Correlated Orienteering Problem and its Application to InformativePath Planning for Persistent Monitoring Tasks

Jingjin Yu Mac Schwager Daniela Rus

Abstract—We propose a novel non-linear extension to theOrienteering Problem (OP), called the Correlated OrienteeringProblem (COP). We use COP to plan informative tours (cyclicpaths) for persistent monitoring of an environment with spatialcorrelations, where the tours are constrained to a fixed lengthor time budget. The main feature of COP is a quadraticutility function that captures spatial correlations among pointsof interest that are close to each other. COP may be solvedusing mixed integer quadratic programming (MIQP) that canplan multiple disjoint tours that maximize the quadratic utilityfunction. We perform extensive characterization of our method toverify its correctness, as well as its applicability to the estimationof a realistic, time-varying, and spatially correlated scalar field.

I. INTRODUCTION

We envision control algorithms for teams of robots wherethe robots use correlations established between different partsof the environment for specific events using historical datato determine more effectively the next action. For example,consider a scenario where a robot senses a specific event,say an intruder at a specific location in the environment.Using correlations and prediction a different robot can adjustits location to intercept the intruder. Toward that goal, inthis paper, we consider how previously learned correlationscan improve the performance and capabilities of teams ofrobots tasked with persistent surveillance and monitoring withlimited travel budgets. Figure 1 provides a graphical exampleillustrating the problem addressed in this work.

In persistent surveillance and monitoring tasks using mobilerobots, such as unmanned aerial vehicles (UAVs), the mobilerobots usually have fixed base stations from which they mustdepart and return. Moreover, these robots have limited traveldistance or time budget. Thus, when a large number of pointsof interest (nodes) must be surveyed, it may well be the casethat only a subset of the nodes can be visited by the robots.Then, choices among the nodes must be made to accommodatetwo conflicting goals: 1. each robot must follow a tour (cyclicpath) whose total cost does not exceed its travel budget,and 2. the robots must visit as many nodes as possible tomaximize the amount of collected information, as measuredby some utility (reward) function. When nodes have utilities

Jingjin Yu is with the Computer Science and Artificial Intelligence Lab atthe Massachusetts Institute of Technology and the Mechanical EngineeringDepartment at Boston University. E-mail: jingjin@csail.mit.edu.

Mac Schwager is with the Mechanical Engineering Department at BostonUniversity. E-mail: schwager@bu.edu.

Daniela Rus is with the Computer Science and Artificial Intelligence Labat the Massachusetts Institute of Technology. E-mail: rus@csail.mit.edu.

This work was supported in part by ONR projects N00014-12-1-1000 andN00014-09-1-1051.

We thank the anonymous reviewers for their constructive comments.

Fig. 1. [top] A surveillance scenario in which an UAV with limited rangeand sensing budgets is faced with the problem of covering a large numberof nodes. [bottom] Using correlations among nodes in a network, it becomespossible to infer at least partial information for nodes that are not directlysurveyed by the UAV. Following the red tour path, which is much shorterthan a traveling salesman (TSP) tour, the UAV can provide at least partialinformation about every node in the network.

that are additive, an Orienteering Problem (OP) [1], a problemclosely related to the well known Traveling Salesman Problem(TSP) [2], arises. Research on OP has yielded many effectivealgorithms for solving many versions of the problem, includingTeam Orienteering Problem (TOP)[3], in which multiple toursmust be planed.

