Elsevier

Automatica

Volume 42, Issue 4, April 2006, Pages 661-668
Automatica

Brief paper
Optimal sensor placement and motion coordination for target tracking

https://doi.org/10.1016/j.automatica.2005.12.018Get rights and content

Abstract

This work studies optimal sensor placement and motion coordination strategies for mobile sensor networks. For a target-tracking application with range sensors, we investigate the determinant of the Fisher Information Matrix and compute it in the 2D and 3D cases, characterizing the global minima in the 2D case. We propose motion coordination algorithms that steer the mobile sensor network to an optimal deployment and that are amenable to a decentralized implementation. Finally, our numerical simulations illustrate how the proposed algorithms lead to improved performance of an extended Kalman filter in a target-tracking scenario.

Introduction

New advancements in the fields of microelectronics and miniaturization have generated a tremendous surge of activity in the development of sensor networks. The envisioned groups of agents are endowed with communication, sensing and computation capabilities, and promise great efficiency in the realization of multiple tasks such as environmental monitoring, exploratory missions and search and rescue operations. However, several fundamental problems need to be solved in order to make this technology possible. One main difficulty is the requirement for decentralized architectures where each agent takes autonomous decisions based on information shared with only a few local neighbors. Ongoing research work focuses on decentralized filters and data-fusing methods for estimation, and on the motion algorithms that guarantee the desired global behavior of the network. Ideally, both the motion control algorithms and estimation processes should be optimally integrated to make the most of the network performance.

In this paper, we investigate the design of distributed motion coordination algorithms that increase the information gathered by a network in static and dynamic target-tracking scenarios. To do this, we define an aggregate cost function encoding a “sensitivity performance measure” and design our algorithms to maximize it. This idea has been widely used in papers on optimum experimental design for dynamical systems with applications to measurement problems. For example, Porat and Nehorai (1996) and Uciński (2004) deal with problems on target tracking and parameter identification of distributed parameter systems. The motion control algorithms proposed in these papers either are computed via some off-line numerical method or are gradient algorithms. Often these algorithms are designed to maximize an appropriate scalar cost function and to choose the best sensor locations from a grid of finite candidates. Unfortunately, these schemes turn out to be not distributed since in order to define the control law for each agent, it is necessary to know all other agents’ positions at each step. A second set of relevant references are those on distributed motion coordination. Our proposed control algorithms are in the same spirit of those of cyclic pursuit (Marshall, Broucke, & Francis, 2004; Wagner & Bruckstein, 1997), flocking (Jadbabaie, Lin, & Morse, 2003), and coverage control (Cortés, Martínez, Karatas, & Bullo, 2004).

The contributions of this paper are the following. Under the assumption of Gaussian noise measurements with diagonal correlation, Section 2 presents closed-form expressions for the determinant of the Fisher Information Matrix for “range-measurement” models in non-random static scenarios, for 2D and 3D state spaces. This determinant plays the role of an objective function: we characterize its critical points in the 2D version and obtain sets of positions that globally maximize its value. If the sensors measure distances to the target, then an optimal configuration is one in which the sensors are uniformly placed in circular fashion around the target, confirming a natural intuition about the problem. Taking this optimal configuration as a starting point in Section 3, we then consider a target-tracking scenario where the sensors move along the boundary of a convex set containing the target. We define discrete-time control laws that, relying only on local information, achieve the uniform configuration around the target (estimate) exponentially fast. In essence, our laws are very intuitive and simple-to-implement interaction behaviors between the sensors along the boundary. Finally, in Section 4, we numerically validate our coordination and optimal deployment laws in a particular dynamic target-tracking scenario. Although the network achieves global optimum configurations for a non-random-static parameter estimation scenario, we simulate a dynamic random scenario. Our simulations illustrate the following reasonable conjecture: optimizing the sensitivity function for the static non-random case improves the performance of a filter (in our case an EKF) for the dynamic random scenario.

Finally, we point out that we assume that the process of estimation is performed by a central site or by a distributed process that we do not implement here. For works dealing with multisensor fusion possibly under communication constraints, we refer to Rao, Durrant-Whyte, and Sheen (1993), Sinopoli et al. (2004) and references therein.

