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# Decomposition of Collaborative Surveillance Tasks for Execution in Marine Environments by a Team of Unmanned Surface VehiclesOPEN ACCESS

[+] Author and Article Information
Shaurya Shriyam, Brual C. Shah

Department of Aerospace & Mechanical Engineering,
University of Southern California,
Los Angeles, CA 90089

Satyandra K. Gupta

Department of Aerospace & Mechanical Engineering,
University of Southern California,
Los Angeles, CA 90089

1Corresponding author.

Manuscript received September 26, 2017; final manuscript received December 14, 2017; published online February 12, 2018. Assoc. Editor: Venkat Krovi.

J. Mechanisms Robotics 10(2), 025007 (Feb 12, 2018) (7 pages) Paper No: JMR-17-1329; doi: 10.1115/1.4038974 History: Received September 26, 2017; Revised December 14, 2017

## Abstract

This paper introduces an approach for decomposing exploration tasks among multiple unmanned surface vehicles (USVs) in congested regions. In order to ensure effective distribution of the workload, the algorithm has to consider the effects of the environmental constraints on the USVs. The performance of a USV is influenced by the surface currents, risk of collision with the civilian traffic, and varying depths due to tides and weather. The team of USVs needs to explore a certain region of the harbor and we need to develop an algorithm to decompose the region of interest into multiple subregions. The algorithm overlays a two-dimensional grid upon a given map to convert it to an occupancy grid, and then proceeds to partition the region of interest among the multiple USVs assigned to explore the region. During partitioning, the rate at which each USV is able to travel varies with the applicable speed limits at the location. The objective is to minimize the time taken for the last USV to finish exploring the assigned area. We use the particle swarm optimization (PSO) method to compute the optimal region partitions. The method is verified by running simulations in different test environments. We also analyze the performance of the developed method in environments where speed restrictions are not known in advance.

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## Introduction

Unmanned surface vehicles (USVs) are being developed for use in a variety of operations such as search and rescue, surveillance, and defense applications. We have also seen significant improvements in their autonomous trajectory planning capabilities [15]. We know that success in multirobot applications is achieved by deploying a team of robots instead of a single robot so as to improve safety, increase operational efficiencies, and significantly reduce costs [69]. Similarly, multiple USVs may be used so that they divide an assigned task into subtasks, which are then independently accomplished by each of them.

Most categories of collaborative tasks involving one or more team(s) of USVs require spatial partitioning of the region of interest. For example, in the tasks like search and rescue, and patrolling we need the USVs to sweep and clear the region of interest as early as possible. In such scenarios, it will be quite difficult for a single USV to accomplish the task in an acceptable amount of time. It is advantageous to use a team of collaborative USVs that perform multiple tasks in parallel.

Decomposition of tasks among team(s) of USVs is a very challenging problem because we have to consider a large variety of factors. Some of these factors include: imperfect information about the environment, changing environmental conditions, varying performance of the USV as a result of its interaction with the environment, risk of collision with external entities (e.g., civilian traffic), etc. The algorithm used to decompose the task has to ensure that the workload has been effectively distributed among all the USVs according to their capabilities. Also, the algorithm should be capable enough to replan dynamically when it receives additional information about the USVs' operating condition as well as environmental conditions.

This paper is a shortened version of our previously published conference paper [10]. This paper addresses the problem of surveying a region around the marine port using a team of USVs. Figure 1 shows an illustrative example of a port region in which the area that needs to be explored (or the region of interest) and the obstacles inside the region are labeled. For efficient exploration of the region to take place, we need to partition the environment among the USVs to ensure that the USVs can perform exploration in parallel and the workload in balanced among the team members.

We propose a method that decomposes the exploration task for a given region assigned to a team of multiple USVs. We have to consider different constraints to USVs' motion in the region of interest. We try to minimize the time taken by the bottleneck USV to cover the assigned area, which in turn lets us achieve workload balancing among all the USVs. We also consider cases where information regarding the environmental conditions are not known in advance.