In practice, however, the information to be collected at thenodes are frequently correlated between nodes that are close toeach other, rendering the total utility a non-linear combinationof individual node utilities. That is, it is often the case that suchinformation can be viewed as forming a spatially continuousfield (that also varies over time). Therefore, surveying a givennode will also yield partial information about its neighbors. For

2014 IEEE/RSJ International Conference onIntelligent Robots and Systems (IROS 2014)September 14-18, 2014, Chicago, IL, USA

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example, the nodes may be cities, city blocks and locationsin reservoirs with the associated quantities being populationdynamics, criminal activities, and water pollutant concentra-tion, respectively. In this paper, assuming that the spatialcorrelation among the nodes are intrinsic (i.e., determinedby local structures and mostly time-invariant) we propose aquadratic extension to OP and TOP, called the CorrelatedOrienteering Problem (COP), to incorporate such correlationsin the informative path planning phase. After formulating COP,we provide mixed integer quadratic programming (MIQP)models for solving the problem for a single robot as well asfor multiple robots. Our simulations indicate that COP and theassociated MIQP capture spatial correlations among the nodesquite well. In a nutshell, COP can be viewed as a relaxationfrom the problem of planning informative tours for multiplerobots under a spatially correlated field; the relaxed problemis then solved exactly. We also note that we do not assumeconvexity or submodularity over the underlying field.

Our work brings together ideas from two relatively disjointbranches of research: (discrete) OP and (mostly continuous)informative path planning for persistent monitoring tasks. OP,as indicated by its name, originates from orienteering games[3], [4]. In such a game, rewards of different sizes are spatiallyscattered. A player (or players in TOP) is faced with the task ofmaximizing the additive rewards under a time constraint thattranslates naturally to a distance constraint. Thus, OP can beviewed as a variation of both the Knapsack Problem (KP) [5]and the Traveling Salesman Problem (TSP) [2]. For a detailedaccount of OP, see [1].

The literature on informative path planning for persistentmonitoring is fairly rich [6]–[19], covering both theories andapplications. On work most closely related to ours, in [8],iterative TSP paths are planned to minimize the maximumlatency across all nodes in a connected network. The authorsshow that the approach yields O(logn) approximation onoptimality in which n is the number of nodes in the network.The problem of generating speed profiles for robots alongpredetermined cyclic (closed) paths for keeping bounded theuncertainty of a varying field is addressed in [16], in whichthe authors characterize appropriate policies for both singleand multiple robots. In [17], decentralized adaptive controllerswere designed to morph the initial closed paths of robots tofocus on regions of high importance.

Sampling based planning methods (e.g. PRM, RRT, RRT!

and their variations [20]–[22]) have also been applied to in-formative path planning problems. In [18], Rapidly-ExploringRandom Graphs (RRG) are combined with branch and boundmethods for planning most informative path for a mobilerobot. In [13], the authors tackle the problem of planning acyclic trajectory for best estimation of a time-varying GaussianRandom Field, using a variation of RRT called Rapidly-Expanding Random Cycles (RRC).

Lastly, our problem, and OP in general, also has a coverageelement. Coverage of a two-dimensional region has beenextensively studied in robotics [23]–[25], as well as in purelygeometric settings, for example, in [26], where the proposed

algorithms compute the shortest closed routes for continuouscoverage of polygonal interiors under an infinite visibilitysensing model. Coverage with limited sensing range was alsoaddressed later [27], [28].

This work brings two main contributions. First, we proposeCOP as a novel non-linear extension to OP to model thespatial correlations that are frequently present in informativepath planning problems for persistent monitoring tasks. Inparticular, our formulation addresses the difficult problemof planning tours (cyclic paths) under limited travel budget.Second, we provide complete mixed integer quadratic pro-gramming (MIQP) models for solving COP. These models,with a MIQP solver, can effectively compute tours for multiplerobots, each with a fixed base node. We then show thatour models are applicable to estimate time-varying, spatiallycorrected scalar field, with computational experiments.

The rest of the paper is organized as follows. Section IIformulates the Correlated Orienteering Problem (COP) thatwe study. MIQP models are then outlined for solving thesingle-robot and multi-robot cases in Section III. In Section IV,we perform extensive computational experiments to evaluatethe applicability and effectiveness of COP and the associatedMIQP models. Section V concludes the paper.

