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    Coverage Maximization with Autonomous Agents in Fast FlowEnvironments

    Andrew Kwok Sonia Martnez

    Communicated by Emilio Frazzoli

    Abstract This work examines the cooperative motion of a group of autonomous ve-hicles in a fast flow environment. The magnitude of the flow velocity is assumed to begreater than the available actuation to each agent. Collectively, the agents wish to max-imize total coverage area defined as the set of points reachable by any agent within Ttime. The reachable set of an agent in a fast flow is characterized using optimal con-trol techniques. Specifically, this work addresses the complementary cases where thestatic flow field is smooth, and where the flow field is piecewise constant. The lattercase arises as a proposed approximation of a smooth flow that remains analyticallytractable. Furthermore, the techniques used in the piecewise constant flow case enabletreatment for obstacles in the environment. In both cases, a gradient ascent method isderived to maximize the total coverage area in a distributed fashion. Simulations showthat such a network is able to maximize the coverage area in a fast flow.

    Keywords Optimal Sensor Coverage Distributed Control of Multi-agent Systems

    Mathematics Subject Classification (2000) MSC 93C99 MSC 49J15

    1 Introduction

    The development of novel schemes that allow for the re-adaptation of unmanned vehiclesto changing environmental conditions is a vigorously studied field [13]. Indeed, the per-vasiveness of miniaturized embedded devices has spurred a great interest in the productionof lower-cost autonomous vehicles. However, the low power capacity of these vehicles canseverely affect the system autonomy and performance. This may be the case when operatingin harsh environments, where the actuation of a vehicle may not be sufficient to deal with

    Work supported by grants NSF CAREER Award CMMI-0643679 and NSF CNS-0930946.

    A. KwokOpera Solutions, San Diego

    S. MartnezMechanical and Aerospace Engineering, 9500 Gilman Drive, La Jolla, CA 92093E-mail:

  • 2 Andrew Kwok, Sonia Martnez

    environmental disturbances. In this work, we consider the case where the environment pre-vents an agent from maintaining a constant position, or even revisiting previous locations.Such scenarios are motivated by applications, such as performing surveillance or monitor-ing operations with lightweight UAVs in a large storm system. Specifically, we will assumethat the agents move in an a priori known flow field, but the field is faster than the absolutemaximum speed of an agent. We wish to enable a group of cooperative vehicles to movesuch that the area of the reachable set of points in the flow environment is maximized.

    Contributions. The main contribution of this work is an algorithmic solution to the au-tonomous deployment of vehicles to maximize coverage in a fast flow environment. Thiscan be used as a high-level routine for the deployment of UAVs in cooperative surveillanceor sensing tasks. We encode this objective through a performance metric that the agents wishto maximize as the total area of all points reachable within a chosen time horizon T . Thismetric contains global information since it is the sum of all the individual vehicles reach-able areas. However, using gradient ascent techniques, we show that local maximization ispossible with knowledge of only a subset of other agent locations.

    We restrict our study to flows that are either affine or piecewise constant, which allows usto provide an analytical characterization of the gradient ascent direction that vehicles mustfollow. Additionally, we allow for the presence of obstacles in the piecewise constant flowcase. The notion of reachable sets motivates the study of time-optimal trajectories withinthese two respective flow fields. Hence, we characterize the behavior of time-optimal trajec-tories for each flow case. While there is a rich body of existing literature regarding motionplanning with and without obstacles in various environments, we provide results that arespecifically adapted to the problem that we are addressing. In particular, we study the differ-ent cases of time-optimal trajectories that may arise in a piecewise constant flows, and usethese cases as a basis of a gradient-based coverage motion algorithm.

    Literature review. The problem of path planning in a flow field has its roots in the clas-sical optimal control problem known as Zermelos navigation problem [4], whose solutionrelies on use of the Pontryagin minimum principle, as in [5]. Motion planning for single ve-hicles in a flow has been lately studied [6,7]. More recently, [810] characterizes the optimaltrajectories for a Dubins-like vehicle in the presence of constant flows, while [11] presentsa robust control strategy for a Dubins vehicle in non-constant flows. The paper [12] pro-vides a generalization of time-optimal course changes for a Dubins vehicle moving across aheterogeneous terrain, and across boundaries where the vehicle dynamics change.