## Related Work

There are two bodies of work related to the problem discussed in this paper. The first body of work belongs to area partitioning using the computational–geometric methods [11,12]. Researchers have also developed area partitioning methods to handle multirobot exploration tasks [13,14]. These types of approaches attempt to divide a given region into a set of regions. It is difficult to account for agent availability and physics-based vehicle constraints in this framework. The second body of work is done in the context of coverage planning using multiple robots [1518]. In this approach, trajectories are generated for multiple robots to explore a region and execution of these trajectories implicitly generates partitions of the given region into smaller regions.

The approach presented in this paper is inspired by the work done by the coverage planning community. However, this problem has several unique characteristics due to the marine domain and therefore, we had to develop a specialized method that explicitly generates partitions by taking into account the domain related requirements. The first main requirements is that we can select the starting location of each USV on the region boundary in the marine domain. Therefore, we can use starting locations of USVs as variables, using which we minimize the task completion time. USVs approach the region to be surveyed from a location exterior to the region being surveyed; therefore starting locations on the boundary serve as the natural decision variables. If all the USVs are not available at the beginning of the task, then our approach easily accounts for this constraint and generates nonuniform sized partitions that account for the USV availability constraints such that the task completion time is minimized. This requirement is currently not addressed by the existing approaches.

The second main requirement is that we do not have perfect information in the beginning of the mission. We may need to update the area partitions once new information becomes available. We explicitly prefer convex boundaries among the regions so that update in partitions simply shifts the partition boundaries by taking into account the location of the USV after the partial task execution. Our approach is capable of handling this requirement. Most previous approaches in coverage planning have not adequately addressed ease of replanning to handle speed limit constraints.

The third main requirement is that contingency situations may require one or more members of the team to be assigned to a different task. This requires adjusting the partition boundaries to deal with the reduced team size. Our method explicitly prefers convex partitions. This means that each USV generates trajectories during execution that try to preserve convexity of the remaining area. This makes it easy to update the partitions in the event of an contingency event. Most previous approaches in coverage planning have not adequately addressed ease of replanning to handle change in team size.

The final requirement is that during collaborative survey, the USVs should spread out and should not interfere with each other's trajectories or require frequent communication. Our method enables explicit generation of partitions by taking into account speed-limit constraints. This enables the USVs to generate independent trajectories and reduces the need for collision avoidance. Most previous approaches in coverage planning have not adequately addressed explicit generation of partition to enable independent operation of team members.

## Problem Formulation

###### Definitions.

Let A be the planar region that we are interested in surveying with the team of USVs. Region $A=[(x,y)|(x,y)∈A⊂ℝ2]$ is the closure of a subset of the Euclidean space $ℝ2$ in the usual topological sense. We assume that the region A is a simply connected region of the Euclidean space $ℝ2$. Simply connected means there exists a path completely contained in A between any two arbitrary locations that lie inside the region. In other words, the planar region A does not consist of any disjoint regions.

Let, $OA$ be the set of all nontraversable or obstacle regions that lie inside the planar region A. These are the land areas that cannot be traversed by the USVs or port areas that must be avoided. Each obstacle $Oi∈O$ is a simply connected region like the region of interest A. Now, the traversable area within the area A that needs to be surveyed by the team of USVs can be computed as $Ap=A∖OA$. The simply connected property of region A remains intact even after removal of the simply connected obstacles O from the region. Thus, we can state that the region Ap is also a simply connected traversable area. The boundary (denoted by $∂Ap$) of the region Ap is assumed to be nontraversable wherever it intersects with obstacles O.

Let the team of USVs be denoted by $Tn$, where n is the total number of available USVs that may be used to survey the region Ap. Each USV in the team is denoted by $Ui∈Tn$. In this paper, we primarily focus on studying a partitioning problem where a team of USVs $Tn$ is assigned to perform a collaborative exploration task of region Ap. Here, we assume that the USVs are homogeneous and have same physical capabilities. We also assume a functioning global communication network exists among the USVs in the team.

