Symmetric Coverage of Dynamic Mapping Error for Mobile Sensor Networks

2011 American Control Conference on O'Farrell Street, San Francisco, CA, USA June 29 - July 01, 2011 Symmetric coverage of dynamic mapping error for mobile sensor networks Carlos H. Caicedo-Nuńez˜ and Naomi Ehrich Leonard Abstract—We present an approach to control design for nonuniform density fields. Related problems include control a mobile sensor network tasked with sampling a scalar field of a mobile sensor network in a noisy, possibly time-varying and providing optimal space-time measurements. The coverage field for minimum-error spatial estimation [9], [10] and metric is derived from the mapping error in objective analysis (OA), an assimilation scheme that provides a linear statistical cooperative exploration of features [11]. In [12] agents are estimation of a sampled field. OA mapping error is an example deployed in a distributed way in order to maximize the of a consumable density field: the error decreases dynamically probability of event detection. The authors of [13] study the at locations where agents move and sample. OA mapping error distributed implementation of maximizing joint entropy in is also a regenerating density field if the sampled field is measurements. However, none of these methods have been time-varying: error increases over time as measurement value decays. The resulting optimal coverage problem presents a chal- designed to specifically address a spatial-temporal density lenge to traditional coverage methods. We prove a symmetric field that changes in response to the motion of the agents. dynamic coverage solution that exploits the symmetry of the In this paper we propose a coverage strategy for a density domain and yields symmetry-preserving coordinated motion field defined by OA mapping error in the case that the of mobile sensors. Our results apply to symmetric sampling sampled field is time-varying. The approach, based on greedy regions that are non-convex and non-simply connected. search, exploits symmetry in the sampling region and yields I. INTRODUCTION symmetric coverage patterns of mobile agents. We show how A mobile sensor network used to observe a scalar field the symmetry group that defines the spatial configuration of over a finite region can be made most efficient if it is the mobile sensors relates to the symmetry of the region, and designed for optimal measurement coverage. If the field we prove that the search strategy preserves the symmetry varies spatially and temporally, sensing agents should be of the spatial configuration. The method does not require a dynamically distributed in space and time to match the spatial convex or simply connected sampling region; we illustrate and temporal scales of the field. This can be formulated as with simulation examples. the problem of designing motion control laws for the agents In Section II, we review OA mapping error. We de- that maximize information in the data collected. fine the sensor network system and the coverage goals in Inspired by ocean sampling field experiments in Monterey Section III. In Section IV we show that a well-known Bay CA [1], we examine a coverage problem in which coverage solution for static problems does not address the information derives from the classic objective analysis (OA) coverage problem for minimizing OA mapping error. We mapping error. OA is linear statistical estimation based on present our symmetry-based approach in Section V and prove specified field statistics, and the mapping error provides invariance of symmetry in the spatial distribution of agents. a measure of the residual uncertainty in the model [2]. We discuss performance and robustness issues in Section VI. Since reduced uncertainty, equivalent to increased entropic In Section VII we discuss future directions. information, implies better measurement coverage, the OA II. OA MAPPING ERROR mapping error provides a useful density field for determining the coverage metric [2], [3]. In this section we review the method of objective analysis In this context, optimal coverage is achieved by seeking (OA); for further details see [2]. maximum reduction of the residual OA mapping error; this OA models a scalar sampling field observed at a point x problem is particularly challenging because mapping error and a time t as a random process T (x, t). It is assumed that changes with the spatial dynamics of the sensing agents. a priori information about this process is available, namely Indeed, error decreases near the locations where agents take the mean value T and the covariance B of fluctuations about