Energy-Aware Coverage Control with Docking for Robot Teams Jason Derenick, Nathan Michael and Vijay Kumar

Abstract— In this paper, we formulate a distributed, energyaware control policy aimed at enabling persistent surveillance of a specified region of interest by teams of networked robots. Central to our formulation is the fundamental idea that as an agent participating in coverage approaches a low energy reserve the team should cooperatively adjust the coverage formation to allow the agent to return to a designated base station, where it can recharge before rejoining the effort. Towards this end, we build upon recent efforts in employing Centroidal Voronoi Tessellation (CVT)-based coverage control laws by defining a policy that exploits a power-dependent weighting scheme that embeds an agent’s trade-off to achieve its coverage mission and to maintain a desired energy reserve to guarantee its own safety. Stability of the proposed approach is considered, and we show that coupling our continuous controller with a straightforward switching mechanism guarantees every agent will return to its base station safely. Simulation results are presented to verify and demonstrate the utility of the proposed control scheme.

I. I NTRODUCTION Developing robust, decentralized control for collaborative persistent surveillance still poses a formidable challenge to the robotics community. In many cases, the duration of the coverage task (e.g., border surveillance, environmental monitoring, etc.) will exceed an agent’s maximum runtime. This realization implies the need for a control-scheme that explicitly accounts for the dynamics of each agent’s energy consumption to ensure its safe, long-term operation. Towards enabling this technology, we present an energyaware coverage controller that combines an agent’s desire to achieve its coverage mission with its desire to achieve a docking station to recharge. Central to our approach is combining a Centroidal Voronoi Tessellation (CVT)-based control policy with a docking control law via an energydriven weighting scheme. Intuitively, as an agent approaches a low energy reserve its desire to achieve its designated base station increases. As such, it will be driven to this location, while still participating in the team’s coverage formation. As the agent moves, its teammates appropriately adjust their coverage to support its needs. To guarantee convergence of each agent to its designated base station, we augment our energy-aware control policy with a fail-safe switching mechanism that permits an agent to “drop-out” of the formation when it determines it can no longer support the coverage task without jeopardizing itself. J. Derenick, N. Michael and V. Kumar are with the GRASP Laboratory, University of Pennsylvania, Philadelphia, PA 19104, USA.

{jasonder,nmichael,kumar}@seas.upenn.edu The authors gratefully acknowledge discussions with Luciano Pimenta and Mac Schwager as well as support from ARO Grant W911NF-05-10219, ONR Grants N00014-07-1-0829 and N00014-08-1-0696, and ARL Grants W911NF-08-2-0004 and W911NF-10-2-0016 .

xBi xBj

xi xEi xEj

Q xj

xk

xEk xBk

Fig. 1. Geometric assumptions and notation for agent state, entry points, and base station. Agent ri enters/leaves the coverage region Q via XEi ∈ ∂Q and then drives to its unique base station xBi . When an agent achieves xBi it is considered docked.

II. R ELATED W ORK Coverage control has garnered much attention from the robotics community. Most notable of these efforts is [1], who exploit the locational optimization paradigm to develop control strategies inspired by Lloyd’s Algorithm to drive agents towards a weighted Voronoi tessellation. Others have also explored the utility of Voronoi-based control strategies to yield impressive results. Among them is [2], who extends the results of [1] to heterogenous robot teams with varying sensing ranges operating in non-convex environments. [3] also considers Voronoi coverage in nonconvex environments and develops a Tangent Bug-style control scheme to ensure agents properly converge. [4] develops a distributed controller where agents posit themselves to optimize the measurement of sensory information. Finally, [5] considers a time-varying density function over the environment and formulates a controller to enable simultaneous coverage and tracking of mobile targets. Despite the large body of literature on coverage control, there are only a small number of papers that explicitly address energy-aware control. Of these, the most relevant is [6], who develops a Voronoi-based control scheme that directs agents to balance energy expenditures across the network by having those with abundant power resources compensate for those without. Additionally, [7] develops an energy-aware control scheme for robot teams charged with completing a task in fixed time. Also relevant to this research are recent efforts in dynamic vehicle routing [8], [9], persistent surveillance [10], [11], and robotic deployment [12], [13]. III. P ROBLEM S TATEMENT Begin by letting R = {r1 , . . . , rn } denote a robot configuration charged with covering a closed, convex, polygonal