While performing the collaborative exploration task, each USV $Ui∈Tn$ will explore a subset of region Ap. Hence, the team of USVs has to divide the exploration tasks among themselves in an optimal manner. Essentially, this particular case reduces to a variant of area partitioning problem. Each USV claims subregions, $Ap1,Ap2,…,Ap|Tn|$, in region Ap such that the disjoint union of each subregion sums up to the original area to be explored.

During the execution of exploration task, the motion of USVs gets affected by surface currents, sea depth, and wakes generated by other civilian vessels and USVs. These factors introduce uncertainty in the positions of the USVs and also lead to noise in the controller. The positional uncertainties and controller noises increase with the increase in the USV's speed. Thus, the USVs operating at high speeds have large uncertainty in their positions and reduced controllability of the vessel, and vice versa. Therefore, the USVs operating at high speeds have a higher probability of collision with other civilian vessels and the static obstacles O. In the regions with high operating speeds, the USV will cover larger areas as compared to the regions with lower operating speeds.

The interplay between such speed constraints and USV's dynamics model plays a significant role in guiding the coverage of a port region. Based on this, we devise velocity maps $V(a)∈ℝ$ that guide the rate at which each USV claims locations $a∈Ap$ for itself. We study the effect of using such maps to guide the area growth rate of individual USVs. This also helps us constitute a heuristic method of area partitioning. We restrict ourselves to decomposing the exploration tasks among multiple USVs, thereby reducing our problem to optimal partitioning of the port region. The incorporation of the velocity maps serves as a heuristic that guides the rate at which each USV claims the areas inside the region of interest for its own exploration. In the present work, the velocity map values are not used as physical USV speeds because area coverage is not considered. We assume that such a velocity map is already given to us and in the present work create custom maps to check our algorithm.

###### State Space Representation.

We discretize the continuous region of interest Ap with a uniform grid of suitable step-size δl. The discretized region of interest is denoted by $Ap,d$. Each cell in $Ap,d$ is represented by state $s=(x,y)∈Z2$, where $1≤x≤xmax$ and $1≤y≤ymax$. Let $S$ be the set of all the states from region $Ap,d$.

Neighbors of each state s are determined by performing a single step of Manhattan moves from s in each direction, i.e., North, South, East, and West. Let $Neigh(s)$ be a function that returns all the feasible neighbors of state s. Finally, the discrete region $Ap,d$ is approximated by a polygon BA such that all the cells of region $Ap,d$ intersect with or lie in the interior region of BA. The discrete velocity map $Vd(s)∈ℕ$ provides the maximum velocity the USV can operate at state s.

###### Problem Statement.

Our goal is to seek an optimal division of area such that the area is partitioned in minimum time and work load is optimally balanced among the USVs. We achieve this by minimizing the time taken by the bottleneck USV based on Minimax style of decision-making.

Formally, given (a) region of interest $Ap,d$, (b) team of USVs, Tn, and (c) velocity map, $Vd(·)$. We compute optimal region of partition $Pi∈P$ for each USV $Ui∈Tn$, where $i∈[1,n]$. Each partitioned region $Pi∈P$ is computed such that the time taken by the team of USVs Tn is minimized to explore the entire region $Ap,d$.

## Approach

The discretized region of interest $Ap,d$ consists of discrete states $s∈S$. Let Ntotal be the total number of discrete states in $S$. In this paper, we want to compute the partitioned region for each USV $Ui∈Tn$ while minimizing the time taken by bottleneck USV during its exploration of the region $Ap,d$. A partition P of the region is a collection of n mutually exclusive and exhaustive subsets of discrete states $S$; here, n is total number of USVs in team Tn.

The boundary of the polygon representing the region Ap is parameterized by variable $ρ∈[0,1)$. Here, ρ = 0 represents the starting point of the closed polygon. Note that ρ = 1 is excluded because it corresponds to the same point as ρ = 0, i.e., the starting point on the boundary. Once each USV has been assigned a boundary point, we readily obtain the starting states for each USV, $si,init$. However, the mapping from ρ-space to states in grid region $Ap,d$ is generally not injective though this also depends on the step size chosen to discretize the grid. Also, those boundary points which are inside obstacles will never be assigned as starting points for any of the USVs.