II. PROBLEM STATEMENT

Table I lists the symbols used in this paper. We study theproblem of using mobile sensors to periodically monitor a setof nodes for quantities of interest that can be measured atthese nodes. In general, these quantities can be representedas a continuous scalar or vector field that changes over spaceand time. One common characteristic of such fields is that theyhave spatial continuity. That is, at any fixed time instance, thevalues do not fluctuate much for nodes that are (spatially) closeto each other. Such relationships can be represented as someform of correlations among the nodes1.

This paper focuses on node networks in which the corre-lations determined by spatial relationships are known and donot vary much over time (i.e., it is time-invariant) whereasthe field itself may vary significantly over time. Then, usingthese correlations, it becomes possible to evaluate the field’svalue at one node using nearby nodes at any given time.In particular, we are interested in using mobile sensors withlimited travel range to “sample” nodes of the network andthen estimate the field’s value at the rest of the nodes usingcorrelations. We assume that the field changes at a slowerpace in comparison to the time it takes a mobile sensor tocomplete a set of measurements at sampled nodes (i.e., thefield remains relatively static during a trip by the mobilesensor(s)). Below, this problem is formulated as a CorrelatedOrienteering Problem (COP). More precisely, our formulationis a quadratic extension to the linear Orienteering Problem(OP), in which one or multiple team members use only pathswith fixed end points to visit sites and gain rewards. Our

1Here, we use the broad meaning of correlation, which could be, but is notnecessarily, the correlation of random variables.

343

TABLE ILIST OF FREQUENTLY USED SYMBOLS AND THEIR INTERPRETATIONS.

V = {vi} Node or point of interest, |V |= nG = (V,E) Node network

pi The two dimensional coordinate of viri Utility of knowing complete information about vi

!(vi, t) Time-varying scalar fieldA = {ak} Mobile robot k, |{ak}|= m

vbk Start and end node for robot akck Travel budget for robot ak" #1, . . . ,#m, a set of robot tours

J(") The cost function for a given set of robot tourswi j The weight measuring vi’s influence on v jxi Binary variable indicating whether vi is on a tourxi j Binary variable indicating whether v j is visited immedi-

ately after vixi jk Binary variable indicating whether v j is visited immedi-

ately after vi, by robot akui Integer variable, 2 ! ui ! n, the order of vi in a tour path,

if useduik Integer variable, 2 ! uik ! n, the order of vi in a tour

path, if used, by robot akdi j Travel cost from vi to v j , maybe non-symmetric

$i j,%i j Linear regression coefficients

particular focus in this paper are tours - cyclic paths withthe same starting and ending nodes.

Let V = {v1, . . . ,vn} be a set of spatially distributed nodesin some workspace W " R2. Each node vi # V is associatedwith coordinates pi # R2. Let

r : V $ R+,vi %$ ri,

represent the importance (utility) of the nodes. Let !(p, t) bea scalar field over W that changes over time. The values on thenodes of V , with a slightly abuse of notation, are written asas !(vi, t),1 ! i ! n. We assume that the spatial relationshipamong the nodes of V , as determined by ! , induces a directedgraph G over V . More precisely, G=(V,E) has an edge (vi,v j)if and only if !(v j, t) is dependent on !(vi, t). That is, letNi = {vi1 , . . . ,vik} be the neighbors of vi (in G, with edgespointing to vi), then for some time-invariant fi,

!(vi, t) = fi(!(vi1 , t), . . . ,!(vik , t)). (1)

We point out that these requirements are quite mild as we donot assume that ! has properties such as convexity.

Let there be m mobile robots (sensors), A = {a1, . . . ,am}.For a single robot ak, let its base (i.e., where it must start andend in a tour) be a node vbk #V . Let

c : A $ R+,ak %$ ck,

represent the maximum distance budget the mobile robots cantravel. In TOP, a team of mobile robots, working together, getto collect a reward (utility) ri for visiting vi the first time. Therobots do not gain more utility for subsequent visits to vi. Thegoal is to find tours for the robots satisfying the travel budgetconstraint (i.e., tour distance for robot ak is no more than ck)so that the total utility is maximized. Given a set of tours(paths) " = {#1, . . . ,#m} taken by the robots, let {x1, . . . ,xn}be n binary variables, with xi = 1 if and only if vi (i.e., pi)

is on some tour #k # ". We emphasize that the robots arenot constrained to stay on the graph G, but can move betweenany two nodes with a cost proportional to the distance betweenthem. The graph G only represents the spatial correlations inthe field.

