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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,
Philadelphia, PA 19104
e-mail: atanasov@seas.upenn.edu

Jerome Le Ny

Department of Electrical Engineering and GERAD,
École Polytechnique de Montréal,
Montréal, QC H3T-1J4, Canada
e-mail: jerome.le-ny@polymtl.ca

George J. Pappas

Department of Electrical and
Systems Engineering,
University of Pennsylvania,
Philadelphia, PA 19104
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.

###### FIGURES IN THIS ARTICLE
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Copyright © 2015 by ASME
Topics: Sensors , Algorithms , Signals
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## References

Lux, R., and Shi, W., 2004, “Chemotaxis-Guided Movements in Bacteria,” Crit. Rev. Oral. Biol. Med., 15(4), pp. 207–220.
Frankel, R., Bazylinski, D., Johnson, M., and Taylor, B., 1997, “Magneto-Aerotaxis in Marine Coccoid Bacteria,” Biophys. J., 73(2), pp. 994–1000. [PubMed]
Ögren, P., Fiorelli, E., and Leonard, N., 2004, “Cooperative Control of Mobile Sensor Networks,” IEEE Trans. Autom. Control, 49(8), pp. 1292–1302.
Sukhatme, G., Dhariwal, A., Zhang, B., Oberg, C., Stauffer, B., and Caron, D., 2007, “Design and Development of a Wireless Robotic Networked Aquatic Microbial Observing System,” Environ. Eng. Sci., 24(2), pp. 205–215.
Rybski, P., Stoeter, S., Erickson, M., Gini, M., Hougen, D., and Papanikolopoulos, N., 2000, “A Team of Robotic Agents for Surveillance,” International Conference on Autonomous Agents, Barcelona, Spain, ACM, New York, pp. 9–16.
Kumar, V., Rus, D., and Singh, S., 2004, “Robot and Sensor Networks for First Responders,” IEEE Pervasive Comput., 3(4), pp. 24–33.
Wu, W., and Zhang, F., 2011, “Experimental Validation of Source Seeking With a Switching Strategy,” IEEE International Conference on Robotics and Automation (ICRA ), pp. 3835–3840.
Li, S., and Guo, Y., 2012, “Distributed Source Seeking by Cooperative Robots: All-to-All and Limited Communications,” IEEE International Conference on Robotics and Automation (ICRA), pp. 1107–1112.
Brinón-Arranz, L., and Schenato, L., 2013, “Consensus-Based Source-Seeking With a Circular Formation of Agents,” European Control Conferenc e, pp. 2831–2836.
Zhang, F., and Leonard, N., 2010, “Cooperative Filters and Control for Cooperative Exploration,” IEEE Trans. Autom. Control, 55(3), pp. 650–663.
Choi, J., Oh, S., and Horowitz, R., 2009, “Distributed Learning and Cooperative Control for Multi-Agent Systems,” Automatica, 45(12), pp. 2802–2814.
Jadaliha, M., Lee, J., and Choi, J., 2012, “Adaptive Control of Multiagent Systems for Finding Peaks of Uncertain Static Fields,” ASME J. Dyn. Syst. Meas. Control, 134(5), p. 051007.
Azuma, S., Sakar, M., and Pappas, G., 2012, “Stochastic Source Seeking by Mobile Robots,” IEEE Trans. Autom. Control, 57(9), pp. 2308–2321.
Zhang, C., Arnold, D., Ghods, N., Siranosian, A., and Krstić, M., 2007, “Source Seeking With Non-Holonomic Unicycle Without Position Measurement and With Tuning of Forward Velocity,” Syst. Control Lett., 56(3), pp. 245–252.
Liu, S., and Krstić, M., 2010, “Stochastic Source Seeking for Nonholonomic Unicycle,” Automatica, 46(9), pp. 1443–1453.
Stanković, M., and Stipanović, D., 2010, “Extremum Seeking Under Stochastic Noise and Applications to Mobile Sensors,” Automatica, 46(8), pp. 1243–1251.
Ghods, N., and Krstić, M., 2011, “Source Seeking With Very Slow or Drifting Sensors,” ASME J. Dyn. Syst. Meas. Control, 133(4), p. 044504.
Atanasov, N., Le Ny, J., Michael, N., and Pappas, G., 2012, “Stochastic Source Seeking in Complex Environments,” IEEE International Conference on Robotics and Automation (ICRA), pp. 3013–3018.