workspace Q ⊂ R2 . Let xi (t) ∈ R2 denote the timebased positional state of ri ∈ R at time t and let X(t) = T (x1 (t), . . . , xn (t)) ∈ R2n denote the team state. Assume each agent is governed by single-integrator dynamics, i.e. x˙ i = ui (1) As each agent’s energy approaches a minimum level during a coverage mission, we would like it to exit the coverage region Q (see Fig. 1) and return to its assigned base station. At the base station, its batteries can be manually exchanged or it can autonomously dock to a charger. To achieve such behavior, we assume ri is capable of measuring its time-dependent voltage reserve, denoted vi (t) ∈ R+ and time-dependent change in this quantity v˙ i . Accordingly, we define a desired voltage level vsi that the agent is to maintain before transitioning to its unique base station, xBi ∈ R2 \ Q for recharging. Furthermore, let the vector V(t) = {v1 (t), . . . , vn (t)} denote the vector of timedependent voltage reserve values. Additionally, with each base station, we associate a unique access point xEi ∈ ∂Q, where ∂Q denotes the boundary of our coverage region. As agents are leaving refueling stations, they travel from xBi to xEi and enter Q from this position, at which point their coverage control takes over. Without loss of generality, we assume xBi − xEi is orthogonal to the boundary segment containing xEi . Figure 1 illustrates our notation and geometric assumptions. Notice that our choice to restrict xBi to lie outside of Q is quite natural. This is especially the case when one considers persistent coverage over potentially unknown or hostile areas. Given these assumptions, the problem addressed by this paper can be stated as follows: Problem 1: Develop an energy-aware control policy that embeds spatial coverage control while guaranteeing lim k xi (t) − xEi k2 = 0, ∀ri ∈ R

vi (t)→0

(2)

Towards solving this problem, we couple the coverage control and docking behaviors via a single continuous control policy that attempts to smoothly transition ri out of coverage as v(t) → vsi . Such an approach is beneficial as it allows teammates of low power agents to smoothly adjust coverage and allow their teammate to transition to its refueling station’s access point in Q. The low-power agent also participates in coverage until it exits the region or until it reaches a critical value (to be discussed shortly) forcing it to abandon coverage and return immediately to its dock. IV. E NERGY- AWARE C OVERAGE C ONTROL As our objective is to, in part, enable spatial coverage control, it is natural to begin by considering a Voronoi-based control scheme. As noted in Section II, such controllers have garnered much interest from the robotics community.

associated with pi is given by Ωi = {qj : k pi − qj k ≤ k pj − qj k, ∀pj ∈ P }

(3)

where qi ∈ Q ⊂ R and k  k denotes the Euclidean distance metric. Note, that similar to previous work in Voronoibased control (e.g., [5]), we utilize a weak inequality in this definition to facilitate analysis. A special kind of Voronoi-tesselation is called the Centroidal Voronoi Tessellation (CVT). In a CVT, the point pi corresponds to the centroid of Ωi , where the centroid and mass of Ωi are respectively defined Z Z 1 qφ(q)dq, MΩi = φ(q)dq (4) CΩi = MΩi Ωi Ωi Observe that in this formulation, we include a time-invariant density function φ(q) : Q → R used to assign weight to points in Q. The higher the value of φ at a particular point, the more important it is to cover. A classic discrete-time algorithm for computing CVT’s is Lloyd’s Algorithm [14], which follows three distributed steps: 1.) Compute Ωi , ∀pi 2.) Compute CΩi 3.) Move pi to CΩi , ∀pi . This process recurses until pi = CΩi , ∀pi . Leveraging this intuition, [1] establishes a continuous-time variation yielding a controller for single-integrator systems (i.e., dynamics given by (1)). The policy is given as follows ui = γ {CΩi − pi }