At each time-step, the USVs can only move at the speed prescribed by the velocity map, $Vd(s)$. We express the speed in units of number of cells per unit time. The value assigned by the velocity map is constant for the entire cell because each cell is considered to be the lowest level of spatial discretization. That is why we do not allow multiple USVs to occupy the same cell. Suppose for the USV starting at grid cell s, we look up the maximum allowed velocity as $Vd(s)$. So, the USV Ui will attempt to claim $Vd(s)$ states to augment its currently claimed area. We denote the area claimed by USV Ui at iteration t by a(t, i).

Initially, all the USVs are assigned states along the boundary polygon BA of the region $Ap,d$. All states are labeled open. At this point, each USV $Ui∈Tn$ has claimed the state $si,init$ at which they are initialized. Each USV has claimed area $a(1,i)=1$. In order to partition the region $Ap,d$ into n convex shaped simply connected regions, it will be advantageous for each USV to select states that are connected and nearest to current state of the USV. Let $CUi$ be the cache of all the possible states that USV Ui can claim in next time-step by maintaining the convexity of the partitioned region $AUi$. Here, $AUi$ is the partitioned region for USV Ui and $Si$ is the set of all states belonging to region $AUi$. In the initial step, $Si$ will contain $si,init$, and $CUi$ will contain all the neighboring states provided by $Neigh(si,init)$. Let $c1,i$ denote the maximum number of states that USV Ui can claim at time-step t = 2. Clearly, $c1,i=Vd(si,init)$. The cache must have at least $c1,i$ states. Initially, cache of each USV, $CU(i)$, is empty so at time t = 1, neighbors of state $si,init$ are added to the cache $CU(i)$.

Once the state $si,init$ has been claimed by one of the USVs and its neighbors are generated, we label the state as closed. Since the starting state is a boundary cell, at least one neighboring state will lie outside the grid region $Ap,d$. But because nonconvex polygons are allowed, we can have $|Neigh(si,init)|≤3$. If $c1,i≤Neigh(si,init)$; then we can proceed with claiming states next time-step from the current cache. States unclaimed at time t = 2 from the cache will remain in cache for future times. But if $c1,i>Neigh(si,init)$, then those states in the cache $CU(i)$ that are open have their neighbors generated and stored in the cache. This will continue till the cache population, $|CU(i)|$, equals or exceeds $c1,i$.

The actual number of states $c2,i′$ that the USV claims for time-step t = 2 depends on conflicts with other USVs and also the availability of free states in its neighborhood to explore, which leads to the relation: $0≤c2,i′≤c1,i$.

We define frontier cells as follows: At any time t, for a USV Ui, it refers to the $ct,i′$ states it actually claims. Thus, the set of states that the USV Ui has claimed at time-step t = 2 can be denoted by $si,2=[si,21,si,22,…,si,2c2,i′]$. These form the frontier cells for the USV Ui at time-step t = 2. These frontier cells augment the current claimed area for the USV to $a(2,i)=a(1,i)+c2,i′$.

The first cell $si,21$ is chosen from the contending open states in the cache such that it augments the current claimed area of USV Ui with minimum loss of convexity. The chosen state should have minimum center-to-center distance from the center of the starting state of USV Ui. Such an expansion in an obstacle-free and unconstrained square lattice invariably leads to a circle-like expansion. So, this heuristic is also expected to give good results for nonconvex regions.

The distance heuristic referred to above is not based on the classical Euclidean metric because the boundary contour itself may be nonconvex or contain nonconvex polygonal obstacles inside it. Euclidean distance works well only in the absence of these concavities and holes in the map world. Instead, we apply Dijkstra's algorithm to every lattice point, thereby computing the lengths of shortest collision-free paths through the region, Ap between each pair of lattice points. To this algorithm, we give the undirected connectivity graph of the grid world as the input along with its weighted edges. Weights are the Euclidean distances between neighboring states; hence, either 1 or $2$. It is prudent here to allow diagonal connectivity as well so as to compute paths more tuned to the local concavities. We use the distances from the look-up table thus generated as the heuristic in guiding the growth of USV's claimed area, a(t, i). The subregions claimed by USVs will not be convex in general because of the nonconvex free region Ap.