measurements, reflecting the new information acquired, but the mean: as time passes, the relevance of past measurements decreases ′ ′ E and the error increases. B (x, t, x , t ) = T (x, t) − T (x, t) Coverage for static density fields has been studied for × T (x′, t′) − T(x′, t′) . (1) convex and non-convex domains; see, e.g., [4]–[8]. In [8] OA is a linear estimator; it provides an estimate for the field convergence is proved in the case of slowly time-varying, as a linear combination of the discrete set of measurements This research was supported in part by ONR grant N00014-09-1-1074 obtained up to time t as and AFOSR grant FA9550-07-1-0-0528. C.H. Caicedo-N. and N.E. Leonard are with the Department of Mechani- P cal and Aerospace Engineering, Princeton University, Princeton, NJ 08544, Tˆ (x, t) = T (x, t)+ ηk (x, t) Mk − T (xk, tk) , (2) USA ({ccaicedo,naomi}@princeton.edu). kX=1 978-1-4577-0079-8/11/$26.00 ©2011 AACC 4661 where P is the number of measurements, Mk is the k-th A coverage metric can be derived from ρ(x, t). For measurement, which is taken at location xk at time instant example, entropic information I(t), computed as minus the tk, and ηk are coefficients that minimize the least square log of the average over D of ρ(x, t) at time t, provides one error of the estimate: natural choice of coverage metric [3]. In the next section, we show that an extension of a A (x, t, x, t) = E T (x, t) − Tˆ (x, t) static coverage algorithm to the density field ρ(x, t) = h A(x, t, x, t) does not address the problem; indeed, the × T (x, t) − Tˆ (x, t) . (3) i coverage algorithm converges to a static configuration. Our An important aspect of OA is that the residual error approach, presented in Section V, uses a kind of greedy A (x, t, x, t) depends on the location and time of the search to prevent the sensors from becoming static and measurements but not on the measured quantity (nor on T¯). exploits symmetry to ensure that the domain is sufficiently The covariance between data points is given by well covered. We discuss performance with respect to the entropic information metric I(t) in Section VI. [C]jl = ηδjl + B (xj, tj, xl, tl) , (4) IV. EXTENSION OF STATIC COVERAGE where η is the measurement noise (given by the physical In this section we apply the static coverage law of characteristic of the sensors) and δ is Kronecker’s delta. ij Cortes´ et al. [4] to the consumable density field ρ(x, t) = As shown in [14], the error (3) can be written as A(x, t, x, t). The approach in [4] has been successfully P P extended to include some time-varying fields, as in [15]. Sub- ′ ′ ′ ′ A (x, t, x , t ) = B (x, t, x , t )− [B (x, t, xj, tj) stituting in the time-varying density ρ(x, t), the cost function kX=1 Xl=1 of [4] to be minimized for optimal coverage becomes −1 ′ ′ × C B (xl, tl, x , t ) . (5) N jl 2 i J(t) = kx − p k ρ(x, t) dx, (8) Z i As is typical in ocean modeling [3], we assume that Xi=1 Vi ′ 2 ′ 2 kx−x k |t−t | − − where Vi is the Voronoi region associated to agent i. The ′ ′ σ τ B (x, t, x , t ) = σ0e „ « „ « , (6) coverage control law of [4] for agent i is given by where σ0 denotes the space-time average for the field covari- p˙ i = − (CVi − pi) , (9) ance B, and σ and τ represent its length and time scales, respectively. where CVi (t) is the centroid of Vi at time t: We define the density field that describes the OA mapping 1 error as ρ(x, t) = A(x, t, x, t). For τ < ∞, the mapping CVi (t) = x ρ(x, t) dx, MVi (t) = ρ(x, t) dx. MV Z Z error is regenerating: if no new measurements are made, then i Vi Vi ρ(x, t) → σ0 as t → ∞. If τ = ∞ the mapping error is not The computation of Voronoi regions is independent of regenerating: if no new measurements are made in any time density ρ, but the control law (9) directs the ith agent to the interval I = [t0, t1], then ρ (x, t) is constant for t ∈ I. density weighted centroid of its Voronoi region. This yields a static configuration of sensors that can initially reduce III. SAMPLING SYSTEM AND GOALS ρ in some locations; however, it falls short of addressing Consider a set of N autonomous agents (mobile sensors), the coverage problem at hand. As described above, con- indexed by i = 1, 2,..., N, deployed in a bounded region tinuous sensor dynamics are necessary for good coverage D ⊂ R2. As is traditional in the field, we model each agent performance. In the next Theorem we prove that in the case i as a first-order system [4], [5], [8], for which its location of infinite τ, the agents converge to a static configuration pi ∈ D evolves in time as independent of how much the density field ρ(x, t) has been reduced.

Load more