(5)

and ensures convergence to a CVT. In this formulation, γ ∈ R+ serves as a tuning parameter. Continuous-time CVT algorithms have a natural relationship to locational optimization, where the objective is to place resources (in our case robots) to minimize a global cost function (e.g., aggregate sensor error). Additional details can be found in [15]. B. An Energy-Aware Controller Our current objective is to develop a continuous-time control policy that aims to transition agents from a CVT coverage formation to a formation enabling low-energy agents to achieve access points. In contrast to a switching scheme, where agents would discretely drop out of coverage, a continuous controller will allow the weakened agent to still contribute to the coverage task as it drives to exit Q, while enabling its teammates to adjust properly. Towards this end, we consider an approach where we perturb ri ’s cell’s centroid, CΩi , over time, drawing it to xEi . This perturbation is given by a convex combination of CΩi and xEi , where the weights are governed by a smooth functional of the agent’s current voltage reserve, vi . Accordingly, we begin our formulation by defining the coverage energy function (denoted with subscript C) EC =

n X

γ k gi (t)CΩi (t) + hi (t)xEi − xi (t) k2

(6)

i=1

A. Voronoi-based Control Given a finite set of points P = {p1 , . . . , pm } ⊂ Q, the associated Voronoi tessellation is defined such that the cell

where gi (t) = αi (vi (t)), hi (t) = 1 − gi (t) with weighting functional αi : R+ → [0, 1], which is a monotonically increasing function of voltage, vi (t). γ is as defined in (5).

composed of linear segments with `ij denoting the segment separating Voronoi neighbors ri and rj . As such, the second term in Equation (10) becomes  T X Z ∂LΩi ∂`ij φ(q)dq (11) = q n ˆ ij ∂xi ∂xi `ij ∀j∈Ni

Fig. 2. α(vi (t)) for various values of γα . In this plot, the fuel supply corresponds to agent voltage with vsi set at 4.25v.

A natural choice for αi is a standard sigmoid (see Fig.2), which is defined by  −1 s αi (vi (t)) = 1 + eγαi (vi (t)−vi ) (7) where γαi ∈ R+ is a positive gain for the transition function. αi provides a continuous mechanism to shift between coverage and docking behaviors. To see this, note that αi (vi ) = 0 converts the objective from one minimizing the disparity from CΩi to xEi . αi (vi ) = 1 has the opposite effect. As EC is time-varying, our initial analysis follows that of [5]. In contrast to our work, their consideration is a timevarying density function to enable simultaneous coverage of Q and tracking of mobile targets (SCAT). Indeed, it can be shown that our results, when combined with theirs, yields an energy-aware formulation of the SCAT controller. To ease derivation, we abandon our functional notation with respect to time, and begin by computing the timederivative of EC . Doing so yields ) ( T n X ∂E ∂E C C v˙i (8) x˙ i + E˙C = ∂xi ∂vi i=1 C Observe that computing the partial ∂E ∂xi in Equation (8) requires computing the partials of the centroid and mass of Ωi with respect to agent position. Intuitively, this is because each shared Voronoi cell boundary is implicitly a function of the pair of Voronoi neighbors Rit separates. Adopting the notation of [5], we define LΩi = Ωi qφ(q)dq and note that (  T ) ∂CΩi 1 ∂LΩi ∂MΩi = − CΩi (9) ∂xi MΩi ∂xi ∂xi

Since CΩi and MΩi are defined in terms of integrals, we leverage the classical technique from multivariate calculus of differentiating under the integral sign [16]. Doing so, we see  T Z Z ∂ ∂(∂Ωi ) ∂LΩi = qφ(q)dq + q n ˆi φ(q)dq ∂xi ∂xi Ωi ∂xi ∂Ωi (10) where the latter term emerges due to the change in the boundary of CΩi , which we denote ∂Ωi with outward facing normal n ˆ i . Noting that the former term does not contain a function of xi , it vanishes. Leveraging this observation, we also note the boundary of Ωi corresponds to the boundary of a convex polygonal region