In the second iteration, there are possibly multiple frontier cells. Each cell shall correspond to a value of maximum speed that can be attained around that cell. So, we take an average of these multiple maximum speeds to determine how many states should the USV Ui claim next time-step.

The states in the region $Ap,d$ are divided among multiple USVs in finite number of time steps, tmax. At each time-step, the USVs search their neighboring unclaimed states and keep expanding their claimed subregion for exploration. The iterations continue until all free states have been claimed. Depending upon the shape and size of the region $Ap,d$, each USV $Ui∈Tn$ will take different amount of time to claim its subregion. The USV, which claims the last unclaimed state, is considered to be the bottleneck agent. In the scenario where multiple USVs finish the final iteration, any one of them can be considered the bottleneck agent. The time taken by each USV to claim all the possible free states depends upon the initial location of the USV on the boundary BA. In this paper, we assume that each USV is allotted a different initial state.

Suppose USV Ui takes time tif to complete claiming states for exploration, then the time-step at which the process of area partitioning terminates and the bottleneck USV $Ui∗$ are determined as follows: Display Formula

(1)$tfinish=maxi(tif) i∗=arg maxi(tif)$

The initial starting states map to an n-dimensional vector, $ρU=[ρ1,…ρn]$. The vector elements should be sufficiently spaced so that two or more USVs do not claim to the same initial state. If the minimum time taken by the bottleneck USV is allowed to only be a function of vector ρU, then our problem can be formulated in terms of optimal initial placement $ρU∗$ as follows: Display Formula

(2)$ρU∗=arg minρUmaxi(tif) tfinish∗=minρUmaxi(tif)$

## Area Partitioning Using Velocity-Maps Known in Advance

###### Overview.

While implementing the partitioning algorithm, we did not implement the USVs as concurrently operating agents that claim states in the region of interest. However, if the USVs sequentially claim states, then there will be bias in the partitions created among the USV for that USV, which is at the top of the list that stores the team of USVs. Instead, we generate a random order at each discrete time-step using which the USVs claim the states from the set of free available states. This is the only source of randomness in this task decomposition technique.

During area partitioning, this protocol will lead to deadlocks for those time steps when multiple USVs lay claim on the same cell. To resolve such deadlocks, we again make use of the random ordering of the USVs assigned to each iteration of the simulation. In case of a deadlock, the USV with higher priority at that time-step wins the bid to its claim.

The scenarios used to evaluate the performance of the developed approach were assigned increasing degree of concavity to represent regions in ports such as narrow channels or small, enclosed regions in the harbor (see Fig. 2). We have selected scenario of size 500 × 500 m and discretized it with a uniform grid-size of 10 m.

We have limited ourselves to two types of velocity maps for testing our implementation in this paper. The simplest example of such a velocity map would be a uniform, constant map such that all states have the same value of maximum speed allowed. Then, we created a map where velocities for each state are defined as a function of its Cartesian (Fig. 2(a)) coordinates. This map was based on using linear gradients to signify speed zones. It is generated applying a linear gradient horizontally and superposing it with another linearly graded risk profile in vertical direction. The horizontal profile prescribes maximum risk on the left, whereas the vertical profile prescribes maximum risk to the top of the area. We also show similar results for a nonlinear map defined in terms of polar coordinates (see Fig. 3).

The time taken by the bottleneck USV is only optimized in the present work over the boundary points assigned to each USV. This decision variable has already been defined by the ρ-vector, $ρU=[ρ1,ρ2,…ρn]$. For a given scenario, the initial choice of n boundary points determines the partitions along with the randomness introduced in prioritizing the USVs at each time-step. Other factors like world size and region of interest are held constant in the current simulation. Thus, we have to minimize over the vector of starting states ρU. The objective function, $ap:(ρU∈[0,1)n)→Z$, is nonlinear. The objective itself is an ill-defined mapping as far as gradient-based optimization methods are concerned.