Our generalization of OP to COP aims at incorporatingcorrelations among the nodes during the tour planning phase.To obtain a problem amenable to mathematical programmingtechniques, we relax Equation (1) in the following way. Let

w : E $ R+,(v j,vi) %$ w ji,

represent the effectiveness (a weighting) of using !(v j, t) toestimate !(vi, t). One may view w ji as representing the amountof information that !(v j, t) has about !(vi, t), independent ofother neighbors of vi. The utility that can be collected over avertex vi is defined as

ri(xi + &v j#Ni

wi jx j(x j & xi)),

in which the quadratic term x j(x j &xi) is non-zero if and onlyif x j = 1 and xi = 0. Obviously, for each 0! i! n, &v j#Ni w ji !1. Note that we do not assume that wi j = w ji, which may bethe case if the field’s values at vi and v j are Gaussian randomvariables. Over a set of tours for the robots, ", the total utilityto be maximized is then

J(") =n

&i=1

(rixi + &v j#Ni

r jwi jxi(xi & x j)). (2)

The tour length constraints and the cost function given byEquation (2) defines a quadratic extension to TOP. In the nextsection, we propose a quadratic integer programming modelwith quadratic cost functions and linear constraints (oftenknown as mixed integer quadratic programming or MIQP) forsolving the problem. Then we will look at several practicalestimation scenarios, involving one or multiple mobile robots,that can be tackled using this fairly general method.

Remark.We note that OP, TOP, and COP are all NP-hardproblems. Restricting to OP (i.e., a one-member team or asingle mobile robot), for a given travel budget, an algorithmfor OP must answer the question of whether the budget isenough for going through all nodes in V . Therefore, OPrequires solving (potentially many) TSPs. For COP, making theweights {wi j} sufficiently small reduces it to an OP, becausethe quadratic cost (the second summation in Equation (2)) thenbecomes negligible.

III. MIXED INTEGER QUADRATIC PROGRAMMINGMODELS FOR QUADRATIC TEAM ORIENTEERING PROBLEM

In this section, we outline MIQP models for COP, startingfrom the case of a single mobile robot and then move to thecase of multiple robots. Then, we show how these models maybe applied to potential applications.

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A. MIQP Model for COP with a Single Tour

We start with the case of a single tour (m = 1). Without lossof generality, let the single robot start from v1. We adapt theconstraints from [1]. Let xi j be a binary variable with xi j = 1if and only if the robot visits v j immediately after it visitsvi. Note that this does not depend on the existence of an edgebetween vi and v j in G. Because it is never beneficial to returnto revisit a vertex, the robot should only enter and leave anyvertex at most once, yielding the following constraints.

n

!i=2

x1i =n

!i=2

xi1 = x1 = 1, (3)

n

!j=1, j !=i

xi j =n

!j=1, j !=i

x ji = xi " 1, #2 " i " n. (4)

The equality in Equation (3) is due to the requirement that thebase node must be used.

Equations (3) and (4) ensure that the robot will take atour starting and ending at v1. They do not, however, preventadditional disjoint tours being created. To prevent this fromhappening, let 2 " ui " n be integer variables for 2 " i " n.The constraints

ui $u j +1 " (n$1)(1$ xi j), 2 " i, j " n, i != j (5)

guarantees that no additional tours not containing vi getcreated.

