; doi: 10.1115/1.4027577 History: Received August 06, 2013; Revised April 26, 2014

With a single stochastic extremum seeking control signal, we steer multiple autonomous vehicles, modeled as nonholonomic unicycles, toward the maximum of an unknown, spatially distributed signal field. The vehicles, whose angular velocities are constant and distinct, travel at the same forward velocity, which is controlled by the stochastic extremum seeking controller. To determine the vehicles’ velocity, the controller uses measurements of the signal field at the respective vehicle positions and excitation based on filtered white noise. The positions of the vehicles are not measured. We prove local exponential convergence, both almost surely and in probability, to a small neighborhood near the source and provide a numerical example to illustrate the effectiveness of the algorithm.

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Figures

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Fig. 1

The notation used in the model for vehicle i, where zi = (xi, yi) is the center of the vehicle, θi the orientation, vi is the forward velocity, and ωi is the angular velocity

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Fig. 2

Stochastic extremum seeking scheme for N unicycles driven by a single forward velocity controller

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Fig. 4

Zoomed in section of U1's trajectory in Fig. 3(a), highlighting the star-shaped trajectory

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Fig. 3

Trajectories of the vehicle centers converging to a neighborhood of the source at (0, 0) from an initial circle formation for (a) N = 5, (b) N = 20, and (c) N = 40 vehicles. The signal field's contour lines are superimposed. The cost J for each scenario is shown in (d).

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Fig. 5

Trajectories of the vehicle centers converging to a neighborhood of the source at (0, 0) from a random clustering about the point (2,2) for (a) N = 5, (b) N = 20, and (c) N = 40 vehicles. The signal field's contour lines are superimposed. The average cost J for each scenario is shown in (d).

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Fig. 6

Linear relationship between the vehicles’ convergence time and the number of vehicles in a swarm initialized in a circle formation (blue circles) and about the point (2,2) (red squares). The value t * on the vertical axis is the time when the average cost J satisfies |J-Q*| < 1.

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Fig. 7

The initial transient, i.e., first 50 s, is shown of (a) the scalar velocity command applied to all vehicles and of (b) each vehicle's signal field measurement and the mean measurement J for the scenario with N = 5 vehicles initialized about the point (2,2)

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Fig. 8

(a) Trajectories of vehicle centers converging to a neighborhood of the source at (0, 0) from an initial circle formation after increasing the control parameter c to 250, and (b) the signal field measurements for each vehicle shown with the mean of the measurements (cost J)

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Fig. 9

(a) Trajectories of the vehicle centers converging to a neighborhood of the source at (0,0) from an initial circle formation when using a weighted cost J, and (b) the signal field measurements for each vehicle shown with the weighted cost J and, for comparison, the mean of all the measurements

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