Since the mapping associates every bound vector to a positive integer, it is not expected to be differentiable. Classical gradient-based methods are expected to perform poorly. Therefore, we employ the particle swarm optimization (PSO) method [19] to minimize the time taken by bottleneck USVs to explore the region. We use PSO as it efficiently handles higher dimensions and does not require the objective function to be differentiable. For our purpose, we chose the cognitive trust parameter to be 1.7 and the social trust parameter to be 2. Based on observations from multiple simulation runs, a total of 12 particles were chosen for optimization over four USVs and PSO was run for 75 iterations. The tolerance for the convergence of the solution was chosen to be 0.001. We chose the uniformly spaced placements as the initial population of USVs along the polygonal contour in terms of ρ parameter, viz, $0, 0.25, 0.5, 0.75$.

We coded the above discussed scheme of decomposing exploration tasks among multiple USVs in matlab software on Ubuntu MATE 16.04.1 operating system. The workstation used was Dell® Precision Tower 3620 including eight Intel® Xeon® E3-1245V5 CPUs having 3.5 GHz speed and a total of 32 GB RAM. Computation time for generating partitioned regions for a team of four USVs exploring the region (see Fig. 2(b)) without obstacles and velocity map of Fig. 2(a) is around 15 s.

###### Results.

In this section, we will be evaluating the effectiveness of the algorithm by varying the number of USVs from 2 to 10. Minimization of time taken by the bottleneck USV requires each USV in the team to finish claiming all the possible remaining unexplored states at the same time as the bottleneck USV. Range is computed by taking a difference between maximum and minimum value of the exploration ending times of the USVs. Theoretically, the range of the final exploration times of the USVs should be zero because of optimal load balancing among the agents but nonconvexity associated with a region leads to deviation from ideal results as can be seen in the simulation data recorded. We show a box plot of these ranges over 200 simulations (see Fig. 4(a)). The values shown on the y-axis are discrete values of time-step. In Fig. 4(a), the median of the data is 2 while the minimum is zero, which is expected. The maximum range is capped at 6 and the interquartile range is 2. These results indicate successful minimization of time taken by the bottleneck USV.

We run 100 simulations by varying the number of robots from 2 to 10 and record the computational times required for partitioning a fixed region and then present the data in the form of a box plot (see Fig. 4(b)). On y-axis, we plot the percentage reduction in computation time with respect to the mean computation time required by the algorithm to compute the optimal partitions for two USVs. Ordinarily, the optimization with more number of USVs should take more computational time. However, with increase in the number of USVs, the areas assigned to each USV reduce. This results in a decrease in the number of time-steps required to compute optimal partitions for each USV. So, the time to compute optimum partitions of the region for each optimization iteration reduces. This effect is more significant and leads to reduction in overall computation time. We also observed that the final optimal solution is not sensitive to the initial locations of the USVs along the region boundary that we chose as input for the PSO algorithm.

## Area Partitioning Using Velocity-Maps Not Known in Advance

###### Overview.

In this section, we want to evaluate the robustness of the area partitioning algorithm when the velocity map is unknown initially. The algorithm does not have perfect knowledge of the velocity map while computing the initial exploration plan. However, while the USVs are exploring the region, they gather more information and obtain the correct velocity map. We have developed two approaches to handle such scenarios of imperfect or unknown velocity maps.

In the first approach, we determine the optimal starting boundary points $ρU∗$ for the USVs based on uniform, constant velocity map. USVs begin claiming the region starting from the boundary points $ρU∗$ but claim states based on speeds from the correct velocity map. This is because even if the USVs do not know about the correct velocity map but while actually exploring the region, USVs will move according to the actual environmental conditions. While the USVs are traversing the areas in real time, we wait for the correct velocity map to be received. The planner then uses the correct velocity map to compute the optimal partitions for the remaining region. During this computation, the planner excludes the states of the region $Ap,d$ that have already been explored by the USVs.