With the definition of the variables {xi j}, the tour distanceconstraint can be easily enforced as

n

!i=1

n

!j=1, j !=i

xi jdi j " c1, (6)

in which di j is the distance between vi to v j and c1 is the tourdistance constraint for the single (first) robot. Note that thedistance di j needs not to be symmetric. Moreover, it is easyto incorporate sensing cost at a node vi by absorbing that costto di j for all j != i. Alternatively, if the sensing cost is notcompatible with the travel cost, an additional cost constraintcan be added as well.

The objective function Equation (2), subject to the constraintEquations (3), (4), (5), and (6), define a complete MIQP modelthat can be solved using an MIQP solver.

B. MIQP Model for COP with Multiple Tours

Once the MIQP model for a single robot is fully specified,extending it for multiple robots is rather straightforward. Toaccommodate m robots, the variables {xi j} and {ui} areextended to xi jk and uik, with 1 " k " m representing therobots. Constrain Equations (3) and (4) become

n

!i=1,i !=bk

xbkik =n

!i=1,i !=bk

xibkk = xbk = 1, 1 " k " m (7)

andn

!j=1, j !=i

xi jk =n

!j=1, j !=i

x jik " 1, #1 " i " n,1 " k " m (8)

m

!k=1

n

!j=1, j !=i

xi jk =m

!k=1

n

!j=1, j !=i

x jik = xi " 1, #1 " i " n. (9)

Equation (4) splits into Equations (8) and (9) because eachvertex should only be used at most once by each robot aswell as by all robots.

The constraints on uik become (for all 1 " k " m)

uik $u jk +1 " (n$1)(1$ xi jk), i, j != bk, i != j,1 " i, j " n.(10)

The traveled distance constraint, Equation (6), becomesn

!i=1

n

!j=1, j !=i

xi jkdi j " ck, 1 " k " m. (11)

Finally, the cost function Equation (2) remains the same.Remark. The above MIQP model for COP does not allow

two robots to start from the same base. We can easily ac-commodate such scenarios via modifying Equations (7), (8),and (9) accordingly. Also, if a fixed base is not required, themodels can be easily modified to accommodate this.

C. Application to Persistent Monitoring TasksWe project that our MIQP models will be useful in ap-

proaching OPs in which the objective is not simple linearsummations of individual utility at the nodes. Here, we il-lustrate COP as a relaxed problem for multi-robot persistentmonitoring tasks. As an example application, we look at planarnetworks. Such networks are fairly prevalent in persistentmonitoring tasks.2 To connect to COP, we assume that thefi’s in Equation (1) take the form

"(vi, t) = #0i + !v j%Ni

# ji"(v j, t), (12)

in which #0i and # ji’s are coefficients. Note that when notall values of " are available for nodes in Ni, Equation (12)can still be applied. To map these coefficients to our relaxedCOP models, for each w ji, we compute two sets of coefficients(using historical data). The first set of coefficients # &

ji’s arecomputed assuming all of Ni are available; the second set,# &&

ji’s, are computed assuming v j is vi’s only neighbor. Wethen compute the weight w ji via

w ji =# &

ji +# &&ji

!k%Ni(#&ki +# &&

ki). (13)

Equation (13) was chosen to balance the impact of singleneighbors as well as the impact of the entire neighborhoodin estimating the value of " of a node using its neighbors.

IV. SIMULATION EXPERIMENTS

In this section, we first evaluate MIQP models for COPover various benchmark examples. Then, we apply the modelsto a realistic synthetic scalar field that varies spatially andtemporally. All computations were performed on a computerwith Intel Core-i7 3930K CPU under an 8GB JavaVM. TheMIQP solver used is Gurobi[29].

2For example, cities and connecting roads form such natural node networks.

345

A. Validity of the MIQP Models for COP

We first verify the validity of the MIQP models, i.e., wecheck whether the models actually maximizes the objectivefunction given by Equation (2). For this task, we begin with asingle robot. Here, 3!3 and 4!4 grid networks are used withunit edge length as test node networks. For such grids, we setthe weights w ji for a vertex i simply as 1/|Ni|. For example,if vertex i has three neighbors, then all w ji’s are set to 1/3.

