Persistent Monitoring of Dynamically Changing Environments Using an Unmanned Vehicle

Persistent Monitoring of Dynamically Changing Environments Using an Unmanned Vehicle

S. K. K. Hari, S. Rathinam, S. Darbha, K. Kalyanam, S. G. Manyam, D. Casbeer Department of Mechanical Engineering, Texas A&M University, College Station, U.S.A. 77845. Control Science Center of Excellence, Autonomous Control Branch, Aerospace Systems Directorate, Wright-Patterson A.F.B., OH 45433.
Abstract

We consider the problem of planning a closed walk for a UAV to persistently monitor a finite number of stationary targets with equal priorities and dynamically changing properties. A UAV must physically visit the targets in order to monitor them and collect information therein. The frequency of monitoring any given target is specified by a target revisit time, , the maximum allowable time between any two successive visits to the target. The problem considered in this paper is the following: Given targets and allowed visits to them, find an optimal closed walk so that every target is visited at least once and the maximum revisit time over all the targets, , is minimized. We prove the following: If , (or simply, ) takes only two values - when is an integral multiple of , and otherwise. This result suggests significant computational savings – one only needs to determine and to construct an optimal solution . We provide MILP formulations for computing and . Furthermore, for any given , we prove that .

I Introduction

Dynamically changing environments are typical in a wide variety of applications ranging from military to agricultural domains. In many applications, an up-to-date knowledge of the changing parameters is crucial [1, 2, 3] for managing uncertainty. Often, the environment to be monitored is vast, and the coverage area of sensors is limited. Since it is practically impossible to sweep the entire region with a sensor’s footprint, the environment is usually divided into a set of disjointed regions represented by stationary targets, the properties of which are indicators of the properties of the environment. Persistent monitoring of the environment or targets can be achieved using a set of vehicles equipped with appropriate sensors that travel across the region to collect the required information.

Monitoring a target requires a vehicle to travel to the target (referred to as a visit to the target), and collect information. The collected information is then sent to a base station, where it is analyzed by a human operator to take appropriate actions [4]. To ensure timely actions, maintaining freshness of information is pivotal. Persistent monitoring requires that the time between successive revisits to a target be as small as possible and that the data received by the operator is as fresh as possible. There are many problem determinants for persistent monitoring applications: for example, the number of vehicles, vehicle-target assignment constraints, communication capabilities of vehicles, persistent monitoring requirements of every target, fuel and refueling constraints. Given the problem determinants, the core problem of persistent monitoring is to find the sequence of targets to be visited by each vehicle so that the maximum revisit time between successive revisits to any target is minimized.

In this study, we assume that the information collected at the targets is instantly transmitted to the base station. In doing so, we implicitly assume that the communication between a vehicle and its base station is unrestricted. When the base station is out of the communication range of the vehicle, information must be physically transported. In such cases, one must pay attention to preserving the freshness of information [5] in addition to the typical requirements of persistence and efficiency of monitoring, which are discussed in the following paragraphs.

Persistence is crucial in dynamic environments, as a stoppage in monitoring leads to a continuous growth of uncertainty in the properties of interest. Persistently monitoring a target requires it to be visited again and again. However, fuel limitations of a vehicle may impose additional constraints on persistent monitoring. For example, if the vehicle has limited fuel on-board, it may require refueling before proceeding with revisiting the set of targets again. In this case, not only is the fuel carried by the vehicle on-board important, but also the refueling time; this is because persistent monitoring is characterized by the maximum time between successive revisits to any target. If the vehicle can carry limited fuel on-board, then there is a frequent need to refuel; in this article, we assume that refueling is done at a depot.

Targets can be closely monitored with multiple vehicles; however, there are a number of issues that arise with multiple vehicles: for example, not all vehicles may have the sensor suite to monitor every target; in this case, vehicle-target assignment constraints must be obeyed. Vehicle-target assignment constraints also subsume the important issue of what constitutes a visit to a target – should it be performed by the same vehicle or by different vehicles? Is there a precedence constraint for these visits? In this article, we sidestep these issues and consider the basic problem of persistent monitoring with a single vehicle; the results from the single vehicle problem can be used to build solutions for persistent monitoring with multiple vehicles. We further assume that there is a single depot and it is a target where the vehicle starts from with the maximum possible fuel it can store and returns to for refueling.