where Ni denotes the set of Voronoi neighbors with respect to agent ri , `ij is as previously defined with outward facing normal n ˆ ij . Additionally, observe, that by definition, it holds that n ˆ ij = −ˆ nji . Similarly, it is straightforward to arrive at X Z ∂`ij ∂MΩi = n ˆ ij φ(q)dq (12) ∂xi ∂xi ∀j∈Ni `ij   ∂` Fortunately, computing these integrals, namely ∂xiji nij , can be efficiently done in a straightforward manner using a parametric formulation as was shown in [5]. The interested reader is referred there for additional details. Taking note of these observations and applying some algebra, Equation (8) can be stated as follows E˙C =

n X

T

−2γ {gi CΩi + hi xEi − Si − xi } x˙ i + Gi (13)

i=1

where we define the term Si as P ∂LΩi gi Si = ∂xi MΩi {gi CΩi + hi xEi − xi } − ∀j∈Ni P ∂LΩi gj  ∂xi MΩj gj CΩj + hj xEj − xj + ∀j∈N P ∂MΩi i gj T  ∂xi MΩj CΩj gj CΩj + hj xEj − xj − ∀j∈Ni P ∂MΩi gi T ∂xi MΩ CΩi {gi CΩi + hi xEi − xi }

(14)

i

∀j∈Ni

C ˙ Additionally, Gi = ∂E ∂vi vi is the additive term introduced due to the time varying nature of agent ri ’s voltage.

Gi = 2γαi Hi

∂αi v˙ i ∂vi

(15)

with Hi being defined as Hi = k CΩi k2 αi (vi ) + CΩTi xEi − 2CΩTi xEi αi (vi ) −CΩTi xi − k xEi k2 + k xEi k2 αi (vi ) + xTEi xi (16) Given Equation (13), we now consider the control policy
pi κ k gi CΩi + hi xEi − xi k2 +Gi (17) uCi = 2γ where κ ∈ R+ serves as a gain and pi is given by pi =

{gi CΩi + hi xEi − Si − xi } k gi CΩi + hi xEi − Si − xi k2

(18)

Doing so, we arrive at the following result Theorem 4.1: The control policy given by (17) results in exponential decay of (6). Proof: Computing the time-derivative of EC , yields (13). Back-substituting (17) into (13), we obtain E˙C = −κ

n X i=1

γ k gi CΩi + hi xEi − xi k2 = −κEC

(19)

It follows that EC (t) = EC (0)e−κt . Theorem 4.1 shows that (17) leads to the exponential convergence of xi to gi CΩi +hi xEi for all robots; however, it should be noted that a theoretical singularity can occur since xi = gi CΩi + hi xEi does not necessarily imply Si 6= 0. A similar singularity was also encountered in [5], who showed the feasibility of such cases. Although this caveat prevents us from establishing the exponential stability of (17) in theory, it must be emphasized that in practice it can be safely mitigated/removed by simply including a small regularization term  ∈ R+ ,  << 1 in the denominator of (18). A key advantage of (17) is that it lends itself to a distributed implementation, which highlights its scalability to larger scale robot teams. This is the case, because each agent only requires the centroid, state and access point positions associated with its Voronoi neighbors. C. A Fail-Safe Docking Mechanism Establishing the theoretical convergence of an agent to its access point with sufficient energy reserves to then achieve its base station, using only (17), is hardly trivial. To understand this, observe that even if αi (vi ) = 0, indicating that ri only desires to reach its base station, its role in coverage still affects its control inputs as (14) includes terms that are solely a function of ri ’s Voronoi neighbors. Additionally, these terms do not necessarily cancel. This interdependence emerges because the centroid associated with ri ’s neighbors is also implicitly a function of ri ’s positional state. Although our results (see Section V) indicate that (17) will successfully drive the agent to achieve its access point with sufficient energy, we choose to formulate a fail-safe switching mechanism to make this a system-wide performance guarantee. Our approach in achieving this result is to consider (17) in the context of a switched system. Intuitively, agents will use the continuous formulation to govern system dynamics for as long as possible before an agent decides that it can no longer wait to achieve its access point. In such a case, it will switch to an explicit docking-only controller and, in doing so, abandon its role in the coverage task. This mechanism serves only as a fail-safe to ensure the agent will be able to safely continue by directly driving to xBi via xEi . Formalizing this approach, we assume each agent also has a minimal, critical voltage level vcrit ∈ R+ , defined i , that represents the minimal amount such that vsi > vcrit i of energy reserve required to guarantee convergence of xi can be expressed as two component to xBi . Note that vcrit i crit crit terms vcrit = v + v i Ei Bi where the former corresponds to the minimal voltage required to achieve xEi form within Q and the latter is similarly defined with respect to xEi and xBi . Given this definition, we begin our analysis by proposing the following switching policy for agent ri  C ui , vi > vcrit i ui = (20) uD i , otherwise where the docking controller, uD i is formalized as κ {xEi − xi } uD i = 2