Fig. 2. Single robot tours for travel budgets 2,3,4,5, and 6, in that order,from left to right.

For the 3!3 grid, we let the single robot start at the middlenode on the top row (the circled node in Figure 2) and let themaximum allowed travel budget vary from 2 to 6 with unitincrements. Each node has a unit utility. The computed toursfor these budges are are illustrated in Figure 2, with utilities as4.0,4.5,5.7,7.3, and 9 (maximum possible), respectively. Onecan easily verify (manually) that these are consistent with thedesign of the MIQP model for a single robot.

Fig. 3. Single robot tours for travel budgets 4,8, and 12, in that order, fromleft to right.

For the 4! 4 grid, under a similar setup, we get the toursas illustrated in Figure 3 for travel budgets 4,8, and 12,respectively. The associated maximum utilities are 6.2,11.5,and 16.0, respectively.

Fig. 4. Two-robot tours for travel budgets (per robot) 3,5, and 7, in thatorder, from left to right.

Next, a two-robot setup is tested on the 4!4grid, with therobots starting at opposite locations as indicated as the redand purple circled nodes in Figure 4, which illustrates thetours with individual travel budgets 3,5, and 7, respectively.The associated maximum utilities are 7.2,12.5, and 16.0,respectively. Each problem instance in this subsection tookat most two seconds to solve.

B. Irregular Node Network

Our second experiment works with the irregular, realisticnode network from Figure 1. The bounding rectangle of thenetwork is roughly 13 units by 8 units. For this network,synthetic weights (wi j’s) are again computed based on thenumber of neighbors. Up to three robots were attempted withthe longest running time being about 100 seconds. The trialresults and the associated parameters are given in Figure 5.The base nodes, indicated as colored circles, were hand picked(only once, i.e., we did not try any other choices and thenselect the best one) to be roughly evenly distributed on thenetwork.

(a) B=15.0, R=7.9 (b) B=25.0, R=14.7 (c) B=35.0, R=18.0

(d) B=9.0, R=10.9 (e) B=13.5, R=14.9 (f) B=18.0, R=18.0

(g) B=7.5, R=11.8 (h) B=10.5, R=15.3 (i) B=13.5, R=18.0

Fig. 5. Results from running MIQP models for COP on the irregular, realisticnode network from Figure 1. The numbers under each picture indicate thebudget (B) per tour/robot and the total utility (R), respectively.

From the result (Figure 5), we see that the MIQP modelalways select tours that do not have spatial overlaps, whichis expected but nevertheless a nice feature to have. Also,regardless of the number of robots and tours, the total travelbudgets (35.0 for one tour, 36.0 for two tours, and 40.5 forthree) to ensure full coverage of the network appear to besimilar. We note that for the cases with multiple tours, someof the individual budgets can be shortened. For example, thetours in Figure 5(f) can be (manually) updated to the tours inFigure 6, with reduced total (actual) travel budget but withoutreducing the collected utility. Varying individual budget issupported in the model by default.

Fig. 6. Updated two-robot tours with the same utility as the tours inFigure 5(f).

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C. Computational PerformanceThe third experiment seeks to establish the performance of

our current MIQP models for solving COP. To evaluate thecomputational performance, we again start with a single robotand attempt grid networks with different sizes. Unlike the lastexperiment, the grid network is perturbed so that the edgelengths are no longer uniform; Figure 7 shows a typical 5!5perturbed node network used in this experiment.

Fig. 7. A purturbed 5!5 grid network.

For a single robot, computation over a 7! 7 grid networkwith 49 nodes for a full range of travel budgets can becompleted within 20 minutes. We list the computational resultsin Table II. We notice from the experiment that, similarto TOP, the computational time can vary greatly dependingon the travel budget for the same node network. The mostcomputationally costly instances are these with utilities (forthe given budget) close to the maximum possible utility. Onthe other hand, if the travel budget is too small to yieldthe maximum possible utility, then the computational cost isrelatively low. Same holds when the travel budget is very large.