The everlasting nature of the persistent monitoring together with the limited fuel capacity of the vehicle, requires the vehicle to be periodically refueled [6]. In this work, we surrogate the vehicle’s fuel capacity by the number of visits after which it must be refueled (say visits). After every visits, the vehicle is refueled at the depot, after which the persistent monitoring task is resumed. There are four other assumptions we make in this paper: (1) triangle inequality is satisfied by travel times between targets, i.e., it is faster for a vehicle to go directly from a target to a target than through some other intermediate target . There is a tacit assumption that a vehicle can travel without any restraint from any target to any other target. (2) the endurance time of a vehicle (time taken by a vehicle to empty its fuel tank or discharge its battery from a fully charged condition) is proportional to the fuel capacity and that there is no loss of generality in surrogating the fuel capacity (fuel carried on-board by a vehicle) with the number of visits. If one can solve the problem of routing vehicles for persistent monitoring with a specified number, , of visits, one can compute the fuel required, say ; by triangle inequality, this function is monotonically increasing in . The main result of this paper can be used to compute this function; knowing one can find the number of visits for any given fuel capacity and number of targets to be visited. (3) Here, we also assume that the time taken to refuel/recharge the vehicle is negligible compared to the travel times between the targets; this is not an unreasonable assumption, especially when discharged battery packs can be removed and charged battery packs can be quickly loaded onto the vehicle and this time can be assumed negligible when compared to the travel time. (4) From the perspectives of convenience and efficiency (as in battery recharging), there is a tacit assumption made: a vehicle can be refueled only when visits have been made (i.e., fuel/charge has been expended fully before recharging).

The problem considered in this paper is the following: Given that there are targets and the vehicle starts at the depot with a fuel capacity equivalent to visits, find a route of visits so that the vehicle (1) visits each target at least once, and return to the depot for its visit and (2) the maximum time between successive revisits to any target is minimized. Note that the vehicle refuels after the visit at the depot; since , the vehicle is allowed to make multiple visits to targets. In graph-theoretic terminology, this route of visits is referred to as a closed walk (more about this will be discussed in Section II). To ensure persistence, this walk is repeated over and over. The maximum revisit time over all the targets, when the walk is repeated continuously, is referred to as the revisit time of the walk . The objective of this work is to identify a closed walk with visits such that the revisit time of the walk , is the minimum. For a given number of visits, the lowest possible revisit time is referred to as the optimal revisit time , and a walk with the minimum revisit time (i.e., ) is referred to as an optimal walk.

When the number of allowable visits is equal to the number of targets, each target is visited exactly once in any walk. In such a case, the revisit time of any target is equal to the time taken to complete a closed tour visiting all the targets once. Hence, finding a walk with the least maximum revisit time is equivalent to finding a traveling salesman tour over all the targets, which is very well known to be NP-Hard. Hence, the persistent monitoring problem is also NP-Hard. The computational difficulty of the problem increases with the number of visits.

The problem of persistent monitoring in the ideal case of unlimited fuel capacity takes the same form as the one considered here; this same problem was considered in [7]; in this work, heuristic algorithms were provided for the closed walk. A generalization of this problem was considered in [8], where some targets are considered more important and their revisit time is weighed differently; the corresponding objective of routing is to minimize the maximum weighted revisit time of targets. In [8], the authors proved that an optimal infinite walk (a walk with infinite number of visits) can be constructed by concatenating a finite closed walk (a walk with the same initial and terminal node) with itself multiple times. Nonetheless, they showed that the size of the finite closed walk can be arbitrarily large. They proved that the problem is APX-Hard, and provided two polynomial time approximation algorithms, with approximation ratios and , where is the ratio of maximum and minimum weights of the targets. In [9], the authors posed the problem as an MILP, and provided an iterative sub-optimal scheme for the problem. The sub-optimal scheme was observed to be an order of magnitude faster than the optimal scheme, without compromising much on the quality of solutions.