In the second approach, the USVs proceed with a noisy estimate of velocity map till the correct velocity map is obtained. The noisy velocity map is assumed to be the best estimate of the correct velocity map based on the sensor data that is available at that moment. In the current work, we sample values for each cell of the velocity map from the uniform distribution ranging across the allowable speeds for USVs (here we used the range $[1,6] m/s$)

###### Results.

In this section, we perform computational experiments by varying the time taken by the USVs to estimate the correct velocity map. In order to simplify the implementation, we have presented the results by varying the percentage area explored by the USVs before it acquires the correct estimate of the velocity map (shown on the x-axis of the graphs in Fig. 5). In Fig. 5, we plot the percentage increase in execution time taken for exploration by the USVs from the mean true-optimal time on the y-axis. The simulation data presented in the form of box plots are taken over 100 simulations with medians denoted by the red lines.

We can see from the graphs (see Fig. 5) that with more delay in estimating the correct values of the velocity map, there is a steady increase in the execution time incurred by the team of USVs. Both the approaches perform similarly when the correct estimates of velocity map are received before the USVs explore about 10% of the area. However, when the USVs explore more than 20% of the area, the latter approach using noisy estimates performs marginally poor compared to the former approach. This suggests that it is better to use the mean estimate of the velocity map computed over time as compared to using currently available noisy estimates. We can see from the graph (shown in Fig. 5) that the percentage increase in execution time taken by the USVs from the true-optimal time dips below zero. This is because we set as zero the median time taken by the team of USVs when correct velocity map is available from the beginning.

## Conclusion and Future Work

We have developed an algorithm that solves the area partitioning problem while taking into account the constraints on the rates at which USVs are able to perform exploration tasks. The algorithm is able to partition a large spatial region among multiple USVs using underlying velocity maps well under 1 min. We show that by increasing the number of USVs, the computational time actually reduces. We have evaluated the performance of the algorithm in scenarios where the velocity map is initially unknown for a given length of time. Subsequently, the accurate velocity maps become available after the USVs have explored part of the desired region. We observe that the percentage increase in the optimal partitioning time remains below 5% even when the accurate velocity map becomes available after $10−20%$ of area has been explored.

Future work shall include recursive decomposition of exploratory tasks among multiple USVs. Minimizing time taken by bottleneck USV in a highly nonconvex region could lead to partitions being highly nonconvex in some regions. This may be remedied by recursive partitioning till acceptable concavity is achieved. Though we restricted ourselves to a single optimization technique in this work, we plan to explore the effect of different optimization techniques on partition quality. Future work would further incorporate heterogeneous capabilities of the USVs. It would be very interesting to record the effect of uncertain maps on the quality of partitioning in more detail and by taking into account different kinds of noise addition techniques. The use of adaptive and nonuniform grid sizes provides significant computational boost. We plan to explore the idea of nonuniform grid like quad trees in our partitioning approach.

## Acknowledgements

Opinions expressed are those of the authors and do not necessarily reflect opinions of the sponsors.

• National Science Foundation (Grant No. 1634433).

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Jager, M. , and Nebel, B. , 2002, “ Dynamic Decentralized Area Partitioning for Cooperating Cleaning Robots,” IEEE International Conference on Robotics and Automation (ICRA), Washington, DC, May 11–15, pp. 3577–3582.
Ahmadi, M. , and Stone, P. , 2006, “ A Multi-Robot System for Continuous Area Sweeping Tasks,” IEEE International Conference on Robotics and Automation (ICRA), Orlando, FL, May 15–19, pp. 1724–1729.
Choset, H. , 2001, “ Coverage for Robotics–A Survey of Recent Results,” Ann. Math. Artif. Intell., 31(1), pp. 113–126.
Zelinsky, A. , Jarvis, R. A. , Byrne, J. , and Yuta, S. , 1993, “ Planning Paths of Complete Coverage of an Unstructured Environment by a Mobile Robot,” International Conference on Advanced Robotics (ICAR), Tokyo, Japan, Nov. 1–2, pp. 533–538.
Galceran, E. , and Carreras, M. , 2013, “ A Survey on Coverage Path Planning for Robotics,” Rob. Auton. Syst., 61(12), pp. 1258–1276.
Zheng, X. , and Koenig, S. , 2007, “ Robot Coverage of Terrain With Non-Uniform Traversability,” IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), San Diego, CA, Oct. 29–Nov. 2, pp. 3757–3764.
Kennedy, J. , and Eberhart, R. , 1995, “ Particle Swarm Optimization,” IEEE International Conference on Neural Networks, Perth, Australia, Nov. 27–Dec. 1, pp. 1942–1948.
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## References