TABLE IICOMPUTATIONAL PERFORMANCE THE MIQP MODEL FOR A SINGLE

ROBOT

Grid size Trial #1 2 3 4

3!3budgettime(s)utility

2.70.044.0

5.40.087.6

8.10.019.0

10.80.019.0

4!4budgettime(s)utility

3.60.025.3

7.20.9211.2

10.80.1615.4

14.40.0116.0

5!5budgettime(s)utility

4.50.196.1

9.08.113.8

13.533.120.0

18.04.7

25.0

6!6budgettime(s)utility

10.831.616.25

16.248.324.9

21.623.632.6

27.00.8936.0

7!7budgettime(s)utility

12.635820.2

18.937631.2

25.233739.8

31.5238746.8

We also tested the case of two and three robots usingthe same node networks; the model worked well with up to6!6 grids. While all cases for 5!5 grids can be computedrelatively quickly, hard instances for the 6!6 grid took overa day to compute, at which point we stopped the trial run.The results are listed in Tables III and IV for the two-robotand three-robot cases, respectively. It is interesting to see that

the computational time does not seem to vary much betweenthose to cases.

TABLE IIICOMPUTATIONAL PERFORMANCE FOR TWO ROBOTS

Grid size Trial #1 2 3 4

3!3budgettime(s)utility

2.40.046.5

3.60.049.0

4.80.049.0

6.00.049.0

4!4budgettime(s)utility

3.20.19.6

4.86.213.8

6.49.016.0

8.03.716.0

5!5budgettime(s)utility

4.00.410.5

6.016616.2

8.01.5K20.8

10.01.3K25.0

6!6budgettime(s)utility

2.40.27.3

4.818.214.3

7.227K20.5

TABLE IVCOMPUTATIONAL PERFORMANCE FOR THREE ROBOTS

Grid size Trial #1 2 3 4

4!4budgettime(s)utility

2.10.59.8

3.20.213.5

4.31.814.6

5.33.216.0

5!5budgettime(s)utility

2.70.411.5

4.01.015.6

5.31.4K19.8

6.720K24.3

6!6budgettime(s)utility

3.21.42114.7

4.815720.3

D. Measuring a Time-Varying Scalar FieldLastly, we perform experiments to verify the effectiveness of

Equation (13) in connecting actual scalar fields to our relaxedCOP problem and models. We focus on the case of a singlerobot as the number of robots do not matter in measuring thequality of the tours produced by the MIQP model.

Our experiments are performed over a synthetic scalarfield generated by three two-dimensional Gaussians. TheseGaussians have fixed centers but varying magnitudes andcovariance matrices over time; we fix the centers to ensurethat the spatial correlations are relatively time-invariant. Thefield is simulated for 200 time steps; the snapshots of the fieldat time steps 0,50,100,150, and 200 are provided in Figure 8.The node network used here is the same 5! 5 randomizedgrids (see, e.g., Figure 7 ) scaled to the dimensions of thesupport of the scalar field. For each fixed travel budget, 100random 5!5 node networks were generated. In each randomlygenerated network, the nodes of the network are given equalimportance (i.e., unit utility). To estimate ! "

i j’s and ! ""i j’s for

computing the weights, data from the first fifty time steps wereused. For running the models, the second diagonal node fromthe top-left corner was used as the base node.

After a tour is produced, the quality of the tour is estimatedas follows. Visiting each node yields one quality point. For

347

t = 0 t = 50 t = 100 t = 150 t = 200

Fig. 8. Snapshots of a synthetic scalar field at time steps 0,50,100,150, and 200, from left to right, respectively. [top row] Three-dimensional views. [bottomrow] Two-dimensional heat-map views.

a node that is not visited but is a neighbor of one or morevisited nodes, the values of the node at time steps 100,150,and 200 are estimated in the following way. Let vi be sucha node and let N!

i be its neighbor set that is actually visitedby the robot. The data for vi and nodes in N!