In [10], the author considered the problem of multi-agent patrolling of equally weighted targets, with a focus on developing efficient team patrolling strategies, based on a TSP tour over the targets. A TSP tour does not allow multiple visits to targets in a cycle, as opposed to closed walks considered in the present article. The author proposed two team patrolling strategies: cyclic based strategy and partition-based strategy and compared their performance based on a metric of minimizing the team revisit time. In [11], the authors extended the problem to weighted targets (which was proved to be NP-hard in [12]). The authors in [11] construct a non-intersecting tour (without repeated visits to targets) based on a minimum spanning tree over the targets, and compare two team patrolling strategies: equal spacing strategy, and equal time spacing strategy over the the tour. Additionally, they propose a distributed control algorithm to equally space robots along a shared trajectory, considering two different communication models (passing communication model and neighborhood-broadcast communication model). The authors of [13] consider a relatively simpler problem of multi-robot frequency coverage, in which the number of visits made to each target is pre-specified. They propose an algorithm that does not require inter-robot communication, and has a low computational requirement. In contrast to the above articles, the authors in [14] considered fixed trajectories, and developed velocity controllers for robots/agents to achieve minimum revisit time.

I-a Contributions

In the present article, we consider the problem of monitoring equally weighted targets, and provide results characterizing the structure of optimal solutions. Given allowable visits for the vehicle, we prove the following:

  1. , ; equality holds when is an integral multiple of .

  2. is bi-modal for ; it takes only two values, if is an integral multiple of , and otherwise.

  3. , .

  4. is a monotonically non-decreasing function of , for , .

  5. , where is not an integral multiple of , and can be expressed in the quotient remainder form as , such that , and .

These results indicate that one needs to solve at most problems (from visits to visits) to completely determine optimal solutions for any given and can lead to a substantial reduction in the computational effort required to solve the problem for higher number of visits.

I-B Organization

The rest of the paper is organized as follows: The exact problem statement is presented in Section II. Section IV provides useful lemmas and theorems that characterizes optimal solutions. Section V provides numerical simulations to corroborate the results proved in Section IV, using an MILP formulation presented in Appendix A. Conclusions and future directions are provided in Section VI.

Ii Problem Statement

Consider a set of spatially distributed targets/nodes that needs to be monitored by a UAV. Let the targets be denoted by the set , and the travel times between any two distinct targets () in the set be denoted by . We assume that the targets are of equal priorities, and the travel times between them satisfy the triangle inequality, i.e., for all .

The UAV tasked with monitoring the targets must be recharged after every visits. The UAV visits the targets in the order specified by the sequence (such that for ); this sequence is repeated after every visits. Here, we assume , which allows each target to be visited at least once and some targets to be visited multiple times in a cycle of visits. Due to repetition of nodes/vertices, this sequence is referred to as a walk. The node is the first or starting node of the walk from which the vehicle starts its mission, and is not counted in the number of visits made to targets. The node is the last visited target in the walk.

We assume that the vehicle starts from and returns to the depot (chosen as of one the targets), after every visits. Consequently, we have , and a walk with the same initial and terminal node is referred to as a closed walk. Note that a visit is counted only if a vehicle travels between two distinct targets. Therefore, we have for . Moreover, the vehicle must visit all the targets in a walk of visits. In contrast with the graph-theoretic terminology[15], we use the term closed walk differently to refer to a walk (a) with the same initial and terminal nodes, (b) in which every target is visited at least once and (c) successive visits correspond to distinct targets. Since we primarily deal with only closed walks in this paper, we use walks and closed walks interchangeably when the context is clear.

For the purpose of illustration, consider Figure 1, which shows a closed walk of eight visits. Since a closed walk is continuously repeated after every visits (in Figure 1, ), it is often depicted using a cyclic representation as shown in Figure 2. The total time required to traverse through all the nodes in a walk is referred to as the duration of the walk.

Fig. 1: Graphical representation of a walk with 8 visits
Fig. 2: A cyclic representation of the walk (shown in Figure 1) with node 5 as the depot

Given a walk, , of visits, the revisit time of a target , denoted by , is the maximum time taken between successive visits to the target, when the walk is repeated. For example, consider the walk shown in Figure 3. Target 2 is visited twice in the walk. Time between successive visits to target 2 are and as shown in the figure. Hence, the revisit time for target 2 is max . Similarly, the revisit time of target 4 is max , where and . The maximum revisit time over all the targets is defined as the revisit time of the walk or walk revisit time . Note that if a target is visited exactly once in a walk, it has the maximum revisit time, which is also equal to the duration of the walk. Therefore, in the example considered above, = = , as targets 1 and 3 are visited exactly once in the walk.