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Sheng, W. , Yang, Q. , Tan, J. , and Xi, N. , 2006, “ Distributed Multi-Robot Coordination in Area Exploration,” Rob. Auton. Syst., 54(12), pp. 945–955.
Burgard, W. , Moors, M. , Stachniss, C. , and Schneider, F. E. , 2005, “ Coordinated Multi-Robot Exploration,” IEEE Trans. Rob., 21(3), pp. 376–386.
Grocholsky, B. , Keller, J. , Kumar, V. , and Pappas, G. , 2006, “ Cooperative Air and Ground Surveillance,” IEEE Rob. Autom. Mag., 13(3), pp. 16–25.
Roy, N. , and Dudek, G. , 2001, “ Collaborative Robot Exploration and Rendezvous: Algorithms, Performance Bounds and Observations,” Auton. Robots, 11(2), pp. 117–136.
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Pavone, M. , Arsie, A. , Frazzoli, E. , and Bullo, F. , 2011, “ Distributed Algorithms for Environment Partitioning in Mobile Robotic Networks,” IEEE Trans. Autom. Control, 56(8), pp. 1834–1848.
Lien, J.-M. , and Amato, N. M. , 2006, “ Approximate Convex Decomposition of Polygons,” Comput. Geom., 35(1–2), pp. 100–123.
Jager, M. , and Nebel, B. , 2002, “ Dynamic Decentralized Area Partitioning for Cooperating Cleaning Robots,” IEEE International Conference on Robotics and Automation (ICRA), Washington, DC, May 11–15, pp. 3577–3582.
Ahmadi, M. , and Stone, P. , 2006, “ A Multi-Robot System for Continuous Area Sweeping Tasks,” IEEE International Conference on Robotics and Automation (ICRA), Orlando, FL, May 15–19, pp. 1724–1729.
Choset, H. , 2001, “ Coverage for Robotics–A Survey of Recent Results,” Ann. Math. Artif. Intell., 31(1), pp. 113–126.
Zelinsky, A. , Jarvis, R. A. , Byrne, J. , and Yuta, S. , 1993, “ Planning Paths of Complete Coverage of an Unstructured Environment by a Mobile Robot,” International Conference on Advanced Robotics (ICAR), Tokyo, Japan, Nov. 1–2, pp. 533–538.
Galceran, E. , and Carreras, M. , 2013, “ A Survey on Coverage Path Planning for Robotics,” Rob. Auton. Syst., 61(12), pp. 1258–1276.
Zheng, X. , and Koenig, S. , 2007, “ Robot Coverage of Terrain With Non-Uniform Traversability,” IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), San Diego, CA, Oct. 29–Nov. 2, pp. 3757–3764.
Kennedy, J. , and Eberhart, R. , 1995, “ Particle Swarm Optimization,” IEEE International Conference on Neural Networks, Perth, Australia, Nov. 27–Dec. 1, pp. 1942–1948.

## Figures

Fig. 1

This is an illustration of a port scenario where a region has been demarcated for exploration by multiple USVs

Fig. 2

(a) Linearly graded velocity map, (b)–(d) application on manually created regions, and (e)–(h) application on regions based on real maps

Fig. 3

(a) Polar velocity map, (b)–(d) application on manually created regions, and (e)–(h) application on regions based on real maps

Fig. 4

(a) Range of exploration completion times for USVs over 200 simulations in the form of box plot and (b) variation of computational time with respect to number of USVs

Fig. 5

Variation of optimal times for area partitioning in the absence of correct velocity map, when we use either (a) constant map or (b) noisy map

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