i from t " [1,50]were used to perform multiple linear regression according to

!(vi, t) = "0i + #v j"N!

i

" ji!(v j, t). (14)

The resulting parameters were then used to estimate !(vi, t)for t = 100,150, and 200. Let the estimated value be ! !(vi, t),we compute the quality of ! !(vi, t) as

#t"{100,150,200}(!(vi, t)# |! !(vi, t)#!(vi, t)|)#t"{100,150,200} !(vi, t)

(15)

To compare to our results, we also exhaustively searchthrough the network for tours starting and ending at thesame base node for the tour that minimizes the same qualitydefined by Equation (15) under the same travel budget. Thisexperiment was limited to travel budgets 6 and 8. Thesebudgets correspond to tours containing up to five nodes. Whileour model can produce tours with many more nodes, forcomparing the result, we have to exhaustively search throughall tours starting from the base node to find the best one,which becomes exponentially costly for tours with more than5 nodes. The quality score obtained this way is denoted as“actual quality”.

TABLE VMODEL FIDELITY OVER A SYNTHETIC SCALAR FIELD

Travel Budget Model Quality Actual quality Relative error6.0 7.16 7.64 0.488.0 8.46 9.38 0.92

The comparison result is given in Table V. Using the givenmetric, the average quality error was less than one, meaningthat it was not more than the error incurred by omitting a single

node. In roughly 30% of the cases, the tour found using ourmethod was identical to the one found using exhaustive toursearch.

!5

0

5

!505!5

0

5

!505!5

0

5

!505

t = 100 t = 150 t = 200

Fig. 9. Estimated scalar field based on the data obtained using the MIQPmodel.

As a secondary measure of the quality of our method, weput a regular 6$ 6 node network fitted over the same field(Figure 8) and run the MIQP model such that we just haveenough budget to obtain a full utility of 36. We let the startnode be the second left most node on the first row. From theoutput we can then estimate ! for all nodes that are not visitedon the tour. We then plot the much sparser survey data overthe same space for time steps 100,150, and 200 as shownin Figure 9. Comparing these figures with the correspondingones from Figure 8, we observe that our models providevery reasonable estimation of the entire synthetic scalar fieldwithout the need to visit all the nodes.

V. CONCLUSION AND FUTURE WORK

In this paper, we introduce COP as an extension to OP witha non-linear cost function, to address the problem of planningtours for surveying a spatially correlated field that also variesover time. Our preliminary computational experiments showthat the MIQP models for COP are effective in capturing thespatial correlation among nearby nodes, indicating that COPand the associated MIQP models are applicable to persistentmonitoring tasks in which the mobile robots have limited travelrange.

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As a first attempt to introduce a new problem, our discussionof COP and its solution in the current paper is certainly limitedin its scope. In the future, we expect to extend this study inseveral directions. Whereas our method is reasonably effective,it cannot yet handle networks with a large number of nodesquickly. We are working toward improving the current modelsto make them more efficient. To improve the computationalefficiency, in addition to perfecting the MIQP model throughtechniques such as reachability analysis, we hope to under-stand from the solutions to the MIQP problems the generalstructures of these (optimal or near optimal) solutions. Fromthere, we can then design heuristic algorithms that produceapproximate solutions quickly. Currently, we are evaluating theperformance of some preliminary heuristic algorithms; earlyresults show that much larger problem instances involvinghundreds of nodes can be solved quickly.

At the same time, we hope to improve our models tobetter capture real-world application scenarios. Toward thisend, we note that the current weight selection criterion haslots of room for improvement. Through a more systematicapproach, perhaps via statistical methods, we hope to derivemore principled procedures for selecting the weights for theMIQP models.

Beyond these immediate steps, this paper only begins toaddress the problem of using correlations in informative pathplanning in a discrete fashion. The dual problem to this esti-mation problem is a learning problem: how can we learn thecorrelations among the nodes so as to apply the methods fromthis paper? Can we do learning and estimation simultaneously?

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