Fig. 3: Figure depicting the revisit times of targets 2 and 4 in the walk considered in Figure 1

Using the aforementioned notation, the problem is stated as follows:

Given allowed visits for a UAV, find a closed walk of visits with the minimum walk revisit time.

A walk with the minimum revisit time is referred to as an optimal walk, and is denoted by , and the minimum possible revisit time for a given is denoted by

Iii Terminology

This section details the terminology used in the rest of the paper for understanding the structure and the construction of an optimal walk.

Cyclic Permutation:

A cyclic permutation (or simply, permutation) of a closed walk shifts the first node of the walk to an intermediate node, say , while retaining the order of visits in ; we represent such a cyclic permutation as . For example, the permutation of that starts at the intermediate node is defined as

Figure 4 illustrates .

Fig. 4: A cyclic permutation () of the walk depicted in Figures 1 and 2, with depot at node 1

Concatenation:

Concatenation of two closed walks and is defined as . Figure 5(b) shows a walk obtained by concatenating walks in Figure 5(a).

(a) Concatenation of two walks with 5 visits each
(b) A walk with 10 visits obtained by concatenation the walks in 5(a)
Fig. 5: Concatenation of two walks with 5 visits each to form a walk with 10 visits

It is easy to see that the revisit time of any node does not change when a closed walk is concatenated with itself multiple times. Concatenation of walks is important in constructing feasible walks to the problem corresponding to higher number of visits.

Shortcut Walk:
Given a closed walk , removal of a visit to any target is referred to as shortcutting a visit to from . In this article, we refer to a shortcut walk of as a closed walk obtained by shortcutting visits from , but retaining (not shortcutting) the last visits to each target. For example, (shown in Figure 1) is a closed walk with 8 visits, with two visits to each of targets 2, 4 & 5, and one visit to each of targets 1 & 3. A shortcut walk, , of is obtained by shortcutting the first visits to targets 2 & 5, and retaining the last visits to all the targets in the order they appear in , as shown in Figure 6. We remind the readers that shortcutting the last revisits to targets 2, 4 & 5 or visits to targets 1 & 3 is not allowed in our definition. In the former case, we cannot shortcut the last revisit to a target; in the latter case, shortcutting does not result in a walk as the shortcut targets are not visited in the walk!

Fig. 6: A shortcut walk obtained by shortcutting visits to targets 2 and 5 from the walk

Subwalk:

A subwalk of a given closed walk is a subsequence obtained by removing certain nodes from the walk, yet retaining the order of visits. A subwalk need not span all the targets and hence, need not be a closed walk. For example, consider the closed walk shown in Figure 7. The subsequences (2,5,1,4), (2,5,3) are subwalks of . A subwalk with the same first and last nodes is referred to as a closed subwalk. Figure 7 shows a closed subwalk of with target 5 as its initial and terminal nodes.

Fig. 7: Figure depicting subwalks and closed subwalks of the walk .

If the terminal node of a subwalk is the initial node of a subwalk , they can be concatenated as . Note may not be possible unless .

The time taken to traverse through all the nodes of a subwalk is referred to as the travel time of the subwalk, and is given by the expression . The travel time of a subwalk is similarly also referred to as the duration of the subwalk.

In some closed subwalks, the end node is not visited in between; we refer to such an end node as a terminus. For example, are closed subwalks of with node as the terminus. Given a node that is visited times in a closed walk , it is easy to see that the cyclic permutation can be decomposed into closed subwalks with each of them having as its terminus, i.e.,

and is the terminus of closed subwalks . Note respectively indicate time between successive revisits to target . The revisit time of target in is

The revisit time of a given walk is the maximum revisit time among all the targets, i.e., . Note that the revisit time of a walk is equal to the revisit time of any of the cyclic permutations of the walk.

A decomposition of a walk with respect to a vertex is also useful in analyzing revisit times when concatenating two different walks; suppose is visited times in a walk , i.e., except for ; the decomposition of with respect to is defined as:

where are subwalks with as terminus and are subwalks with different initial and terminal nodes. Let be a walk of targets with 13 visits; decomposition of with respect to target is .