(21)

with κ ∈ R+ being defined as in (17). Accordingly, we consider the aggregate system energy as it evolves between switches that occur as agents leave coverage after reaching vcrit i . Our result can be formulated as follows Theorem 4.2: Let CR ⊆ R denote the set of agents with C vi > vcrit i (i.e., agents using ui ) and let DR = R\CR denote D the set of agents using ui . Let T denote the time interval before the next agent switches from CR to DR . Using said control scheme results in exponential decay of the aggregate energy function E = EC + ED over T , where EC is as in (6) with the caveat that the summation is now over CR and X γ k xEi − xi k2 (22) ED = ∀ri ∈DR

where γ ∈ R+ is as defined in (6). Proof: By Theorem 4.1, it holds that E˙C = −κEC . BackD ˙ i we see E˙D = −κED . As substituting (21) into E˙D = ∂E ∂xi x E E C D such, E˙ = ∂xi x˙ i + ∂xi x˙ i = −κ(EC + ED ) = −κE. Thus, it follows that E(t) = (EC (0) + ED (0))e−κt . An implication of this result is that the controller given in (21) ensures exponential convergence of xi to xEi to within a desired tolerance in finite time More rigorously, xi converges to within a ball of radius 2i in time   i tEi = −2κ−1 log (23) i (0) where i (0) denotes the initial distance between the agent’s position and xEi . Similarly, we can define the time tBi that it takes ri to travel from xEi + 2i to within 2i units of xBi . Although this result is sufficient to guarantee reaching xBi in time tEi + tBi as a function of the agent’s time-dependent position, it fails to capture the temporal constraint imposed by the robot’s remaining power. To embed this constraint, we assume that a continuos voltage model Vi : R+ → R+ , has been generated to map agent runtimes to voltage values. As the model reflects energy consumption, it is monotonically decreasing with agent runtime, before converging to zero. Given this model, we can establish the following result Theorem 4.3: Let t denote the current runtime of agent ri ∈ R having dynamics (1). Assume vi (t) ≥ vcrit (i.e., i as assume feasibility) and perfect model Fi . Choosing vcrit i follows
−1 vcrit (24) i ≥ Fi Fi (0) − tEi − tBi guarantees convergence to within a desired tolerance of xBi via (21), by first traveling through xEi . Proof: Employing the controller in (21), we see that convergence to within a desired tolerance of xEi will be obtained at time tEi as per Equation (23). Similarly, the time it takes to converge from xEi + 2 to within a desired tolerance of xBi is given by tBi . As such, the total runtime required to achieve xBi is given by tEi + tBi . Given the model Fi , it follows that F −1 (0) corresponds to the maximal runtime of ri . It is assured that F −1 (0) exists given that it is a monotonically decreasing polynomial. As such, Fi−1 (0)−tEi −tBi represents the time instant where all subsequent energy reserves must be dedicated to achieving

the base station to ensure docking. Applying the model to this value, the result follows. As such, choosing vcrit i with equality in (24) gives the minimal possible voltage required before the agent must abandon coverage altogether for the purposes of self-preservation. V. S IMULATIONS Although our formulation was general, we are specifically interested in leveraging our results to enable cooperative teams of autonomous aerial vehicles to perform persistent surveillance. We envisage coplanar rotorcrafts (see [17]), enforcing a z-level constraint, with downward facing sensing capabilities providing persistent coverage of some region of interest. It is in this context that we frame our simulations. A. Generating a Voltage Model To derive an accurate voltage model for use in our Matlabbased simulation framework, we conducted experiments with the AscTec Hummingbird quadrotor from Ascending Technologies, GmbH [17]. In this experiment, a single quadrotor hovered for ≈ 1050 seconds before fully draining its battery. Figure 3 shows the resulting ninth order polynomial which yielded a root mean square error of 0.012.