Section snippets

Optimal placement of sensors

Here, we present the assumptions on our sensor network and target models in (1) (non-random) static estimation scenarios and (2) (random) dynamic parameter estimation scenarios. Other assumptions like those on the discrete motion of the sensors are given in Section 3. In this section, we obtain the corresponding fisher information matrices (FIMs) for the estimation models and analyze the global minima of their determinant as a means to guarantee increased sensitivity with respect to the

Motion coordination algorithms for sensor reconfiguration about static targets

This section presents a family of decentralized control laws that steers the sensors to a set of points of maximum for a particular class of costs functions previously defined. Specifically, we focus here on functions corresponding to measurement models with h(r)=r. Our analysis is related to the approaches in Cortés et al. (2004), Jadbabaie et al. (2003), Marshall et al. (2004) and Wagner and Bruckstein (1997). We make the following assumptions on the defining elements of our problem:

  • (i)

    a static

Target-tracking simulations with Kalman filtering and motion coordination algorithms

Here, we combine the developments of the former sections and we define the Active Target Tracking algorithm for collective improved sensing performance. We numerically simulate the algorithm to validate our approach. It is assumed that the estimation step is carried out after a round of communication has taken place to propagate all the measurements taken among the agents. The algorithm is summarized in the following table.

Name:ACTIVE TARGET TRACKING ALGORITHM 
Goal:Decentralized motion

Conclusions and future work

We have presented novel decentralized control laws for the optimal positioning of sensor networks that track a target. It would be of clear interest to modify our model by including upper bounds on the motion and detection range of the sensors. Broader future research lines include (1) the consideration of heterogeneous collections of sensors, (2) the dynamic assignment of sensors to different targets and (3) investigation of decentralized estimation and fusion schemes.

Acknowledgments

This material is based upon work supported in part by ONR YIP Award N00014-03-1-0512 and NSF SENSORS Award IIS-0330008. Sonia Martínez's work was partially supported by a Fulbright Postdoctoral Fellowship from the Spanish Ministry of Education and Culture.

Sonia Martinez received the Licenciatura degree in mathematics from the Universidad de Zaragoza, Spain, in 1997, and the Ph.D. degree in engineering mathematics from the Universidad Carlos III de Madrid, Spain, in 2002. She has been a Postdoctoral Fulbright Fellow both at the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, and at the Center for Control, Dynamical Systems, and Computation, University of California, Santa Barbara (from 2003 to 2005). She is currently

References (13)

  • G. Benet et al.

    Using infrared sensors for distance measurement in mobile robots

    Robotics and Autonomous Systems

    (2002)
  • Aranda, S., Martínez, S., & Bullo, F. (2004). On optimal sensor placement and motion coordination for target tracking....
  • Y. Bar-Shalom et al.

    Estimation with applications to tracking and navigation

    (2001)
  • J. Cortés et al.

    Coverage control for mobile sensing networks

    IEEE Transactions on Robotics and Automation

    (2004)
  • A. Jadbabaie et al.

    Coordination of groups of mobile autonomous agents using nearest neighbor rules

    IEEE Transactions on Automatic Control

    (2003)
  • J.A. Marshall et al.

    Formations of vehicles in cyclic pursuit

    IEEE Transactions on Automatic Control

    (2004)
There are more references available in the full text version of this article.

Cited by (507)

View all citing articles on Scopus

Sonia Martinez received the Licenciatura degree in mathematics from the Universidad de Zaragoza, Spain, in 1997, and the Ph.D. degree in engineering mathematics from the Universidad Carlos III de Madrid, Spain, in 2002. She has been a Postdoctoral Fulbright Fellow both at the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, and at the Center for Control, Dynamical Systems, and Computation, University of California, Santa Barbara (from 2003 to 2005). She is currently an Assistant Professor at the Department of Mechanical and Aerospace Engineering, University of California at San Diego. Her main research interests include systems and information theory, nonlinear control theory and robotics. Her current research focuses on the development of distributed coordination algorithms for the deployment of sensor networks and highly autonomous vehicle systems.

Francesco Bullo received the Laurea degree in Electrical Engineering from the University of Padova, Italy, in 1994, and the Ph.D. degree in Control and Dynamical Systems from the California Institute of Technology in 1999. From 1998 to 2004, he was an Assistant Professor with the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign. He is currently an Associate Professor at the Mechanical and Environmental Engineering Department, University of California, Santa Barbara. His research interests include motion planning and coordination for autonomous vehicles, motion coordination for multi-agent networks, and geometric control of mechanical systems. He is the coauthor (with Andrew D. Lewis) of the book “Geometric Control of Mechanical Systems” (New York: Springer, 2004, 0-387-22195-6). He is currently serving on the editorial board of the IEEE Transactions on Automatic Control, SIAM Journal of Control and Optimization, and ESAIM: Control, Optimization, and the Calculus of Variations.

This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Henk Nijmeijer under the direction of Editor Hassan Khalil.

View full text