Before we proceed to the next section, we introduce the notion of a binding subwalk: Let denote a feasible solution and represent an optimal solution to the persistent surveillance problem with visits.

Definition (Binding Subwalk): A closed subwalk is defined to be a binding subwalk if there is a node such that

  • is the terminus of , and

It is easy to see that

The next section is dedicated to the task of studying the properties of optimal walks.

Iv Properties of optimal walks

First, we show that any binding subwalk of an optimal walk must contain visits to all the targets.

Lemma 1

Let be a binding subwalk of for . Each target is visited at least once in .

Proof:

If a target is not visited in , then the revisit time for must be greater than the time required to traverse as shown in Figure 8. This is because we can find a subwalk with as its terminus that contains . However, this contradicts being a binding subwalk. Therefore, each target must be visited at least once in .
\qed

Fig. 8: A subwalk (shown in yellow) with a missing node cannot be binding, since there exists another subwalk (with as its terminus) with a larger travel time

In essence, itself is a walk with atmost visits.

This result leads to a lower bound on the optimal revisit time. In the following lemma, we prove that the optimal revisit time is lower bounded by the cost of an optimal TSP tour over the targets.

Lemma 2

where denotes the cost of an optimal TSP tour visiting all the targets.

Proof:

Any feasible solution to the TSP is also a feasible solution to the persistent surveillance problem with visits where each target is visited exactly once, and vice versa. Therefore, .

From Lemma 1, for , each target is visited at least once in a binding subwalk . Shortcut any target that is visited more than once in to obtain a tour, . As the travel times satisfy the triangle inequality, short cutting any visit will not increase the revisit time, , . Therefore, .
\qed

From Lemma 2, is a lower bound for every walk revisit time. In some cases, this bound can be tightened, especially when the number of visits, , is not an integral multiple of .

Lemma 3

Consider a walk with visits. One can express in the quotient-remainder form as , where , , and . Then, contains a closed subwalk with a terminus, with at least visits. Consequently, .

Proof:

Suppose the maximum number of visits in a closed subwalk with a terminus is at most . Then, we show that each target must be visited at least times in the walk as follows: In any given walk, the maximum number of visits between consecutive revisits to a target is lower bounded by the average number of visits between consecutive revisits to the same target. If a target is visited at most times, the maximum number of visits between consecutive revisits to is at least ). However, if each target is visited at least times, the total number of visits in the walk, , is at least , which contradicts the hypothesis.

Hence, there exists a closed subwalk with a terminus, that has at least visits. Without loss of generality, one can assume that spans all the targets. (If not, one can find a target , such that it is the terminus of another closed subwalk that contains and spans all the targets). By definition, , and from triangle inequality, it follows that . Therefore, . Since this is true for any walk with visits, we have .

\qed

Remark: If , it is clear that and consequently, if is not an integral multiple of and , we have . We will use this fact in Theorem 2.

In the subsequent sections, we will construct feasible walks by concatenating smaller walks in such a way that the maximum revisit time equals the lower bound found in this section, thereby demonstrating optimality. The key result in the next section is that concatenating a binding subwalk with some of its shortcut walks and subsequently replacing the binding walk with the concatenated walk is not going to increase the maximum revisit time. This result is important in that walks with higher number of visits can be constructed without increasing the maximum revisit time.

Iv-a Properties of Concatenating Feasible Solutions

The procedure to extend a walk into a larger feasible solution is given by the following lemma, and is depicted in Figure 9.

Theorem 1

Let be a binding subwalk and , be subwalks of such that . Let be a closed subwalk obtained by shortcutting visits from , but retaining the last visits to targets. Let a closed walk be formed by concatenating with as follows: . Then, .

Fig. 9: A closed walk is shown on the left, with its binding subwalk colored yellow. A subwalk (colored orange) is constructed from , by shortcutting visits to target (retaining the last visit to ). Figure on the right shows the concatenated walk , as discussed in Theorem 1.
Proof:

See Appendix B.

Next, we present two lemmas involving the cyclic permutation and concatenation of walks. The first of the two deals with the revisit time of a closed walk concatenated with itself an arbitrary number of times.