Fig. 3. A ninth order polynomial fit to raw voltage values obtained from our hover experiment with an AscTec Hummingbird (RMSE = 0.012).

Although generating a voltage model from only a hover state does not capture voltage dissipation as a result of linear motion, we believe that such a model is a reasonable approximation based upon the observation that the battery life is nearly equivalent if the robot is stationary or mobile. B. Results a) Verifying Exponential Decay: To verify the exponential convergence of energy function E = EC +ED over interval time T , we implemented the proposed hybrid controller in Matlab using the voltage model of Section V-A and generated a random configuration of eight robots. Of those eight, 5 operated above a specified vcrit = 10 with the transitional i voltage vsi = 11. The configuration was randomly placed in a rectangular region and the controller was run for a fixed number of iterations. Figure 4 shows the evolution of the Lyapunov function as well as the theoretical bound given by E(t) = (EC (0) + ED (0))e−κt for comparison. The trend closely corresponds with our theoretical bound.

Fig. 4. The evolution of the Lyapunov (dashed) function closely follows the theoretical bound (solid), verifying our analysis.

b) Utility for Persistent Surveillance: In a second simulation, we implemented our results in order gauge the utility of our switched control scheme (20) for enabling persistent surveillance. Specifically, we consider a scenario where a team of seven robots was tasked with maintaining coverage over a specified region of interest for approximately four hours. The team was modeled as coplanar rotorcrafts whose downward facing sensing capabilities projected a coverage area of unit radius, which represented ≈ 20% the total area of our rectangular coverage region Q. To complete this experiment, we formulated a straightforward persistent coverage algorithm that introduced six of the seven agents into Q in a staggered fashion (≈ one agent every two minutes). The seventh robot remained on standby/docked until a docking agent, ri achieved xEi . At this point our algorithm checked if any agents were available for deployment, and, if so, released one to fill the coverage void left by ri . It was assumed that once an agent achieved its base station that its batteries were efficiently swapped. The time for battery exchange was set at 25 seconds, which is modeled based upon our experience with the platform. We chose γ = 1, κ = 1 and γα = 10 and used the voltage model of Section V-A. The algorithm continued until each robot cycled 12 times in and out of Q, yielding a mission length of ≈ 4 hours. Figure 5 shows the behavior of our control scheme during progression of said algorithm. To quantify the coverage quality over the duration of the simulated run, we define the following normalized metric [ A (SR ) CQ = , SR = {Si ∩ Q} (25) A(Q) ∀ri ∈R

where Si denotes agent ri ’s unit-disc sensor cover and A : R → R represents the area function. Figure 6 shows the evolution of (25). At t = 606.62 seconds, the sixth agent enters the environment. From this point until the first agent leaves Q for the final time (t = 13, 404.23), the mean area coverage is 95% (σ = 0.03%). Finally, it should also be noted that all 84 docking attempts were successful using controller (17) without any agent having to leverage the fail-safe docking controller. VI. C ONCLUSION In this paper, we present an energy-aware control policy that embeds an agent’s trade-off to achieve its coverage

(a) t = 705.04

(b) t= 940.03

(c) t = 1010.03

(d) t = 1908.06

(e) t= 2148.09

(f) t = 2504.53

Fig. 5. Evaluating the Utility of our Energy-Aware Controller for Persistent Surveillance: (a) Six robots (with one robot on standby) establish coverage of Q. (b) Agent r1 (Note: ri is labeled i in the figures and xBi is labeled Bi ) drives to its access point while the team appropriately adjusts coverage. (c) Agent r7 enters Q to help restore the coverage lost with r1 ’s departure. (d-f) Respectively, agents 7, 2 and 5 achieving their access points.