Charrow, B., Kumar, V., and Michael, N., 2013, “Approximate Representations for Multi-Robot Control Policies That Maximize Mutual Information,” Robotics: Science and Systems (RSS), Berlin, Germany.
Hoffmann, G., and Tomlin, C., 2010, “Mobile Sensor Network Control Using Mutual Information Methods and Particle Filters,” IEEE Trans. Autom. Control, 55(1), pp. 32–47.
Dames, P., Schwager, M., Kumar, V., and Rus, D., 2012, “A Decentralized Control Policy for Adaptive Information Gathering in Hazardous Environments,” IEEE Conference on Decision and Control (CDC ), Dec., pp. 2807–2813.
Julian, B., Angermann, M., Schwager, M., and Rus, D., 2012, “Distributed Robotic Sensor Networks: An Information-Theoretic Approach,” Int. J. Rob. Res., 31(10), pp. 1134–1154.
Yin, G., Yuan, Q., and Wang, L., 2013, “Asynchronous Stochastic Approximation Algorithms for Networked Systems: Regime-Switching Topologies and Multiscale Structure,” SIAM Multiscale Model. Simul., 11(3), pp. 813–839.
Yu, W., Zheng, W., Chen, G., Ren, W., and Cao, J., 2011, “Second-Order Consensus in Multi-Agent Dynamical Systems With Sampled Position Data,” Automatica, 47(7), pp. 1496–1503.
Derenick, J., and Spletzer, J., 2007, “Convex Optimization Strategies for Coordinating Large-Scale Robot Formations,” IEEE Trans. Rob., 23(6), pp. 1252–1259.
Fornberg, B., Lehto, E., and Powell, C., 2013, “Stable Calculation of Gaussian-Based RBF-FD Stencils,” Comput. Math. Appl., 65(4), pp. 627–637.
Kushner, H., and Yin, G., 2003, Stochastic Approximation and Recursive Algorithms and Applications, 2 ed., Springer-Verlag, New York.
Borkar, V., 2008, Stochastic Approximation: A Dynamical Systems Viewpoint, Cambridge University Press, Cambridge, UK.
Ljung, L., 1977, “Analysis of Recursive Stochastic Algorithms,” IEEE Trans. Autom. Control, 22(4), pp. 551–575.
Spall, J., 2003, Introduction to Stochastic Search and Optimization, John Wiley & Sons, Hoboken, NJ.
Schwager, M., Dames, P., Rus, D., and Kumar, V., 2011, “A Multi-Robot Control Policy for Information Gathering in the Presence of Unknown Hazards,” Proceedings of International Symposium on Robotics Research, Aug.
Thrun, S., Burgard, W., and Fox, D., 2005, Probabilistic Robotics, MIT, Cambridge, MA.
Rad, K., and Tahbaz-Salehi, A., 2010, “Distributed Parameter Estimation in Networks,” IEEE Conference on Decision and Control (CDC), pp. 5050–5055.
Shahrampour, S., and Jadbabaie, A., 2013, “Exponentially Fast Parameter Estimation in Networks Using Distributed Dual Averaging,” IEEE Conference on Decision and Control (CDC ), pp. 6196–6201.
Tahbaz-Salehi, A., and Jadbabaie, A., 2010, “Consensus Over Ergodic Stationary Graph Processes,” IEEE Trans. Autom. Control, 55(1), pp. 225–230.
Atanasov, N., Le Ny, J., and Pappas, G., 2014, “Distributed Algorithms for Stochastic Source Seeking With Mobile Robot Networks: Technical Report,” preprint arXiv: 1402.0051.
Spanos, D., Olfati-Saber, R., and Murray, R., 2005, “Dynamic Consensus on Mobile Networks,” 16th IFAC World Congress, International Federation of Automatic Control, Prague, Czech Republic.
Capulli, F., Monti, C., Vari, M., and Mazzenga, F., 2006, “Path Loss Models for IEEE 802.11a Wireless Local Area Networks,” 3rd International Symposium on Wireless Communication Systems, pp. 621–624.
Durrett, R., 2010, Probability: Theory and Examples, Vol. 4, Cambridge University, Cambridge University Press, New York.

## Figures

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. 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. 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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