Lemma 4

Let a closed walk be formed by concatenating with itself times as follows:

Then, for any node

  • ,

and thus

Proof:

See Appendix B.
\qed

The next lemma involves concatenating a walk with its shortcut walk and is crucial for piecing together smaller feasible walks to build a bigger walk while not increasing the maximum walk revisit time.

Lemma 5

Let be a closed walk with visits, and be a target that is visited exactly once in . Let be a walk formed by shortcutting a revisit from . Consider a closed walk formed by concatenating for times and for times as follows:

Then,

Proof:

See Appendix B
\qed

Remark: This lemma can be used to construct a feasible walk with visits as follows: Let for some integers and being the number of targets. If , then , where . Set , and so that ; using the above lemma, one can construct a feasible walk for visits from the feasible walk of visits and its shortcut walk of . The following lemma shows that building such a bigger walk can be done without increasing the maximum walk revisit time.

Lemma 6

Let be an optimal solution with visits, and be a shortcut walk of . Let a closed walk be formed by concatenating for times and for times as follows:

Then the maximum revisit time for is equal to i.e.,

Proof:

Since , there is at least one node, , in that is visited exactly once. Consider a cyclic permutation of . Let a closed walk be formed by concatenating for times as follows:

From Lemma 4 it follows that . As is visited exactly once in , we have . Moreover, is the terminus of . Therefore, is a binding subwalk of itself. Because , is also a binding subwalk of .

Now consider , which is formed by shortcutting a revisit from . Note that is visited exactly once in . , is a shortcut walk of , which is a binding subwalk of . It follows from Theorem 1 that the revisit time of a walk formed by concatenating with , is . Since , is also a binding subwalk of . Therefore, concatenating with does not change the revisit time, and the process can be repeated for any number of times (say ). Hence, if is defined as,

we have From Lemma 5, is a cyclic permutation of , and thus

\qed

With this, we have enough tools to analyze the properties of optimal revisit time. To study the properties of optimal solutions, we split the number of visits into 3 categories: , and . We begin with the case, , and prove that the optimal revisit time is a periodic function of , with a period .

Iv-B Bi-modal Property of Optimal Revisit Time

The focus of this subsection is to prove the main result of this article, that after a finite number of visits (), the optimal revisit is bi-modal with a period . The two values it takes are: when is an integral multiple of , and when is not an integral multiple of .

Lemma 7

for where is any positive integer.

Proof:

Consider the walk . is feasible to the problem with visits. Using Lemma 4, . Therefore, from Lemma 2, it follows that is also an optimal solution to the problem with visits.
\qed

Lemma 8

for and is not an integral multiple of .

Proof:

Consider the optimal walk with visits. Since exactly one target is visited twice in , shortcut one of the visits to this target to get a new walk . Let , and . Consider the following feasible solution to the problem with visits:

Note that the total number of visits in is . Using Lemma (6), we obtain . Therefore, .
\qed

Theorem 2

For , the optimal revisit time for the persistent surveillance problem can be only one of two values, ,

Proof:

The theorem follows from Lemmas (7) and (8), and the remark following Lemma (3). \qed

Iv-C Monotonicity of Optimal Revisit Time

In this subsection, we show that the optimal revisit time is a monotonic function of the number of visits, for

Lemma 9

for .

Proof:

Consider the optimal solution . As , there is at least one node in that is visited exactly once. Let denote one of these nodes. Since is visited exactly once, . Also, as , there is at least one node (say ) that is visited more than once in . Shortcut one of the revisits to from to obtain a feasible walk to the problem with visits. As the travel times satisfy the triangle inequality, . Therefore, . \qed

Iv-D Bounds on Optimal Revisit Time

In this subsection, we provide a procedure to construct feasible walks for the case , using the optimal solutions for . The revisit times of these walks match the lower bounds presented earlier in this section, proving optimality of the constructed solutions.

Lemma 10

Let be not an integral multiple of , i.e., for some integers , . Then .

Proof:

It is sufficient to construct a feasible solution with visits, such that . Consider an optimal walk with visits, and a corresponding walk with visits as follows:

  1. is obtained by shortcutting a visit to a repeated target if .