Fig. 6. Evolution of the coverage-quality metric (25) for our persistence coverage experiment corresponding to Figure 5.

mission and to maintain energy reserves to guarantee its own safety. Central to our results is utilizing an energyaware weighting scheme that allows the team to continuously transition between coverage-directed and docking-directed behaviors. As a fail-safe to ensure each agent converges to its assigned base station, we formulate a fail-safe switching mechanism that has an agent abandon coverage when it reaches a critical voltage level. Accordingly, we provided simulation results verifying our analysis and demonstrating the utility of our control scheme for persistent coverage. R EFERENCES [1] J. Cortes, S. Martinez, T. Karatas, and F. Bullo, “Coverage control for mobile sensing networks,” vol. 20, no. 2, pp. 243–255, 2004. [2] L. C. A. Pimenta, V. Kumar, R. C. Mesquita, and G. A. S. Pereira, “Sensing and coverage for a network of heterogeneous robots,” in CDC, 2008, pp. 3947–3952. [3] A. Breitenmoser, M. Schwager, J. C. Metzger, R. Siegwart, and D. Rus, “Voronoi coverage of non-convex environments with a group of networked robots,” in Proc. of the International Conference on Robotics and Automation (ICRA 10), May 2010, pp. 4982–4989.

[4] M. Schwager, D. Rus, and J. J. Slotine, “Decentralized, adaptive coverage control for networked robots,” International Journal of Robotics Research, vol. 28, no. 3, pp. 357–375, March 2009. [5] L. C. A. Pimenta, M. Schwager, Q. Lindsey, V. Kumar, D. Rus, R. C. Mesquita, and G. A. S. Pereira, “Simultaneous coverage and tracking (scat) of moving targets with robot networks,” in Proceedings of the Eighth International Workshop on the Algorithmic Foundations of Robotics (WAFR 08), December 2008. [6] A. Kwok and S. Martinez, “Energy-balancing cooperative strategies for sensor deployment,” in Proceedings of the 46th IEEE International Conference on Decision and Control, New Orleans, Dec 2007. [7] Y. Mei, Y. hsiang Lu, Y. C. Hu, and C. S. G. Lee, “Deployment of mobile robots with energy and timing constraints,” IEEE Transactions on Robotics, vol. 22, pp. 507–522, 2006. [8] A. Arsie and E. Frazzoli, “Efficient routing of multiple vehicles with no explicit communications,” International Journal of Robust and Nonlinear Control, vol. 18, no. 2, pp. 154–164, January 2007. [9] S. L. Smith, M. Pavone, F. Bullo, and E. Frazzoli, “Dynamic vehicle routing with priority classes of stochastic demands,” SIAM Journal of Control and Optimization, vol. 48, no. 5, pp. 3224–3245, 2010. [10] N. Nigam and I. Kroo, “Persistent surveillance using multiple unmanned air vehicles,” in Proc. of IEEE Aerospace Conference, 2008. [11] A. Matlock, R. Holsapple, C. Schumacher, J. Hansen, and A. Girard, “Cooperative defensive surveillance using unmanned aerial vehicles,” in Proc. of the 2009 American Control Conference (ACC ’09), 2009. [12] J. L. Ny and G. J. Pappas, “Sensor-based robot deployment algorithms,” in Proceedings of the 49th IEEE International Conference on Decision and Control, Atlanta, GA, Dec 2010. [13] M. Schwager, D. Rus, and J. J. Slotine, “Unifying geometric, probabilistic, and potential field approaches to multi-robot deployment,” International Journal of Robotics Research, vol. 30, no. 3, pp. 371– 383, March 2011. [14] S. P. Lloyd, “Least squares quantization in pcm,” IEEE Transactions on Information Theory, vol. 28, pp. 129–137, 1982. [15] A. Okabe, B. Boots, K. Sugihara, and S. N. Chiu, Spatial tessellations: Concepts and applications of Voronoi diagrams, 2nd ed., ser. Probability and Statistics. NYC: Wiley, 2000, 671 pages. [16] H. Flanders, “Differentiation under the integral sign,” American Mathematical Monthly, vol. 80, pp. 615–627, 1973. [17] “Ascending Technologies, GmbH,” http://www.asctec.de.