    A rich body of literature exists for obstacle avoidance, such as [1316]. The approachthat we take in this work most resembles the results found in [1315], where different possi-ble cases of minimum-time trajectories are considered and characterized. In particular, [14,15] deal with the scenario where the environment exhibits piecewise constant regions withdiffering trasversal costs, which results in a type of Snells law for trajectories. However, theresulting behavior of time-optimal trajectories is different from the results we present herefor fast flows. Namely, with hilly terrain as in [14] one finds that having switchback typepaths may be time-optimal, while in the flow environment we consider, only straight pathsare feasible. In [15], the authors examine the case where refracted trajectories may movealong interfaces between regions; similar to the results described in this paper, except thatin [15], the piecewise constant regions are isotropic with respect to desired travel direction.

    Regarding cooperative motion of multiple vehicles in a flow environment, the authorsin [17] have demonstrated stable formation control in the presence of an external time-invariant flow field. However, that work assumes that individual vehicles have full controlla-bility. Similarly, [18] addresses the case of maintaining a stable formation for sampling in an

  • Coverage Maximization with Autonomous Agents in Fast Flow Environments 3



    bT T

    Fig. 1 On the left, illustration of the , and quantities. The dashed circle represents the set of inputs uiwith magnitude 1. On the right, the reachable set RT (pi0) for pi0 under a constant horizontal flow.

    ocean environment with exogenous currents, while [19] combines time-optimal trajectoriesin a constant flow field with formation stabilization.

    With the exception of [9] and [20], previous work has considered the motion controlproblem of vehicles in a slow flow. That is, the vehicles had sufficient actuation to moveagainst the flow. We utilize a simple kinematic model, as in [17], to consider flows whichare both spatially varying and fast. Furthermore, much previous work, such as [9], considerconstant flow fields. In the interest of obtaining analytic solutions, we consider piecewiseconstant flow fields as an approximation of smooth spatially varying flows, such as in [20].Such a consideration introduces several interesting effects regarding time-optimal trajecto-ries over discontinuous flows. The work of [12] considers a similar situation, except that theeffect of the environment only imposes a change in maximum velocity of a vehicle. This isnot the case for this work due to the directionality of flows in the environment.

    2 Problem Statements and Definitions

    We consider the following kinematic model:

    pi = ui +V (pi) , (1)

    where ui(t) and V (pi) are piecewise smooth, ui 1, and V > 1, and the drift V (pi)models an environmental flow field. Because of this, an agent cannot maintain a constantposition, and is always pushed along the flow.

    Definition 2.1 (Reachable set) The reachable set, R(pi0), of an agent at position pi0 isthe set of points x X that an agent can reach in finite time starting from pi0 and using apiecewise smooth control input ui(t) with ui 1. The T -limited reachable set, RT (pi0),of an agent at position pi0, is the set of points that an agent can reach within time T using apiecewise smooth control input ui(t) with ui 1.

    The definition of the T -limited reachable set motivates the study of time-optimal trajectoriesin Sections 35. It can be shown, see [5], page 77, that maximum actuation is a necessarycondition for time optimality. Thus, for time optimal trajectory calculations, we assume thatui = (cosi,sini)T.

    To better relate flow field and agent motion, we utilize the following two parametersto describe the flow field. Referencing Figure 1, first (x) = atan2(V2(x),V1(x)) , is theflow direction. Next, : X ]0, 2 [ denotes the maximum feasible angle away from (x)that the agent can move in. More precisely, given V (x) and ui = 1, the endpoint of anagents velocity vector (1) lies on the dashed circle of Figure 1. Using trigonometry, one can

  • 4 Andrew Kwok, Sonia Martnez

    show that (x) = arcsin(

    1V (x)


    . Additionally, the choice of headings that result in thesemaximal travel directions are = (x) ( (x)+ 2 ). The feasible set of directions that anagent at pi would be able to travel in is then the interval [(pi) (pi),(pi)+ (pi)].

    Problem 1 Our main objective will be to define a distributed algorithm for the deploymentof agents in a flow environment. Specifically, we wish to maximize the area coverage metric:

    H (p1, . . . , pn) :=