Let for some integers and . For the sake of convenience, let us refer to as , and as . Then, a feasible solution to the problem can be constructed by concatenating times and times as follows:

Note that the total number of visits in is ; the equality can be inferred from the following cases:

  1. If , . Hence, .

  2. If , we have , and . Therefore, .

  3. If , we have and . Hence, .

Since , from Lemmas 4 and 6, we can conclude .
\qed

Theorem 3

For any given number of visits , , where , , and .

Proof:

The theorem follows from Lemmas 10, 3 and 7. \qed

A consequence of this result is that one needs to solve at most distinct problems to construct optimal walks for any given .

We conclude this section with another general result that holds for any . Given an optimal walk with visits, this result enables one to construct feasible solutions of visits, without increasing the revisit time. This translates to the fact that the optimal revisit time for a problem with visits is upper bounded by that for a problem with visits. Hence, for every visits, the optimal revisit time either reduces or stays the same, but does not increase.

Corollary 1 (of Theorem 1)

The optimal revisit time for the persistent surveillance problem with visits is an upper bound to the one with visits. That is, , .

Proof:

Consider an optimal walk with visits, and any of its binding subwalk, say . Figure 10 shows (colored yellow). Let be formed by shortcutting all except the last visits to targets from , such that it has exactly visits. is shown in Figure 10 (colored orange). Now form a closed walk with visits, by inserting in , right next to as shown in Figure 10. From Theorem 1, we have . Since is a feasible solution to the problem with visits, the optimal revisit time for a problem with visits is at most , i.e., .

Fig. 10: Figure showing construction of a solution to the problem with visits, given an optimal solution for the problem with visits.
\qed

Given an optimal walk with visits, one can construct a feasible solution with visits such that , by concatenating with its shortcut walk of visits, as discussed above.

V Numerical Simulations

Numerical simulations were performed on 30 instances to corroborate the results proved in the previous section. All the instances contain equally weighted targets, and the number of targets range from 4 to 6. The coordinates of the targets were randomly generated, and the Euclidean distances between them were used as a proxy for the travel times of the UAV. For any instance, an optimal walk can be computed by solving the MILP formulation presented in Appendix A, using commercial solvers.

For every instance, optimal solutions were computed using the IBM ILOG CPLEX solver, by varying the allowable number of visits from to 45, and their corresponding optimal values were plotted against . Plots for sample instances with 4, 5 and 6 targets are shown in Figures 11(a), 11(b) and 11(c) respectively.

We consider an instance with 4 targets whose coordinates are shown in Figure 12 to demonstrate the structure of optimal solutions. The time taken by a UAV to travel between the targets is listed in Table I. Optimal solutions are found by varying the number of allowed visits from 4 to 16, and the corresponding optimal solutions are presented in Table II. Figure 11(a) shows the optimal values plotted against the number of visits.

Target Target Target Target
Target 0 13.89 10 10.82
Target 13.89 0 7.28 13.34
Target 10 7.28 0 6.08
Target 10.82 13.34 6.08 0
TABLE I: Travel times between the targets for the sample instance with 4 targets

For , there is at least one target that is visited exactly once in a walk (a walk must contain at least visits to allow multiple visits to all the targets).

(a) Four equally weighted targets
(b) Five equally weighted targets
(c) Six equally weighted targets
Fig. 11: The optimal revisit time as a function of the number of visits in the walk.
Fig. 12: Coordinates of an instance with 4 targets

So, monotonically increases with the increase in number of visits from 4 to 7 as shown in Figure 11(a). If is an integral multiple of , the optimal revisit time is equal to , and an optimal solution can be obtained by repeating TSP solutions.

For , optimal revisit time takes only two values: when is an integral multiple of , and otherwise (Figure 11(a)). In fact any optimal solution can be constructed solely using and when . Any can be expressed as where , and and are integers. When , is an integral multiple of , and can be obtained by concatenating for times. For example, when , can be constructed as . Since , is the sequence , as shown in Table II. On the other hand, if , can be constructed by concatenating for times, and for times, where is a shortcut walk of , obtained by removing a repeated visit from . For example, consider , which can be expressed in the quotient remainder form as 3(4)+2. Since , and , an optimal solution can be obtained by concatenating for 3 times and its corresponding shortcut walk for 2 times. That is, can be constructed as , forming the sequence