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Research Papers

# Distributed Algorithms for Stochastic Source Seeking With Mobile Robot Networks

[+] Author and Article Information
Nikolay A. Atanasov

Department of Electrical and
Systems Engineering,
University of Pennsylvania,
e-mail: atanasov@seas.upenn.edu

Jerome Le Ny

Department of Electrical Engineering and GERAD,
École Polytechnique de Montréal,
e-mail: jerome.le-ny@polymtl.ca

George J. Pappas

Department of Electrical and
Systems Engineering,
University of Pennsylvania,
e-mail: pappasg@seas.upenn.edu

The assumption is made only to simplify the presentation of the gradient ascent approach in the model-free case. The approach generalizes to signals of higher dimension.

While Assumptions (A1)–(A5) are sufficient to prove the convergence in our application, they are by no means the weakest possible. If necessary some can be relaxed using the results in stochastic approximation [27,28].

Given two metric spaces $(X,dx)$ and $(G,dg)$, a function $g:X→G$ is Lipschitz continuous if there exists a real constant 0 ≤ L <  such that: $dg(g(x1),g(x2))≤Ldx(x1,x2),∀x1,x2∈X$.

Weak consistency means that the estimates $θ∧i(k)$ converge in probability to θ*, i.e. $limk→∞P(‖θ∧i(k)-θ*‖≥ɛ)=0$ for any ε > 0 and all i.

Mean-square consistency means that the estimates $θ∧i(k)$ converge in mean-square to θ*, i.e. $limk→∞E[‖θ∧i(k)-θ*‖2]=0$ for all i.

The time-scales of the relative state measurements and the signal measurements might be different but for simplicity we keep them the same.

Since all sensors have the same observation model h(⋅, ⋅), each sensor can simulate measurements $zVi,t$ as long as it knows the configurations $zVi,t$.

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received January 31, 2014; final manuscript received June 10, 2014; published online October 21, 2014. Assoc. Editor: Dejan Milutinovic.

J. Dyn. Sys., Meas., Control 137(3), 031004 (Oct 21, 2014) (9 pages) Paper No: DS-14-1055; doi: 10.1115/1.4027892 History: Received January 31, 2014; Revised June 10, 2014

## Abstract

Autonomous robot networks are an effective tool for monitoring large-scale environmental fields. This paper proposes distributed control strategies for localizing the source of a noisy signal, which could represent a physical quantity of interest such as magnetic force, heat, radio signal, or chemical concentration. We develop algorithms specific to two scenarios: one in which the sensors have a precise model of the signal formation process and one in which a signal model is not available. In the model-free scenario, a team of sensors is used to follow a stochastic gradient of the signal field. Our approach is distributed, robust to deformations in the group geometry, does not necessitate global localization, and is guaranteed to lead the sensors to a neighborhood of a local maximum of the field. In the model-based scenario, the sensors follow a stochastic gradient of the mutual information (MI) between their expected measurements and the expected source location in a distributed manner. The performance is demonstrated in simulation using a robot sensor network to localize the source of a wireless radio signal.

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Topics: Sensors , Algorithms , Signals

## Figures

Fig. 1

Joint position and gradient estimation at a single measurement location (on the fast time-scale). The first plot shows the true sensor positions (red circles), initial position estimates (blue circles), and the true gradient of the signal field (red arrow). The second plot shows the position estimates after 40 iterations (blue circles) and the gradient estimate of sensor 1 (blue arrow). The third column shows the root mean squared error (RMSE) of the position (top) and centroid (bottom) estimates of all sensors averaged over 50 independent repetitions. The fourth column shows the RMSE of the gradient magnitude and orientation estimates.

Fig. 2

The paths followed by the sensors after 30 iterations of the model-free source-seeking algorithm in an obstacle-free environment. The white circles indicate sensor 1's estimates of the source position over time. The plots on the right show the average error of the source position estimates and its standard deviation averaged over 50 independent repetitions.

Fig. 3

The paths followed by the sensors after 30 iterations of the model-based source-seeking algorithm in an environment without obstacles (left) and with obstacles (right). The white circles indicate sensor 1's estimates of the source position over time. The plots show the average error of the source position estimates and its standard deviation averaged over 50 independent repetitions. The evolution of sensor 1's distributed particle filter is shown in each scenario (bottom row).

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